%%Page: 1 1 1 0 bop Black 0 TeXcolorgray Black Black Black Black 10800 5269 a @beginspecial 0 @llx 0 @lly 99 @urx 16 @ury 2880 @rwi @setspecial%%BeginDocument: logo129.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: logo128.eps %%Creator: fig2dev Version 3.2.3 Patchlevel %%CreationDate: Thu Nov 8 16:13:04 2001 %%For: pope@fry.research.att.com (Sue Pope) %%BoundingBox: 0 0 99 16 %%Magnification: 0.1500 %%EndComments /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save newpath 0 16 moveto 0 0 lineto 99 0 lineto 99 16 lineto closepath clip newpath -12.0 26.0 translate 1 -1 scale /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y tr xrad yrad sc 0 0 1 startangle endangle arc closepath savematrix setmatrix } def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def $F2psBegin %%Page: 1 1 10 setmiterlimit 0.00900 0.00900 sc 7.500 slw % Ellipse n 2213 1988 856 856 0 360 DrawEllipse gs col4 1.00 shd ef gr gs col4 s gr % Ellipse n 2213 1991 813 813 0 360 DrawEllipse gs col8 1.00 shd ef gr gs col8 s gr % Ellipse n 1602 1879 177 177 0 360 DrawEllipse gs col11 1.00 shd ef gr gs col11 s gr % Ellipse n 2001 2575 177 177 0 360 DrawEllipse gs col11 1.00 shd ef gr gs col11 s gr % Ellipse n 2831 1877 177 177 0 360 DrawEllipse gs col11 1.00 shd ef gr gs col11 s gr % Ellipse n 2618 1509 177 177 0 360 DrawEllipse gs col11 1.00 shd ef gr gs col11 s gr % Ellipse n 1806 1519 177 177 0 360 DrawEllipse gs col11 1.00 shd ef gr gs col11 s gr % Ellipse n 2205 1364 177 177 0 360 DrawEllipse gs col11 1.00 shd ef gr gs col11 s gr % Ellipse n 1669 2290 177 177 0 360 DrawEllipse gs col11 1.00 shd ef gr gs col11 s gr % Ellipse n 2762 2305 177 177 0 360 DrawEllipse gs col11 1.00 shd ef gr gs col11 s gr % Ellipse n 2439 2579 177 177 0 360 DrawEllipse gs col11 1.00 shd ef gr gs col11 s gr % Ellipse n 2220 1992 440 440 0 360 DrawEllipse gs col4 1.00 shd ef gr gs col4 s gr % Ellipse n 2618 1509 141 141 0 360 DrawEllipse gs col30 1.00 shd ef gr gs col30 s gr % Ellipse n 1805 1515 141 141 0 360 DrawEllipse gs col30 1.00 shd ef gr gs col30 s gr % Ellipse n 1669 2290 141 141 0 360 DrawEllipse gs col30 1.00 shd ef gr gs col30 s gr % Ellipse n 2205 1360 141 141 0 360 DrawEllipse gs col30 1.00 shd ef gr gs col30 s gr % Ellipse n 2762 2305 141 141 0 360 DrawEllipse gs col30 1.00 shd ef gr gs col30 s gr % Ellipse n 1600 1875 141 141 0 360 DrawEllipse gs col30 1.00 shd ef gr gs col30 s gr % Ellipse n 2439 2579 141 141 0 360 DrawEllipse gs col30 1.00 shd ef gr gs col30 s gr % Ellipse n 2831 1877 141 141 0 360 DrawEllipse gs col30 1.00 shd ef gr gs col30 s gr % Ellipse n 2001 2573 141 141 0 360 DrawEllipse gs col30 1.00 shd ef gr gs col30 s gr /Times-Bold ff 225.00 scf sf 1997 2652 m gs 1 -1 sc (23) dup sw pop 2 div neg 0 rm col0 sh gr /Times-Bold ff 225.00 scf sf 2438 2659 m gs 1 -1 sc (11) dup sw pop 2 div neg 0 rm col0 sh gr /Times-Roman ff 480.00 scf sf 3375 2205 m gs 1 -1 sc (Article 03.3.4) col0 sh gr /Times-Roman ff 480.00 scf sf 3375 1650 m gs 1 -1 sc (Journal of Integer Sequences, Vol. 6 \(2003\),) col0 sh gr /Times-Bold ff 225.00 scf sf 2618 1592 m gs 1 -1 sc (2) dup sw pop 2 div neg 0 rm col0 sh gr /Times-Bold ff 225.00 scf sf 2830 1956 m gs 1 -1 sc (3) dup sw pop 2 div neg 0 rm col0 sh gr /Times-Bold ff 225.00 scf sf 2761 2386 m gs 1 -1 sc (6) dup sw pop 2 div neg 0 rm col0 sh gr % Ellipse n 1805 1517 42 42 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr % Polyline n 2554 1797 m 2554 1796 l 2554 1790 l 2554 1779 l 2554 1765 l 2553 1753 l 2552 1743 l 2550 1736 l 2548 1730 l 2544 1725 l 2540 1721 l 2534 1717 l 2528 1713 l 2521 1711 l 2514 1709 l 2507 1708 l 2499 1707 l 2492 1707 l 2483 1707 l 2474 1707 l 2465 1708 l 2456 1710 l 2447 1712 l 2438 1714 l 2431 1717 l 2423 1720 l 2416 1724 l 2408 1728 l 2401 1734 l 2394 1739 l 2388 1745 l 2382 1751 l 2377 1756 l 2372 1762 l 2368 1768 l 2363 1775 l 2359 1783 l 2356 1791 l 2352 1799 l 2350 1807 l 2348 1814 l 2346 1823 l 2345 1831 l 2345 1841 l 2345 1852 l 2346 1863 l 2349 1873 l 2352 1884 l 2356 1894 l 2360 1902 l 2365 1910 l 2371 1919 l 2378 1928 l 2385 1937 l 2393 1946 l 2401 1955 l 2409 1963 l 2417 1970 l 2425 1977 l 2435 1985 l 2446 1993 l 2457 2000 l 2467 2008 l 2477 2015 l 2487 2021 l 2494 2027 l 2501 2033 l 2506 2038 l 2510 2043 l 2513 2049 l 2515 2055 l 2516 2061 l 2516 2067 l 2516 2073 l 2514 2079 l 2510 2087 l 2505 2095 l 2498 2104 l 2490 2113 l 2481 2120 l 2473 2125 l 2465 2129 l 2458 2131 l 2449 2133 l 2440 2134 l 2431 2134 l 2422 2133 l 2414 2131 l 2406 2128 l 2396 2122 l 2385 2115 l 2375 2107 l 2366 2101 l 2359 2097 l 2353 2096 l 2350 2098 l 2347 2101 l 2346 2107 l 2344 2114 l 2344 2122 l 2344 2130 l 2345 2138 l 2346 2145 l 2347 2151 l 2349 2158 l 2352 2164 l 2355 2169 l 2360 2174 l 2364 2179 l 2370 2182 l 2375 2186 l 2382 2188 l 2390 2190 l 2399 2193 l 2409 2194 l 2419 2195 l 2429 2196 l 2438 2196 l 2448 2196 l 2456 2195 l 2465 2194 l 2475 2192 l 2484 2189 l 2493 2187 l 2501 2183 l 2509 2180 l 2516 2177 l 2524 2172 l 2533 2166 l 2541 2159 l 2548 2152 l 2555 2144 l 2560 2137 l 2565 2129 l 2569 2121 l 2573 2111 l 2576 2102 l 2579 2093 l 2581 2084 l 2583 2076 l 2584 2068 l 2584 2058 l 2584 2049 l 2583 2040 l 2582 2032 l 2580 2023 l 2577 2014 l 2572 2004 l 2568 1994 l 2563 1984 l 2558 1976 l 2552 1968 l 2546 1960 l 2538 1952 l 2531 1945 l 2524 1938 l 2517 1932 l 2509 1926 l 2501 1921 l 2493 1914 l 2484 1908 l 2476 1902 l 2468 1897 l 2461 1890 l 2453 1883 l 2445 1876 l 2437 1869 l 2431 1862 l 2427 1856 l 2423 1849 l 2420 1842 l 2418 1835 l 2417 1830 l 2417 1825 l 2417 1820 l 2417 1816 l 2418 1812 l 2419 1808 l 2419 1806 l 2420 1803 l 2421 1801 l 2422 1798 l 2424 1796 l 2425 1794 l 2427 1792 l 2429 1789 l 2431 1787 l 2433 1785 l 2436 1783 l 2437 1781 l 2439 1780 l 2442 1779 l 2444 1778 l 2446 1778 l 2448 1777 l 2450 1777 l 2452 1777 l 2455 1777 l 2457 1777 l 2460 1776 l 2462 1776 l 2465 1775 l 2468 1775 l 2470 1775 l 2473 1775 l 2476 1775 l 2479 1776 l 2482 1776 l 2484 1777 l 2487 1777 l 2489 1777 l 2492 1778 l 2495 1779 l 2498 1780 l 2502 1782 l 2506 1785 l 2510 1788 l 2514 1790 l 2517 1792 l 2520 1794 l 2522 1795 l 2524 1796 l 2525 1798 l 2527 1799 l 2529 1800 l 2531 1802 l 2534 1805 l 2536 1807 l 2539 1809 l 2540 1811 l 2542 1812 l 2544 1813 l 2546 1814 l 2547 1814 l 2549 1813 l 2550 1811 l 2552 1808 l 2554 1804 l 2555 1801 l 2556 1798 l 2556 1796 l 2557 1795 l 2557 1793 l 2557 1792 l gs 0.00 setgray ef gr gs col0 s gr % Polyline n 1871 1753 m 1902 1705 l 2090 1705 l 2075 1753 l 2075 2255 l 2012 2317 l 1933 2317 l 2012 2255 l 2012 1753 l 1871 1753 l 1918 1737 l cp gs 0.00 setgray ef gr gs col0 s gr % Polyline n 2169 1705 m 2278 1705 l 2263 1720 l 2247 1753 l 2247 2129 l 2278 2191 l 2152 2191 l 2169 2176 l 2185 2129 l 2185 2098 l 2185 1753 l cp gs 0.00 setgray ef gr gs col0 s gr % Ellipse n 1600 1877 42 42 0 360 DrawEllipse gs 0.00 setgray ef gr gs col0 s gr /Times-Bold ff 225.00 scf sf 2197 1439 m gs 1 -1 sc (1) dup sw pop 2 div neg 0 rm col0 sh gr /Times-Bold ff 225.00 scf sf 1667 2376 m gs 1 -1 sc (47) dup sw pop 2 div neg 0 rm col0 sh gr $F2psEnd rs %%EndDocument @endspecial Black Black 9043 11312 a Ft(Ho)-72 b(w)863 b(to)e(Di\256eren)-72 b(tiate)863 b(a)e(Num)-72 b(b)72 b(er)20727 15704 y Fs(Victor)519 b(Ufnaro)-43 b(vski)15328 17697 y(Cen)g(tre)520 b(for)h(Mathematical)e(Sciences)16835 19689 y(Lund)i(Institute)e(of)i(T)-130 b(ec)-43 b(hnology)22270 21682 y(P)-130 b(.O.)519 b(Bo)-43 b(x)520 b(118)21316 23674 y(SE-221)i(00)f(Lund)24264 25667 y(Sw)-43 b(eden)p 0 1 0 0 TeXcolorcmyk 20960 27659 a(ufn@maths.lth.se)p [[261 468 366 480] [1 1 1 [3 3]] [0 0 1]] (mailto:ufn@maths.lth.se) pdfm Black 22454 30452 a(Bo)24860 30170 y(\272)24860 30452 y(Ahlander)23419 32445 y(KTH/2IT)22237 34437 y(Electrum)519 b(213)22504 36430 y(164)i(40)g(Kista)24264 38422 y(Sw)-43 b(eden)p 0 1 0 0 TeXcolorcmyk 20743 40415 a(ahlb)43 b(oa@isk.kth.se)p [[259 353 368 365] [1 1 1 [3 3]] [0 0 1]] (mailto:ahlboa@isk.kth.se) pdfm Black 800 45308 a Fr(Abstract.)740 b Fq(W)-108 b(e)487 b(de\257ne)f(the)g(deriv)-72 b(ativ)-36 b(e)488 b(of)f(an)g(in)-36 b(teger)486 b(to)h(b)36 b(e)487 b(the)f(map)h(sending)f(ev)-36 b(ery)487 b(prime)g(to)g(1)800 46913 y(and)416 b(satisfying)h(the)f (Leibnitz)f(rule.)573 b(The)416 b(aim)g(of)h(the)f(article)g(is)h(to)f (consider)g(the)f(basic)i(prop)36 b(erties)415 b(of)800 48518 y(this)409 b(map)h(and)f(to)g(sho)-36 b(w)410 b(ho)-36 b(w)409 b(to)h(generalize)g(the)f(notion)h(to)f(the)g(case)h(of)h (rational)f(and)f(arbitrary)h(real)800 50123 y(n)-36 b(um)g(b)36 b(ers.)556 b(W)-108 b(e)370 b(mak)-36 b(e)370 b(some)h(conjectures)f(and)f(\257nd)g(some)h(connections)g(with)g (Goldbac)-36 b(h's)370 b(Conjecture)800 51728 y(and)310 b(the)g(Twin)h(Prime)g(Conjecture.)537 b(Finally)-108 b(,)336 b(w)-36 b(e)311 b(solv)-36 b(e)312 b(the)e(easiest)h(asso)36 b(ciated)311 b(di\256eren)-36 b(tial)311 b(equations)800 53333 y(and)433 b(calculate)i(the)e(generating)g(function.)800 57751 y Fp(1)2152 b(A)717 b(deriv)-120 b(ativ)-60 b(e)716 b(of)h(a)h(natural)e(n)-60 b(um)g(b)60 b(er)800 60671 y Fq(Let)529 b Fo(n)g Fq(b)36 b(e)530 b(a)f(p)36 b(ositiv)-36 b(e)530 b(in)-36 b(teger.)865 b(W)-108 b(e)530 b(w)-36 b(ould)529 b(lik)-36 b(e)530 b(to)g(de\257ne)e(a)h(deriv)-72 b(ativ)-36 b(e)530 b Fo(n)39833 60189 y Fn(0)40674 60671 y Fq(suc)-36 b(h)528 b(that)h(\()p Fo(n;)221 b(n)49257 60189 y Fn(0)49569 60671 y Fq(\))531 b(=)h(1)800 62276 y(if)466 b(and)e(only)i(if)g Fo(n)f Fq(is)g(square-free)g(\(as)h(is)f (the)f(case)i(for)f(p)36 b(olynomials\).)674 b(It)465 b(w)-36 b(ould)465 b(b)36 b(e)465 b(nice)g(to)g(preserv)-36 b(e)800 63882 y(some)404 b(natural)g(prop)36 b(erties,)410 b(for)404 b(example)h(\()p Fo(n)23307 63400 y Fm(k)23876 63882 y Fq(\))24382 63400 y Fn(0)25062 63882 y Fq(=)368 b Fo(k)45 b(n)27939 63400 y Fm(k)24 b Fn(\241)p Fl(1)29711 63882 y Fo(n)30487 63400 y Fn(0)30798 63882 y Fo(:)404 b Fq(Because)g(1)37158 63400 y Fl(2)38053 63882 y Fq(=)369 b(1)404 b(w)-36 b(e)405 b(should)e(ha)-36 b(v)g(e)404 b(1)50089 63400 y Fn(0)50769 63882 y Fq(=)369 b(0)800 65487 y(and)428 b Fo(n)4100 65005 y Fn(0)4780 65487 y Fq(=)369 b(\(1)286 b(+)f(1)221 b Fk(\242)g(\242)g(\242)287 b Fq(+)e(1\))14060 65005 y Fn(0)14739 65487 y Fq(=)369 b(0,)430 b(if)g(w)-36 b(e)429 b(w)-36 b(an)g(t)429 b(to)g(preserv)-36 b(e)428 b(linearit)-36 b(y)-108 b(.)578 b(But)428 b(if)h(w)-36 b(e)430 b(ignore)f(linearit)-36 b(y)430 b(and)800 67092 y(use)481 b(the)g(Leibnitz)g(rule)g(only)-108 b(,)494 b(w)-36 b(e)482 b(will)h(\257nd)d(that)h(it)h(is)g(su\261cien)-36 b(t)481 b(to)g(de\257ne)g Fo(p)40618 66610 y Fn(0)41409 67092 y Fq(for)i(primes)e Fo(p:)g Fq(Let)g(us)800 68697 y(try)434 b(to)f(de\257ne)g Fo(n)9055 68215 y Fn(0)9799 68697 y Fq(b)-36 b(y)434 b(using)f(t)-36 b(w)g(o)434 b(natural)f(rules:)p Black 2737 71296 a Fk(\262)p Black 651 w Fo(p)4705 70814 y Fn(0)5384 71296 y Fq(=)369 b(1)434 b(for)g(an)-36 b(y)433 b(prime)h Fo(p;)p Black 26475 74617 a Fq(1)p Black eop %%Page: 2 2 2 1 bop Black 0 TeXcolorgray Black Black Black 2737 1424 a Fk(\262)p Black 651 w Fq(\()p Fo(ab)p Fq(\))6300 942 y Fn(0)6979 1424 y Fq(=)368 b Fo(a)9042 942 y Fn(0)9353 1424 y Fo(b)295 b Fq(+)g Fo(ab)12744 942 y Fn(0)13488 1424 y Fq(for)434 b(an)-36 b(y)433 b Fo(a;)221 b(b)370 b Fk(2)e Fr(N)434 b Fq(\(Leibnitz)f(rule\).)800 4136 y(F)-108 b(or)433 b(instance,)14091 5741 y(6)14741 5193 y Fn(0)15421 5741 y Fq(=)368 b(\(2)296 b Fk(\242)f Fq(3\))20073 5193 y Fn(0)20753 5741 y Fq(=)368 b(2)22783 5193 y Fn(0)23389 5741 y Fk(\242)295 b Fq(3)h(+)f(2)g Fk(\242)g Fq(3)28565 5193 y Fn(0)29245 5741 y Fq(=)369 b(1)295 b Fk(\242)h Fq(3)f(+)g(2)g Fk(\242)h Fq(1)369 b(=)g(5)p Fo(:)800 8066 y Fq(Here)434 b(is)f(a)h(list)g(of)h(the)e(\257rst)g(18)h(p)36 b(ositiv)-36 b(e)434 b(in)-36 b(tegers)434 b(and)f(their)g(\257rst,)g (second)g(and)g(third)g(deriv)-72 b(ativ)-36 b(es:)p 6055 9807 41490 45 v 6033 11413 45 1606 v 7019 10931 a Fo(n)p 8738 11413 V 1518 w Fq(1)1108 b(2)f(3)g(4)g(5)h(6)f(7)1432 b(8)h(9)1107 b(10)h(11)f(12)h(13)g(14)f(15)1433 b(16)g(17)1108 b(18)p 47523 11413 V 6055 11457 41490 45 v 6033 13062 45 1606 v 6864 12580 a Fo(n)7640 12098 y Fn(0)p 8738 13062 V 9313 12580 a Fq(0)g(1)f(1)g(4)g(1)h(5)f(1)g(12)h(6)1432 b(7)1758 b(1)1432 b(16)h(1)1758 b(9)f(8)h(32)g(1)1433 b(21)p 47523 13062 V 6033 14667 V 6737 14185 a Fo(n)7513 13703 y Fn(00)p 8738 14667 V 9313 14185 a Fq(0)1108 b(0)f(0)g(4)g(0)h (1)f(0)g(16)h(5)1432 b(1)1758 b(0)1432 b(32)h(0)1758 b(6)1432 b(12)h(80)1758 b(0)1433 b(10)p 47523 14667 V 6033 16272 V 6609 15791 a Fo(n)7385 15309 y Fn(000)p 8738 16272 V 9313 15791 a Fq(0)1108 b(0)f(0)g(4)g(0)h(0)f(0)g(32)h(1) 1432 b(0)1758 b(0)1432 b(80)h(0)1758 b(5)1432 b(16)1108 b(176)1433 b(0)1758 b(7)p 47523 16272 V 6055 16316 41490 45 v 2751 18678 a(It)502 b(lo)36 b(oks)503 b(quite)e(un)-36 b(usual)501 b(but)f(\257rst)h(of)h(all)h(w)-36 b(e)501 b(need)g(to)h(c)-36 b(hec)g(k)501 b(that)g(our)g(de\257nition)g(mak)-36 b(es)502 b(sense)800 20283 y(and)433 b(is)h(w)-36 b(ell-de\257ned.)p Black 800 22995 a Fr(Theorem)499 b(1)p Black 651 w Fj(The)442 b(derivative)e Fo(n)17880 22513 y Fn(0)18634 22995 y Fj(c)-66 b(an)441 b(b)-66 b(e)441 b(wel)66 b(l-de\257ne)-66 b(d)442 b(as)h(fol)66 b(lows:)587 b(if)442 b Fo(n)369 b Fq(=)39832 21999 y Fi(Q)41086 22349 y Fm(k)41086 23382 y(i)p Fl(=1)42885 22995 y Fo(p)43538 22403 y Fm(n)44109 22538 y Fh(i)43538 23335 y Fm(i)44958 22995 y Fj(is)442 b(a)h(factoriza-)800 24600 y(tion)464 b(in)g(prime)g(p)-66 b(owers,)465 b(then)22923 27377 y Fo(n)23699 26829 y Fn(0)24379 27377 y Fq(=)369 b Fo(n)27460 25717 y Fm(k)26758 26115 y Fi(X)26956 28915 y Fm(i)p Fl(=1)29031 26479 y Fo(n)29807 26678 y Fm(i)p 29031 27072 1153 54 v 29093 28289 a Fo(p)29746 28488 y Fm(i)30316 27377 y Fo(:)p Black 20461 w Fq(\(1\))p Black 800 30852 a Fj(It)464 b(is)h(the)g(only)f(way)i(to)f(de\257ne)f Fo(n)17320 30370 y Fn(0)18096 30852 y Fj(that)g(satis\257es)h(desir)-66 b(e)g(d)464 b(pr)-66 b(op)g(erties.)800 33564 y(Pr)g(o)g(of.)594 b Fq(Because)427 b(1)10435 33082 y Fn(0)11115 33564 y Fq(=)368 b(\(1)281 b Fk(\242)f Fq(1\))15737 33082 y Fn(0)16417 33564 y Fq(=)368 b(1)18447 33082 y Fn(0)19039 33564 y Fk(\242)280 b Fq(1)h(+)f(1)g Fk(\242)h Fq(1)24141 33082 y Fn(0)24820 33564 y Fq(=)369 b(2)281 b Fk(\242)f Fq(1)28431 33082 y Fn(0)28742 33564 y Fo(;)427 b Fq(w)-36 b(e)426 b(ha)-36 b(v)g(e)427 b(only)g(one)f(c)-36 b(hoice)426 b(for)h(1)46141 33082 y Fn(0)46878 33564 y Fq(:)575 b(it)427 b(should)800 35169 y(b)36 b(e)378 b(zero.)560 b(Induction)377 b(and)g(Leibnitz)g(rule)h(sho)-36 b(w)378 b(that)f(if)h(the)g(deriv)-72 b(ativ)-36 b(e)378 b(is)g(w)-36 b(ell-de\257ned,)389 b(it)378 b(is)g(uniquely)800 36774 y(determined.)669 b(It)465 b(remains)f(to)g(c)-36 b(hec)g(k)464 b(that)g(the)g(equation)h (\()p 0 1 0 0 TeXcolorcmyk(1)p (#equation.1) [[348 386 354 398] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))f(is)h(consisten)-36 b(t)463 b(with)i(our)f(conditions.)670 b(It)800 38379 y(is)501 b(eviden)-36 b(t)501 b(for)h(primes)e(and)h (clear)g(that)g(\()p 0 1 0 0 TeXcolorcmyk(1)p (#equation.1) [[273 372 278 384] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))g(can)g(b)36 b(e)500 b(used)h(ev)-36 b(en)501 b(when)f(some)h Fo(n)41857 38578 y Fm(i)42734 38379 y Fq(are)h(equal)f(to)g(zero.)800 39984 y(Let)433 b Fo(a)369 b Fq(=)5563 38988 y Fi(Q)6817 39339 y Fm(k)6817 40372 y(i)p Fl(=1)8616 39984 y Fo(p)9269 39392 y Fm(a)9769 39527 y Fh(i)9269 40325 y Fm(i)10609 39984 y Fq(and)433 b Fo(b)369 b Fq(=)15441 38988 y Fi(Q)16695 39339 y Fm(k)16695 40372 y(i)p Fl(=1)18495 39984 y Fo(p)19148 39392 y Fm(b)19551 39527 y Fh(i)19148 40325 y Fm(i)19957 39984 y Fo(:)434 b Fq(Then)f(according)h(to)g(\()p 0 1 0 0 TeXcolorcmyk(1)p (#equation.1) [[361 357 367 369] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))f(the)g(Leibnitz)g(rule)g(lo)36 b(oks)436 b(as)14167 44219 y Fo(ab)16327 42559 y Fm(k)15624 42957 y Fi(X)15822 45757 y Fm(i)p Fl(=1)17897 43320 y Fo(a)18580 43519 y Fm(i)19251 43320 y Fq(+)295 b Fo(b)21111 43519 y Fm(i)p 17897 43914 3590 54 v 19178 45130 a Fo(p)19831 45329 y Fm(i)21988 44219 y Fq(=)23369 41948 y Fi(\303)24420 44219 y Fo(a)26027 42559 y Fm(k)25325 42957 y Fi(X)25523 45757 y Fm(i)p Fl(=1)27598 43320 y Fo(a)28281 43519 y Fm(i)p 27598 43914 1059 54 v 27613 45130 a Fo(p)28266 45329 y Fm(i)28789 41948 y Fi(!)30062 44219 y Fo(b)g Fq(+)g Fo(a)33121 41948 y Fi(\303)34173 44219 y Fo(b)35650 42559 y Fm(k)34947 42957 y Fi(X)35145 45757 y Fm(i)p Fl(=1)37270 43320 y Fo(b)37823 43519 y Fm(i)p 37220 43914 1029 54 v 37220 45130 a Fo(p)37873 45329 y Fm(i)38381 41948 y Fi(!)800 48251 y Fq(and)433 b(the)g(consistency)h(is)g(clear.)p 17333 48251 630 630 v 2751 50520 a(F)-108 b(or)433 b(example)9663 54144 y(\(60\))11975 53595 y Fn(0)12655 54144 y Fq(=)368 b(\(2)15191 53595 y Fl(2)16012 54144 y Fk(\242)296 b Fq(3)f Fk(\242)g Fq(5\))19442 53595 y Fn(0)20122 54144 y Fq(=)369 b(60)295 b Fk(\242)23763 52271 y Fi(\265)24873 53245 y Fq(2)p 24873 53838 651 54 v 24873 55055 a(2)25952 54144 y(+)27391 53245 y(1)p 27391 53838 V 27391 55055 a(3)28470 54144 y(+)29909 53245 y(1)p 29909 53838 V 29909 55055 a(5)30692 52271 y Fi(\266)32039 54144 y Fq(=)369 b(60)296 b(+)e(20)i(+)f(12)370 b(=)e(92)p Fo(:)2751 57819 y Fq(W)-108 b(e)310 b(can)g(extend)f(our)h(de\257nition)f(to)h(0)21193 57337 y Fn(0)21872 57819 y Fq(=)369 b(0,)335 b(and)310 b(it)f(is)i(easy)f(to)g(c)-36 b(hec)g(k)310 b(that)f(this)h(do)36 b(es)310 b(not)f(con)-36 b(tradict)800 59424 y(the)433 b(Leibnitz)g(rule.)2751 61029 y(Note)538 b(that)f(linearit)-36 b(y)538 b(do)36 b(es)538 b(not)f(hold)g(in)g(general;)591 b(for)538 b(man)-36 b(y)537 b Fo(a;)221 b(b)538 b Fq(w)-36 b(e)538 b(ha)-36 b(v)g(e)538 b(\()p Fo(a)365 b Fq(+)h Fo(b)p Fq(\))46425 60547 y Fn(0)47281 61029 y Fk(6)p Fq(=)545 b Fo(a)49521 60547 y Fn(0)50198 61029 y Fq(+)365 b Fo(b)52128 60547 y Fn(0)52439 61029 y Fo(:)800 62634 y Fq(F)-108 b(urthermore)527 b(\()p Fo(ab)p Fq(\))10676 62152 y Fn(00)11773 62634 y Fk(6)p Fq(=)k Fo(a)13999 62152 y Fn(00)14924 62634 y Fq(+)360 b(2)p Fo(a)17629 62152 y Fn(0)17940 62634 y Fo(b)18493 62152 y Fn(0)19163 62634 y Fq(+)g Fo(b)21088 62152 y Fn(00)22183 62634 y Fq(b)36 b(ecause)529 b(w)-36 b(e)529 b(need)f(linearit)-36 b(y)530 b(to)f(pro)-36 b(v)g(e)529 b(this.)864 b(It)529 b(w)-36 b(ould)529 b(b)36 b(e)800 64239 y(in)-36 b(teresting)368 b(to)h(describ)36 b(e)369 b(all)g(the)g(pairs)g(\()p Fo(a;)221 b(b)p Fq(\))368 b(that)h(solv)-36 b(e)370 b(the)e(di\256eren) -36 b(tial)368 b(equation)i(\()p Fo(a)163 b Fq(+)g Fo(b)p Fq(\))47186 63757 y Fn(0)47864 64239 y Fq(=)369 b Fo(a)49928 63757 y Fn(0)50401 64239 y Fq(+)163 b Fo(b)52129 63757 y Fn(0)52439 64239 y Fo(:)800 65844 y Fq(W)-108 b(e)506 b(can)f(\257nd)f(one)i(of)g(the)f(solutions,)524 b(\(4)p Fo(;)221 b Fq(8\))507 b(in)e(our)g(table)h(ab)36 b(o)-36 b(v)g(e.)795 b(This)506 b(solution)g(can)f(b)36 b(e)505 b(obtained)800 67449 y(from)434 b(the)f(solution)h(\(1)p Fo(;)221 b Fq(2\))434 b(b)-36 b(y)434 b(using)f(the)g(follo)-36 b(wing)436 b(result.)p Black 26475 74617 a(2)p Black eop %%Page: 3 3 3 2 bop Black 0 TeXcolorgray Black Black Black 800 1424 a Fr(Theorem)499 b(2)p Black 651 w Fj(If)464 b Fq(\()p Fo(a)295 b Fq(+)f Fo(b)p Fq(\))13745 942 y Fn(0)14425 1424 y Fq(=)368 b Fo(a)16488 942 y Fn(0)17094 1424 y Fq(+)295 b Fo(b)18954 942 y Fn(0)19264 1424 y Fo(;)466 b Fj(then)e(for)g(any)h(natur)-66 b(al)465 b Fo(k)45 b(;)465 b Fj(we)g(have)19481 4319 y Fq(\()p Fo(k)45 b(a)295 b Fq(+)g Fo(k)45 b(b)p Fq(\))24773 3771 y Fn(0)25452 4319 y Fq(=)369 b(\()p Fo(k)45 b(a)p Fq(\))29249 3771 y Fn(0)29854 4319 y Fq(+)295 b(\()p Fo(k)45 b(b)p Fq(\))33447 3771 y Fn(0)33757 4319 y Fo(:)800 7214 y Fj(The)465 b(same)g(holds)h (for)e(the)h(ine)-66 b(qualities)13742 10110 y Fq(\()p Fo(a)295 b Fq(+)g Fo(b)p Fq(\))17592 9561 y Fn(0)18271 10110 y Fk(\270)370 b Fo(a)20357 9561 y Fn(0)20962 10110 y Fq(+)295 b Fo(b)22822 9561 y Fn(0)23501 10110 y Fk(\))370 b Fq(\()p Fo(k)45 b(a)295 b Fq(+)f Fo(k)45 b(b)p Fq(\))30490 9561 y Fn(0)31170 10110 y Fk(\270)369 b Fq(\()p Fo(k)45 b(a)p Fq(\))34988 9561 y Fn(0)35593 10110 y Fq(+)295 b(\()p Fo(k)45 b(b)p Fq(\))39186 9561 y Fn(0)39496 10110 y Fo(;)13742 13005 y Fq(\()p Fo(a)295 b Fq(+)g Fo(b)p Fq(\))17592 12456 y Fn(0)18271 13005 y Fk(\267)370 b Fo(a)20357 12456 y Fn(0)20962 13005 y Fq(+)295 b Fo(b)22822 12456 y Fn(0)23501 13005 y Fk(\))370 b Fq(\()p Fo(k)45 b(a)295 b Fq(+)f Fo(k)45 b(b)p Fq(\))30490 12456 y Fn(0)31170 13005 y Fk(\267)369 b Fq(\()p Fo(k)45 b(a)p Fq(\))34988 12456 y Fn(0)35593 13005 y Fq(+)295 b(\()p Fo(k)45 b(b)p Fq(\))39186 12456 y Fn(0)39496 13005 y Fo(:)800 15313 y Fj(Mor)-66 b(e)g(over,)463 b(al)66 b(l)466 b(these)e(c)-66 b(an)464 b(b)-66 b(e)464 b(extende)-66 b(d)464 b(for)g(line)-66 b(ar)464 b(c)-66 b(ombinations,)462 b(for)j(example:)11863 18367 y Fq(\()12369 17105 y Fi(X)14509 18367 y Fo(\260)15184 18566 y Fm(i)15559 18367 y Fo(a)16242 18566 y Fm(i)16617 18367 y Fq(\))17123 17818 y Fn(0)17803 18367 y Fq(=)19183 17105 y Fi(X)21323 18367 y Fo(\260)21998 18566 y Fm(i)22374 18367 y Fq(\()p Fo(a)23563 18566 y Fm(i)23938 18367 y Fq(\))24444 17818 y Fn(0)25123 18367 y Fk(\))370 b Fq(\()p Fo(k)28269 17105 y Fi(X)30409 18367 y Fo(\260)31084 18566 y Fm(i)31460 18367 y Fo(a)32143 18566 y Fm(i)32518 18367 y Fq(\))33024 17818 y Fn(0)33703 18367 y Fq(=)35084 17105 y Fi(X)37224 18367 y Fo(\260)37899 18566 y Fm(i)38274 18367 y Fq(\()p Fo(k)45 b(a)40184 18566 y Fm(i)40560 18367 y Fq(\))41066 17818 y Fn(0)41376 18367 y Fo(:)800 21526 y Fj(Pr)-66 b(o)g(of.)582 b Fq(The)387 b(pro)36 b(of)387 b(is)g(the)f(same)g(for)h(all)h(the)e(cases,)397 b(so)386 b(it)h(is)g(su\261cien)-36 b(t)386 b(to)g(consider)g(only)i (one)e(of)h(them,)800 23131 y(for)434 b(example)g(the)f(case)i Fk(\270)e Fq(with)h(t)-36 b(w)g(o)434 b(summands:)12931 26026 y(\()p Fo(k)45 b(a)295 b Fq(+)f Fo(k)45 b(b)p Fq(\))18222 25478 y Fn(0)18902 26026 y Fq(=)368 b(\()p Fo(k)45 b Fq(\()p Fo(a)295 b Fq(+)g Fo(b)p Fq(\)\))25865 25478 y Fn(0)26544 26026 y Fq(=)369 b Fo(k)28646 25478 y Fn(0)28956 26026 y Fq(\()p Fo(a)295 b Fq(+)g Fo(b)p Fq(\))g(+)g Fo(k)45 b Fq(\()p Fo(a)295 b Fq(+)f Fo(b)p Fq(\))38978 25478 y Fn(0)39658 26026 y Fq(=)10003 28922 y Fo(k)10724 28373 y Fn(0)11034 28922 y Fo(a)h Fq(+)g Fo(k)14040 28373 y Fn(0)14351 28922 y Fo(b)g Fq(+)g Fo(k)45 b Fq(\()p Fo(a)295 b Fq(+)g Fo(b)p Fq(\))21077 28373 y Fn(0)21756 28922 y Fk(\270)369 b Fo(k)23879 28373 y Fn(0)24190 28922 y Fo(a)295 b Fq(+)f Fo(k)27195 28373 y Fn(0)27506 28922 y Fo(b)h Fq(+)g Fo(k)45 b(a)31065 28373 y Fn(0)31671 28922 y Fq(+)294 b Fo(k)45 b(b)34251 28373 y Fn(0)34931 28922 y Fq(=)369 b(\()p Fo(k)45 b(a)p Fq(\))38728 28373 y Fn(0)39333 28922 y Fq(+)295 b(\()p Fo(k)45 b(b)p Fq(\))42926 28373 y Fn(0)43236 28922 y Fo(:)p 800 31230 630 630 v Black 800 34883 a Fr(Corollary)500 b(1)p Black 6344 37779 a Fq(\(3)p Fo(k)45 b Fq(\))8727 37230 y Fn(0)9406 37779 y Fq(=)369 b Fo(k)11508 37230 y Fn(0)12114 37779 y Fq(+)295 b(\(2)p Fo(k)45 b Fq(\))15804 37230 y Fn(0)16114 37779 y Fq(;)221 b(\(2)p Fo(k)45 b Fq(\))19079 37230 y Fn(0)19759 37779 y Fk(\270)370 b Fq(2)p Fo(k)22533 37230 y Fn(0)22843 37779 y Fq(;)221 b(\(5)p Fo(k)45 b Fq(\))25808 37230 y Fn(0)26489 37779 y Fk(\267)369 b Fq(\(2)p Fo(k)45 b Fq(\))30274 37230 y Fn(0)30880 37779 y Fq(+)294 b(\(3)p Fo(k)45 b Fq(\))34569 37230 y Fn(0)34880 37779 y Fq(;)221 b(\(5)p Fo(k)45 b Fq(\))37845 37230 y Fn(0)38525 37779 y Fq(=)369 b(\(2)p Fo(k)45 b Fq(\))42289 37230 y Fn(0)42895 37779 y Fq(+)294 b(3\()p Fo(k)45 b Fq(\))46584 37230 y Fn(0)46895 37779 y Fo(:)800 40779 y Fj(Pr)-66 b(o)g(of.)12450 42384 y Fq(3)13100 41836 y Fn(0)13780 42384 y Fq(=)369 b(1)15811 41836 y Fn(0)16417 42384 y Fq(+)295 b(2)18374 41836 y Fn(0)18685 42384 y Fq(;)221 b(2)19917 41836 y Fn(0)20597 42384 y Fk(\270)369 b Fq(1)22649 41836 y Fn(0)23256 42384 y Fq(+)294 b(1)25212 41836 y Fn(0)25523 42384 y Fq(;)221 b(5)26755 41836 y Fn(0)27436 42384 y Fk(\267)369 b Fq(2)29488 41836 y Fn(0)30094 42384 y Fq(+)295 b(3)32051 41836 y Fn(0)32362 42384 y Fq(;)221 b(5)33594 41836 y Fn(0)34274 42384 y Fq(=)369 b(2)36305 41836 y Fn(0)36911 42384 y Fq(+)295 b(3)g Fk(\242)g Fq(1)40477 41836 y Fn(0)40788 42384 y Fo(:)p 800 44692 V 2751 46951 a Fq(Here)434 b(is)f(the)h(list)g (of)g(all)g(\()p Fo(a;)221 b(b)p Fq(\))434 b(with)g Fo(a)368 b(<)h(b)g Fk(\267)g Fq(100)p Fo(;)221 b Fq(gcd)r(\()p Fo(a;)g(b)p Fq(\))369 b(=)f(1)p Fo(;)435 b Fq(for)f(whic)-36 b(h)433 b(\()p Fo(a)295 b Fq(+)g Fo(b)p Fq(\))45883 46469 y Fn(0)46562 46951 y Fq(=)369 b Fo(a)48626 46469 y Fn(0)49231 46951 y Fq(+)295 b Fo(b)51091 46469 y Fn(0)51771 46951 y Fq(:)11812 49846 y(\(1)p Fo(;)221 b Fq(2\))p Fo(;)g Fq(\(4)p Fo(;)g Fq(35\))p Fo(;)g Fq(\(4)p Fo(;)g Fq(91\))p Fo(;)g Fq(\(8)p Fo(;)g Fq(85\))p Fo(;)g Fq(\(11)p Fo(;)g Fq(14\))p Fo(;)g Fq(\(18)p Fo(;)g Fq(67\))p Fo(;)g Fq(\(26)p Fo(;)g Fq(29\))p Fo(;)10187 52741 y Fq(\(27)p Fo(;)g Fq(55\))p Fo(;)g Fq(\(35)p Fo(;)g Fq(81\))p Fo(;)g Fq(\(38)p Fo(;)g Fq(47\))p Fo(;)g Fq(\(38)p Fo(;)g Fq(83\))p Fo(;)g Fq(\(50)p Fo(;)g Fq(79\))p Fo(;)g Fq(\(62)p Fo(;)g Fq(83\))p Fo(;)g Fq(\(95)p Fo(;)g Fq(99\))p Fo(:)2751 55049 y Fq(A)434 b(similar)g(result)f(is)p Black 800 57729 a Fr(Theorem)499 b(3)p Black 651 w Fj(F)-100 b(or)465 b(any)g(natur)-66 b(al)465 b Fo(k)414 b(>)368 b Fq(1)p Fo(;)20736 60624 y(n)21512 60075 y Fn(0)22192 60624 y Fk(\270)h Fo(n)g Fk(\))g Fq(\()p Fo(k)45 b(n)p Fq(\))28945 60075 y Fn(0)29625 60624 y Fo(>)369 b(k)45 b(n:)800 63519 y Fj(Pr)-66 b(o)g(of.)18313 65124 y Fq(\()p Fo(k)45 b(n)p Fq(\))20822 64576 y Fn(0)21501 65124 y Fq(=)369 b Fo(k)23603 64576 y Fn(0)23914 65124 y Fo(n)295 b Fq(+)g Fo(k)45 b(n)27789 64576 y Fn(0)28469 65124 y Fo(>)369 b(k)45 b(n)31347 64576 y Fn(0)32027 65124 y Fk(\270)369 b Fo(k)45 b(n:)p 800 67432 V 2751 69691 a Fq(The)348 b(follo)-36 b(wing)351 b(theorem)d(sho)-36 b(ws)348 b(that)g(ev)-36 b(ery)349 b Fo(n)369 b(>)g Fq(4)349 b(that)e(is)i(divisible)g(b)-36 b(y)348 b(4)h(satis\257es)f(the)g (condition)800 71296 y Fo(n)1576 70814 y Fn(0)2256 71296 y Fo(>)369 b(n:)p Black 26475 74617 a Fq(3)p Black eop %%Page: 4 4 4 3 bop Black 0 TeXcolorgray Black Black Black 800 1424 a Fr(Theorem)499 b(4)p Black 651 w Fj(If)480 b Fo(n)399 b Fq(=)f Fo(p)13150 942 y Fm(p)13985 1424 y Fk(\242)307 b Fo(m)481 b Fj(for)f(some)h(prime)f Fo(p)h Fj(and)g(natur)-66 b(al)480 b Fo(m)398 b(>)g Fq(1)p Fo(;)482 b Fj(then)e Fo(n)41736 942 y Fn(0)42445 1424 y Fq(=)398 b Fo(p)44508 942 y Fm(p)45037 1424 y Fq(\()p Fo(m)307 b Fq(+)f Fo(m)49444 942 y Fn(0)49754 1424 y Fq(\))481 b Fj(and)800 3029 y Fq(lim)2607 3228 y Fm(k)24 b Fn(!1)5279 3029 y Fo(n)6055 2547 y Fl(\()p Fm(k)g Fl(\))7725 3029 y Fq(=)368 b Fk(1)p Fo(:)800 5741 y Fj(Pr)-66 b(o)g(of.)590 b Fq(According)413 b(to)g(the)g(Leibnitz)f(rule)h(and)f(\()p 0 1 0 0 TeXcolorcmyk(1)p (#equation.1) [[301 665 307 677] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\),)417 b Fo(n)28119 5259 y Fn(0)28799 5741 y Fq(=)369 b(\()p Fo(p)31339 5259 y Fm(p)31868 5741 y Fq(\))32374 5259 y Fn(0)32937 5741 y Fk(\242)253 b Fo(m)g Fq(+)f Fo(p)36867 5259 y Fm(p)37649 5741 y Fk(\242)h Fo(m)39409 5259 y Fn(0)40088 5741 y Fq(=)369 b Fo(p)42122 5259 y Fm(p)42650 5741 y Fq(\()p Fo(m)253 b Fq(+)g Fo(m)46950 5259 y Fn(0)47260 5741 y Fq(\))369 b Fo(>)f(n)414 b Fq(and)800 7346 y(b)-36 b(y)434 b(induction)e Fo(n)9163 6864 y Fl(\()p Fm(k)24 b Fl(\))10833 7346 y Fk(\270)369 b Fo(n)296 b Fq(+)f Fo(k)45 b(:)p 16130 7346 630 630 v 2751 9616 a Fq(The)434 b(situation)f(c)-36 b(hanges)434 b(when)f(the)g(exp)36 b(onen)-36 b(t)433 b(of)h Fo(p)g Fq(is)g(less)g(than)e Fo(p:)p Black 800 12328 a Fr(Theorem)499 b(5)p Black 651 w Fj(L)-66 b(et)556 b Fo(p)11520 11846 y Fm(k)12646 12328 y Fj(b)-66 b(e)556 b(the)h(highest)g(p)-66 b(ower)557 b(of)g(prime)g Fo(p)g Fj(that)g(divides)f(the)h(natur)-66 b(al)557 b(numb)-66 b(er)556 b Fo(n:)i Fj(If)800 13933 y Fq(0)369 b Fo(<)g(k)414 b(<)369 b(p;)461 b Fj(then)g Fo(p)10685 13451 y Fm(k)24 b Fn(\241)p Fl(1)12917 13933 y Fj(is)461 b(the)g(highest)g(p)-66 b(ower)462 b(of)f Fo(p)g Fj(that)g(divides)g Fo(n)34924 13451 y Fn(0)35234 13933 y Fo(:)h Fj(In)f(p)-66 b(articular,)461 b(al)66 b(l)462 b(the)f(numb)-66 b(ers)800 15538 y Fo(n;)221 b(n)2934 15056 y Fn(0)3246 15538 y Fo(;)g(n)4604 15056 y Fn(00)5171 15538 y Fo(;)g(:)g(:)g(:)j(;)d(n)8860 15056 y Fl(\()p Fm(k)24 b Fl(\))10626 15538 y Fj(ar)-66 b(e)465 b(distinct.)800 18250 y(Pr)-66 b(o)g(of.)580 b Fq(Let)378 b Fo(n)370 b Fq(=)e Fo(p)10257 17768 y Fm(k)10826 18250 y Fo(m)p Fq(.)560 b(Then)379 b Fo(n)17003 17768 y Fn(0)17683 18250 y Fq(=)368 b Fo(k)45 b(p)20437 17768 y Fm(k)24 b Fn(\241)p Fl(1)22208 18250 y Fo(m)184 b Fq(+)g Fo(p)25379 17768 y Fm(k)25947 18250 y Fo(m)27085 17768 y Fn(0)27764 18250 y Fq(=)369 b Fo(p)29798 17768 y Fm(k)24 b Fn(\241)p Fl(1)31569 18250 y Fq(\()p Fo(k)45 b(m)184 b Fq(+)g Fo(pm)37105 17768 y Fn(0)37414 18250 y Fq(\),)390 b(and)378 b(the)h(expression)g (inside)800 19855 y(paren)-36 b(theses)433 b(is)g(not)h(divisible)g(b) -36 b(y)434 b Fo(p:)p 19763 19855 V Black 800 23563 a Fr(Corollary)500 b(2)p Black 650 w Fj(A)465 b(p)-66 b(ositive)463 b(inte)-66 b(ger)463 b Fo(n)i Fj(is)g(squar)-66 b(e-fr)g(e)g(e)463 b(if)h(and)h(only)g(if)f Fq(\()p Fo(n;)221 b(n)38905 23081 y Fn(0)39217 23563 y Fq(\))368 b(=)h(1)p Fo(:)800 26607 y Fj(Pr)-66 b(o)g(of.)597 b Fq(If)434 b Fo(p)6773 26125 y Fl(2)7298 26607 y Fk(j)p Fo(n;)h Fq(then)d Fo(p)p Fk(j)p Fo(n)13999 26125 y Fn(0)14744 26607 y Fq(and)h(\()p Fo(n;)221 b(n)19913 26125 y Fn(0)20224 26607 y Fq(\))369 b Fo(>)g Fq(1)p Fo(:)434 b Fq(On)f(the)g(other)g(hand,)g(if)h Fo(p)p Fk(j)p Fo(n)g Fq(and)f Fo(p)p Fk(j)p Fo(n)43093 26125 y Fn(0)43837 26607 y Fq(then)g Fo(p)47453 26125 y Fl(2)47979 26607 y Fk(j)p Fo(n:)p 49919 26607 V 800 31709 a Fp(2)2152 b(The)716 b(equation)g Fg(n)18320 31015 y Fk(0)19272 31709 y Ff(=)531 b Fg(n)800 34629 y Fq(Let)295 b(us)h(solv)-36 b(e)296 b(some)g(di\256eren)-36 b(tial)296 b(equations)g(\(using)f(our)g(de\257nition)g(of)i(deriv)-72 b(ativ)-36 b(e\))296 b(in)f(p)36 b(ositiv)-36 b(e)297 b(in)-36 b(tegers.)p Black 800 38946 a Fr(Theorem)499 b(6)p Black 651 w Fj(The)509 b(e)-66 b(quation)509 b Fo(n)17235 38464 y Fn(0)17998 38946 y Fq(=)451 b Fo(n)510 b Fj(holds)h(if)e(and)h(only)f Fo(n)453 b Fq(=)e Fo(p)34201 38464 y Fm(p)34730 38946 y Fo(;)510 b Fj(wher)-66 b(e)509 b Fo(p)h Fj(is)f(any)h(prime)f(numb)-66 b(er.)800 40551 y(In)464 b(p)-66 b(articular,)464 b(it)g(has)i(in\257nitely)c(many)j (solutions)g(in)f(natur)-66 b(al)465 b(numb)-66 b(ers.)800 43264 y(Pr)g(o)g(of.)588 b Fq(If)408 b(prime)f Fo(p)g Fq(divides)g Fo(n;)h Fq(then)e(according)i(to)f(Theorem)p 0 1 0 0 TeXcolorcmyk 407 w(5)p (#Thm.5) [[364 328 370 340] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)413 b(at)407 b(least)h Fo(p)39148 42781 y Fm(p)40084 43264 y Fq(should)e(divide)i Fo(n)f Fq(or)h(else)800 44869 y Fo(n)1576 44387 y Fn(0)2268 44869 y Fk(6)p Fq(=)381 b Fo(n:)442 b Fq(But)e(Theorem)p 0 1 0 0 TeXcolorcmyk 441 w(4)p (#Thm.4) [[192 313 198 325] [1 1 1 [3 3]] [0 0 1]] pdfm Black 441 w(implies)h(that)f(in)h(this)f(case)i Fo(n)381 b Fq(=)g Fo(p)31699 44387 y Fm(p)32228 44869 y Fo(;)441 b Fq(whic)-36 b(h)441 b(according)g(to)g(\()p 0 1 0 0 TeXcolorcmyk(1)p (#equation.1) [[474 313 480 325] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))f(is)i(eviden)-36 b(tly)800 46474 y(equal)434 b(to)g Fo(n)6598 45992 y Fn(0)6909 46474 y Fo(:)p 7704 46474 V 2751 48743 a Fq(Th)-36 b(us,)465 b(considering)459 b(the)f(map)h Fo(n)412 b Fk(\241)-221 b(!)412 b Fo(n)22961 48261 y Fn(0)23731 48743 y Fq(as)459 b(a)h(dynamical)f(system,)466 b(w)-36 b(e)459 b(ha)-36 b(v)g(e)459 b(a)g(quite)g(in)-36 b(teresting)800 50348 y(ob)72 b(ject.)784 b(Namely)-108 b(,)520 b(w)-36 b(e)502 b(ha)-36 b(v)g(e)502 b(in\257nitely)g(man)-36 b(y)502 b(\257xed)g(p)36 b(oin)-36 b(ts,)519 b(0)502 b(is)g(a)g(natural)g(attractor,)519 b(b)36 b(ecause)502 b(all)800 51953 y(the)407 b(primes)f(after)i(t)-36 b(w)g(o)407 b(di\256eren)-36 b(tiations)407 b(b)36 b(ecome)407 b(zero.)570 b(No)-36 b(w)407 b(it)g(is)h(time)f(to)g(form)-36 b(ulate)408 b(the)e(\257rst)g(op)36 b(en)800 53558 y(problem.)p Black 800 56602 a Fr(Conjecture)499 b(1)p Black 651 w Fj(Ther)-66 b(e)658 b(exist)h(in\257nitely)e(many)i(c)-66 b(omp)g(osite)659 b(numb)-66 b(ers)658 b Fo(n)j Fj(such)f(that)f Fo(n)46107 56120 y Fl(\()p Fm(k)24 b Fl(\))48138 56602 y Fq(=)729 b(0)660 b Fj(for)800 58207 y(su\261ciently)464 b(lar)-66 b(ge)464 b(natur)-66 b(al)465 b Fo(k)45 b(:)2751 61252 y Fq(As)566 b(w)-36 b(e)566 b(will)h(see)f(later,)599 b(the)565 b(Twin)i(Prime)e(Conjecture)h(w)-36 b(ould)566 b(fail)h(if)g(this)e(conjecture)h(is)g(false.)800 62857 y(Preliminary)393 b(n)-36 b(umerical)393 b(exp)36 b(erimen)-36 b(ts)392 b(sho)-36 b(w)392 b(that)g(for)h(non-\257xed)f(p)36 b(oin)-36 b(ts)392 b(either)g(the)g(deriv)-72 b(ativ)-36 b(es)393 b Fo(n)51499 62375 y Fl(\()p Fm(k)24 b Fl(\))800 64462 y Fq(tend)433 b(to)g(in\257nit)-36 b(y)433 b(or)h(b)36 b(ecome)434 b(zero;)g(ho)-36 b(w)g(ev)g(er,)434 b(w)-36 b(e)434 b(do)g(not)f(kno)-36 b(w)434 b(ho)-36 b(w)434 b(to)f(pro)-36 b(v)g(e)434 b(this.)p Black 800 67174 a Fr(Conjecture)499 b(2)p Black 651 w Fj(Exactly)573 b(one)h(of)f(the)h(fol)66 b(lowing)573 b(c)-66 b(ould)574 b(happ)-66 b(en:)815 b(either)573 b Fo(n)39844 66692 y Fl(\()p Fm(k)24 b Fl(\))41715 67174 y Fq(=)571 b(0)j Fj(for)f(su\261ciently)800 68779 y(lar)-66 b(ge)464 b Fo(k)45 b(;)466 b Fj(or)f Fq(lim)8969 68978 y Fm(k)24 b Fn(!1)11641 68779 y Fo(n)12417 68297 y Fl(\()p Fm(k)g Fl(\))14087 68779 y Fq(=)369 b Fk(1)p Fo(;)465 b Fj(or)h Fo(n)369 b Fq(=)g Fo(p)22479 68297 y Fm(p)23472 68779 y Fj(for)465 b(some)g(prime)f Fo(p:)p Black 26475 74617 a Fq(4)p Black eop %%Page: 5 5 5 4 bop Black 0 TeXcolorgray Black Black 800 1424 a Fq(According)589 b(to)g(Theorem)p 0 1 0 0 TeXcolorcmyk 589 w(4)p (#Thm.4) [[203 704 209 716] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)629 b(it)589 b(is)g(su\261cien)-36 b(t)588 b(to)h(pro)-36 b(v)g(e)589 b(that,)628 b(for)589 b(some)g Fo(k)45 b Fq(,)629 b(the)588 b(deriv)-72 b(ativ)-36 b(e)590 b Fo(n)50036 942 y Fl(\()p Fm(k)24 b Fl(\))51926 1424 y Fq(is)800 3029 y(divisible)520 b(b)-36 b(y)518 b Fo(p)8531 2547 y Fm(p)9579 3029 y Fq(\(for)h(example)h(b)-36 b(y)518 b(4\))p Fo(:)h Fq(In)g(particularly)g(w)-36 b(e)519 b(do)g(not)g(exp)36 b(ect)519 b(p)36 b(erio)g(dic)519 b(p)36 b(oin)-36 b(t)518 b(except)800 4634 y(\257xed)433 b(p)36 b(oin)-36 b(ts)434 b Fo(p)8506 4152 y Fm(p)9034 4634 y Fq(.)p Black 800 7595 a Fr(Conjecture)499 b(3)p Black 651 w Fj(The)464 b(di\256er)-66 b(ential)463 b(e)-66 b(quation)464 b Fo(n)25031 7113 y Fl(\()p Fm(k)24 b Fl(\))26701 7595 y Fq(=)368 b Fo(n)466 b Fj(has)f(only)g(trivial)f(solutions)h Fo(p)44405 7113 y Fm(p)45399 7595 y Fj(for)f(primes)h Fo(p:)2751 10556 y Fq(Theorem)p 0 1 0 0 TeXcolorcmyk 590 w(5)p (#Thm.5) [[148 622 153 634] [1 1 1 [3 3]] [0 0 1]] pdfm Black 589 w(giv)-36 b(es)591 b(some)e(restrictions)h(for)f(p)36 b(ossible)590 b(non)-36 b(trivial)590 b(p)36 b(erio)g(ds:)890 b(if)590 b Fo(p)43935 10074 y Fm(k)45093 10556 y Fq(divides)g Fo(n)g Fq(the)800 12161 y(p)36 b(erio)g(d)434 b(m)-36 b(ust)432 b(b)36 b(e)434 b(at)f(least)h Fo(k)341 b Fq(+)295 b(1)p Fo(:)2751 13766 y Fq(Conjecture)p 0 1 0 0 TeXcolorcmyk 399 w(3)p (#Cnj.3) [[156 593 162 605] [1 1 1 [3 3]] [0 0 1]] pdfm Black 399 w(is)399 b(not)g(trivial)h(ev)-36 b(en)398 b(in)h(sp)36 b(ecial)400 b(cases.)567 b(Supp)36 b(ose,)405 b(for)399 b(example,)407 b(that)398 b Fo(n)h Fq(has)g(p)36 b(erio)g(d)800 15372 y(2)p Fo(;)519 b Fq(i.e.)833 b Fo(m)513 b Fq(=)g Fo(n)8776 14889 y Fn(0)9600 15372 y Fk(6)p Fq(=)g Fo(n)518 b Fq(and)g Fo(m)16171 14889 y Fn(0)16994 15372 y Fq(=)513 b Fo(n:)519 b Fq(According)g(to)f(Theorem)p 0 1 0 0 TeXcolorcmyk 518 w(4)p (#Thm.4) [[375 579 381 591] [1 1 1 [3 3]] [0 0 1]] pdfm Black 519 w(and)g(Theorem)p 0 1 0 0 TeXcolorcmyk 518 w(5)p (#Thm.5) [[459 579 465 591] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)540 b(b)36 b(oth)518 b Fo(n)h Fq(and)e Fo(m)800 16977 y Fq(should)419 b(b)36 b(e)419 b(the)g(pro)36 b(duct)419 b(of)h(distinct)f(primes:)571 b Fo(n)369 b Fq(=)27138 15980 y Fi(Q)28392 16331 y Fm(k)28392 17364 y(i)p Fl(=1)30192 16977 y Fo(p)30845 17176 y Fm(i)31220 16977 y Fo(;)420 b(m)369 b Fq(=)34888 15980 y Fi(Q)36143 16331 y Fm(l)36143 17364 y(j)51 b Fl(=1)38054 16977 y Fo(q)38631 17176 y Fm(j)39117 16977 y Fo(;)420 b Fq(where)f(all)i(primes)e Fo(p)50271 17176 y Fm(i)51066 16977 y Fq(are)800 18582 y(distinct)433 b(from)h(all)h Fo(q)10959 18781 y Fm(j)11445 18582 y Fo(:)f Fq(Therefore,)g(our)g(conjecture)f(in)h(this)f(case)h (is)g(equiv)-72 b(alen)-36 b(t)434 b(to)g(the)f(follo)-36 b(wing:)p Black 800 21543 a Fr(Conjecture)499 b(4)p Black 651 w Fj(F)-100 b(or)465 b(any)g(p)-66 b(ositive)463 b(inte)-66 b(gers)463 b Fo(k)45 b(;)221 b(l)29 b(;)467 b Fj(the)d(e)-66 b(quation)19934 23304 y Fi(\303)21689 23915 y Fm(k)20986 24314 y Fi(X)21184 27113 y Fm(i)p Fl(=1)23448 24677 y Fq(1)p 23259 25270 1029 54 v 23259 26487 a Fo(p)23912 26686 y Fm(i)24420 23304 y Fi(!)221 b(\303)27559 23915 y Fm(l)26745 24314 y Fi(X)26887 27113 y Fm(j)51 b Fl(=1)29224 24677 y Fq(1)p 29018 25270 1064 54 v 29018 26487 a Fo(q)29595 26686 y Fm(j)30214 23304 y Fi(!)31635 25576 y Fq(=)368 b(1)800 29707 y Fj(has)466 b(no)e(solutions)i(in)e(distinct)f(primes.)800 34133 y Fp(3)2152 b(The)716 b(equation)g Fg(n)18320 33439 y Fk(0)19272 34133 y Ff(=)531 b Fg(a)800 37054 y Fq(W)-108 b(e)434 b(start)f(with)h(t)-36 b(w)g(o)433 b(easy)i(equations.)p Black 800 39703 a Fr(Theorem)499 b(7)p Black 651 w Fj(The)465 b(di\256er)-66 b(ential)462 b(e)-66 b(quation)464 b Fo(n)23704 39221 y Fn(0)24384 39703 y Fq(=)369 b(0)465 b Fj(has)g(only)g(one)g(p) -66 b(ositive)463 b(inte)-66 b(ger)463 b(solution)i Fo(n)369 b Fq(=)g(1)p Fo(:)800 42353 y Fj(Pr)-66 b(o)g(of.)597 b Fq(F)-108 b(ollo)-36 b(ws)434 b(immediately)h(from)f(\()p 0 1 0 0 TeXcolorcmyk(1)p (#equation.1) [[256 336 262 348] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\).)p 22512 42353 630 630 v Black 800 45957 a Fr(Theorem)499 b(8)p Black 651 w Fj(The)528 b(di\256er)-66 b(ential)526 b(e)-66 b(quation)527 b Fo(n)23894 45475 y Fn(0)24691 45957 y Fq(=)486 b(1)528 b Fj(in)g(natur)-66 b(al)527 b(numb)-66 b(ers)528 b(has)h(only)f(primes)f(as)i(solu-)800 47562 y(tions.)800 50523 y(Pr)-66 b(o)g(of.)580 b Fq(If)380 b(the)e(n)-36 b(um)g(b)36 b(er)378 b(is)h(comp)36 b(osite)380 b(then)e(according)h(to)g(Leibnitz)g(rule)g(and)f(the)h(previous)g (theorem,)800 52129 y(the)433 b(deriv)-72 b(ativ)-36 b(e)434 b(can)g(b)36 b(e)433 b(written)h(as)g(the)f(sum)g(of)h(t)-36 b(w)g(o)434 b(p)36 b(ositiv)-36 b(e)434 b(in)-36 b(tegers)434 b(and)f(is)h(greater)g(than)e(1.)p 50731 52129 V 2751 54377 a(All)i(other)f(equations)h Fo(n)14878 53895 y Fn(0)15558 54377 y Fq(=)369 b Fo(a)433 b Fq(ha)-36 b(v)g(e)434 b(only)g(\257nitely)g(man)-36 b(y)434 b(solutions,)g(if)g(an)-36 b(y)-108 b(.)p Black 800 57338 a Fr(Theorem)499 b(9)p Black 651 w Fj(\([)p 0 1 0 0 TeXcolorcmyk(1)p (#cite.Bar) [[157 201 163 213] [1 1 1 [3 3]] [0 0 1]] pdfm Black(]\))465 b(F)-100 b(or)465 b(any)g(p)-66 b(ositive)463 b(inte)-66 b(ger)463 b Fo(n)22957 60857 y(n)23733 60309 y Fn(0)24413 60857 y Fk(\267)25948 59959 y Fo(n)221 b Fq(log)28626 60272 y Fl(2)29373 59959 y Fo(n)p 25948 60552 4202 54 v 27723 61769 a Fq(2)30282 60857 y Fo(:)p Black 20495 w Fq(\(2\))p Black 800 64057 a Fj(If)464 b Fo(n)h Fj(is)g(not)g(a)g(prime,)e(then)23924 65662 y Fo(n)24700 65114 y Fn(0)25379 65662 y Fk(\270)370 b Fq(2)27432 64641 y Fk(p)p 28539 64641 777 54 v 1021 x Fo(n:)p Black 21462 w Fq(\(3\))p Black 800 67956 a Fj(Mor)-66 b(e)464 b(gener)-66 b(al)66 b(ly,)464 b(if)g Fo(n)h Fj(is)g(a)g(pr)-66 b(o)g(duct)464 b(of)h Fo(k)510 b Fj(factors)465 b(lar)-66 b(ger)464 b(than)g Fq(1)p Fj(,)h(then)23532 70964 y Fo(n)24308 70415 y Fn(0)24988 70964 y Fk(\270)369 b Fo(k)45 b(n)28020 70042 y Fh(k)14 b Fe(\241)p Fd(1)p 28020 70214 1499 40 v 28543 70762 a Fh(k)29707 70964 y Fo(:)p Black 21070 w Fq(\(4\))p Black Black 26475 74617 a(5)p Black eop %%Page: 6 6 6 5 bop Black 0 TeXcolorgray Black Black 800 1424 a Fj(Pr)-66 b(o)g(of.)597 b Fq(If)434 b Fo(n)369 b Fq(=)8646 428 y Fi(Q)9900 778 y Fm(k)9900 1812 y(i)p Fl(=1)11700 1424 y Fo(p)12353 832 y Fm(n)12924 967 y Fh(i)12353 1765 y Fm(i)13330 1424 y Fq(,)434 b(then)18406 5659 y Fo(n)369 b Fk(\270)21545 3998 y Fm(k)20953 4397 y Fi(Y)21041 7196 y Fm(i)p Fl(=1)22872 5659 y Fq(2)23522 5110 y Fm(n)24093 5245 y Fh(i)24869 5659 y Fk(\))g Fq(log)28246 5972 y Fl(2)28993 5659 y Fo(n)h Fk(\270)32244 3998 y Fm(k)31541 4397 y Fi(X)31739 7196 y Fm(i)p Fl(=1)33681 5659 y Fo(n)34457 5858 y Fm(i)34833 5659 y Fo(:)800 9764 y Fq(According)434 b(to)f(\()p 0 1 0 0 TeXcolorcmyk(1)p (#equation.1) [[154 629 160 641] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))h(w)-36 b(e)434 b(no)-36 b(w)434 b(ha)-36 b(v)g(e)16110 13944 y Fo(n)16886 13395 y Fn(0)17566 13944 y Fq(=)369 b Fo(n)20647 12283 y Fm(k)19944 12682 y Fi(X)20142 15481 y Fm(i)p Fl(=1)22217 13045 y Fo(n)22993 13244 y Fm(i)p 22217 13638 1153 54 v 22279 14855 a Fo(p)22932 15054 y Fm(i)23871 13944 y Fk(\267)25406 13038 y Fo(n)26403 12042 y Fi(P)27806 12393 y Fm(k)27806 13426 y(i)p Fl(=1)29606 13038 y Fo(n)30382 13237 y Fm(i)p 25406 13638 5352 54 v 27757 14855 a Fq(2)31260 13944 y Fk(\267)32794 13045 y Fo(n)221 b Fq(log)35472 13359 y Fl(2)36219 13045 y Fo(n)p 32794 13638 4202 54 v 34570 14855 a Fq(2)37129 13944 y Fo(:)800 17976 y Fq(If)434 b Fo(n)370 b Fq(=)e Fo(n)5402 18175 y Fl(1)5928 17976 y Fo(n)6704 18175 y Fl(2)7231 17976 y Fo(n)8007 18175 y Fl(3)8754 17976 y Fk(\242)221 b(\242)g(\242)h Fo(n)11301 18175 y Fm(k)12304 17976 y Fq(then,)433 b(according)h(to)g(the)f(Leibnitz)g(rule,)6225 20909 y Fo(n)7001 20361 y Fn(0)7680 20909 y Fq(=)369 b Fo(n)9837 20361 y Fn(0)9837 21237 y Fl(1)10363 20909 y Fo(n)11139 21108 y Fl(2)11666 20909 y Fo(n)12442 21108 y Fl(3)13189 20909 y Fk(\242)221 b(\242)g(\242)h Fo(n)15736 21108 y Fm(k)16601 20909 y Fq(+)295 b Fo(n)18684 21108 y Fl(1)19210 20909 y Fo(n)19986 20361 y Fn(0)19986 21237 y Fl(2)20512 20909 y Fo(n)21288 21108 y Fl(3)22036 20909 y Fk(\242)221 b(\242)g(\242)h Fo(n)24583 21108 y Fm(k)25447 20909 y Fq(+)295 b Fo(n)27530 21108 y Fl(1)28056 20909 y Fo(n)28832 21108 y Fl(2)29358 20909 y Fo(n)30134 20361 y Fn(0)30134 21237 y Fl(3)30882 20909 y Fk(\242)221 b(\242)g(\242)h Fo(n)33429 21108 y Fm(k)34294 20909 y Fq(+)294 b Fo(:)221 b(:)g(:)297 b Fq(+)e Fo(n)39505 21108 y Fl(1)40031 20909 y Fo(n)40807 21108 y Fl(2)41333 20909 y Fo(n)42109 21108 y Fl(3)42857 20909 y Fk(\242)221 b(\242)g(\242)h Fo(n)45404 20361 y Fn(0)45404 21237 y Fm(k)46342 20909 y Fk(\270)7704 23842 y Fo(n)8480 24041 y Fl(2)9006 23842 y Fo(n)9782 24041 y Fl(3)10308 23842 y Fo(n)11084 24041 y Fl(4)11832 23842 y Fk(\242)f(\242)g(\242)h Fo(n)14379 24041 y Fm(k)15243 23842 y Fq(+)295 b Fo(n)17326 24041 y Fl(1)17852 23842 y Fo(n)18628 24041 y Fl(3)19154 23842 y Fo(n)19930 24041 y Fl(4)20678 23842 y Fk(\242)221 b(\242)g(\242)h Fo(n)23225 24041 y Fm(k)24090 23842 y Fq(+)294 b Fo(n)26172 24041 y Fl(1)26699 23842 y Fo(n)27475 24041 y Fl(2)28001 23842 y Fo(n)28777 24041 y Fl(4)29524 23842 y Fk(\242)221 b(\242)g(\242)i Fo(n)32072 24041 y Fm(k)32936 23842 y Fq(+)295 b Fo(:)221 b(:)g(:)297 b Fq(+)d Fo(n)38147 24041 y Fl(1)38674 23842 y Fo(n)39450 24041 y Fl(2)40197 23842 y Fk(\242)221 b(\242)g(\242)h Fo(n)42744 24041 y Fm(k)24 b Fn(\241)p Fl(1)44885 23842 y Fq(=)5497 27330 y Fo(n)6494 25457 y Fi(\265)7932 26431 y Fq(1)p 7606 27024 1303 54 v 7606 28241 a Fo(n)8382 28440 y Fl(1)9336 27330 y Fq(+)11101 26431 y(1)p 10776 27024 V 10776 28241 a Fo(n)11552 28440 y Fl(2)12506 27330 y Fq(+)295 b Fo(:)221 b(:)g(:)296 b Fq(+)17422 26431 y(1)p 17074 27024 1346 54 v 17074 28241 a Fo(n)17850 28440 y Fm(k)18552 25457 y Fi(\266)19899 27330 y Fk(\270)369 b Fo(n)296 b Fk(\242)f Fo(k)23979 25457 y Fi(\265)25416 26431 y Fq(1)p 25090 27024 1303 54 v 25090 28241 a Fo(n)25866 28440 y Fl(1)26820 27330 y Fk(\242)27943 26431 y Fq(1)p 27617 27024 V 27617 28241 a Fo(n)28393 28440 y Fl(2)29274 27330 y Fk(\242)221 b(\242)g(\242)31525 26431 y Fq(1)p 31178 27024 1346 54 v 31178 28241 a Fo(n)31954 28440 y Fm(k)32656 25457 y Fi(\266)33790 25392 y Fd(1)p 33766 25553 453 40 v 33766 26102 a Fh(k)34776 27330 y Fq(=)369 b Fo(k)340 b Fk(\242)295 b Fo(n)h Fk(\242)f Fo(n)40482 26408 y Fe(\241)p Fd(1)p 40482 26580 1046 40 v 40779 27129 a Fh(k)42085 27330 y Fq(=)369 b Fo(k)340 b Fk(\242)295 b Fo(n)46055 26408 y Fh(k)14 b Fe(\241)p Fd(1)p 46055 26580 1499 40 v 46578 27129 a Fh(k)47742 27330 y Fo(:)800 30345 y Fq(Here)434 b(w)-36 b(e)433 b(ha)-36 b(v)g(e)434 b(replaced)g(the)f(arithmetic)g(mean)h(b)-36 b(y)433 b(the)g(geometric)h(mean.)p 39856 30345 630 630 v 2751 32614 a(Note)g(that)f(b)36 b(ounds)432 b(\()p 0 1 0 0 TeXcolorcmyk(2)p (#equation.2) [[196 423 202 435] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))i(and)f(\()p 0 1 0 0 TeXcolorcmyk(4)p (#equation.4) [[237 423 243 435] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))g(are)h(exact)h(for)f Fo(n)369 b Fq(=)g(2)30719 32132 y Fm(k)31288 32614 y Fo(:)p Black 800 35658 a Fr(Corollary)500 b(3)p Black 650 w Fj(If)434 b(the)h(di\256er)-66 b(ential)433 b(e)-66 b(quation)433 b Fo(n)24753 35176 y Fn(0)25433 35658 y Fq(=)369 b Fo(a)435 b Fj(has)g(any)g(solution)g(in)f(natur)-66 b(al)434 b(numb)-66 b(ers,)440 b(then)434 b(it)800 37263 y(has)466 b(only)e(\257nitely)g(many)g(solutions)h(if)f Fo(a)369 b(>)g Fq(1)p Fj(.)800 40307 y(Pr)-66 b(o)g(of.)591 b Fq(The)416 b(n)-36 b(um)g(b)36 b(er)415 b Fo(n)h Fq(cannot)g(b)36 b(e)416 b(a)h(prime.)572 b(According)416 b(to)g(\()p 0 1 0 0 TeXcolorcmyk(3)p (#equation.3) [[368 354 374 366] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))g(the)g(solutions)h(m)-36 b(ust)415 b(b)36 b(e)416 b(no)g(greater)800 41912 y(than)3968 41389 y Fm(a)4468 41077 y Fd(2)p 3968 41607 962 54 v 4213 42370 a Fl(4)5062 41912 y Fo(:)p 5856 41912 630 630 v 2751 44182 a Fq(What)434 b(ab)36 b(out)433 b(the)g(existence)h(of)g(solutions?)579 b(W)-108 b(e)434 b(start)f(with)h(the)f(ev)-36 b(en)433 b(n)-36 b(um)g(b)36 b(ers.)p Black 800 47226 a Fr(Conjecture)499 b(5)p Black 651 w Fj(The)653 b(di\256er)-66 b(ential)652 b(e)-66 b(quation)652 b Fo(n)25597 46744 y Fn(0)26627 47226 y Fq(=)718 b(2)p Fo(b)654 b Fj(has)h(a)e(p)-66 b(ositive)653 b(inte)-66 b(ger)651 b(solution)j(for)f(any)800 48831 y(natur)-66 b(al)465 b(numb)-66 b(er)463 b Fo(b)369 b(>)g Fq(1)p Fo(:)2751 51875 y Fq(A)434 b(motiv)-72 b(ation)434 b(for)g(this)g(is)f(the)g(famous)p Black 800 54919 a Fr(Conjecture)499 b(6)p Black 651 w(\(Goldbac)-42 b(h)460 b(Conjecture\))434 b Fj(Every)f(even)f(numb)-66 b(er)432 b(lar)-66 b(ger)432 b(than)h Fq(3)g Fj(is)g(a)g(sum)h(of)f(two)800 56524 y(primes.)2751 59568 y Fq(So,)i(if)g(2)p Fo(b)371 b Fq(=)f Fo(p)295 b Fq(+)g Fo(q)48 b(;)435 b Fq(then)e Fo(n)371 b Fq(=)f Fo(pq)481 b Fq(is)435 b(a)g(solution)f(that)g(w)-36 b(e)435 b(need.)580 b(Inequalit)-36 b(y)435 b(\()p 0 1 0 0 TeXcolorcmyk(3)p (#equation.3) [[452 181 458 193] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))g(helps)f(us)g(easy)h(to)800 61173 y(pro)-36 b(v)g(e)484 b(that)f(the)g(equation)i Fo(n)15651 60691 y Fn(0)16416 61173 y Fq(=)454 b(2)484 b(has)g(no)g(solutions.)729 b(What)484 b(ab)36 b(out)483 b(o)36 b(dd)484 b(n)-36 b(um)g(b)36 b(ers)482 b(larger)i(than)g(1?)800 62778 y(It)431 b(is)h(easy)g(to)g(c)-36 b(hec)g(k)431 b(with)h(the)f(help)g (of)h(\()p 0 1 0 0 TeXcolorcmyk(3)p (#equation.3) [[265 152 271 164] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))f(that)g(the)g(equation)h Fo(n)34187 62296 y Fn(0)34867 62778 y Fq(=)368 b(3)432 b(has)f(no)g(solutions.)578 b(F)-108 b(or)431 b Fo(a)369 b Fq(=)g(5)800 64384 y(w)-36 b(e)434 b(ha)-36 b(v)g(e)434 b(one)f(solution)h(and)f(more)h(general)g (ha)-36 b(v)g(e)434 b(a)g(theorem:)p Black 800 67428 a Fr(Theorem)499 b(10)p Black 651 w Fj(L)-66 b(et)408 b Fo(p)h Fj(b)-66 b(e)409 b(a)g(prime)f(and)h Fo(a)369 b Fq(=)g Fo(p)171 b Fq(+)g(2)p Fj(.)579 b(Then)409 b Fq(2)p Fo(p)g Fj(is)g(a)h(solution)f(for)g(the)g(e)-66 b(quation)407 b Fo(n)49695 66946 y Fn(0)50375 67428 y Fq(=)369 b Fo(a:)p Black 26475 74617 a Fq(6)p Black eop %%Page: 7 7 7 6 bop Black 0 TeXcolorgray Black Black 800 1424 a Fj(Pr)-66 b(o)g(of.)597 b Fq(\(2)p Fo(p)p Fq(\))7135 942 y Fn(0)7814 1424 y Fq(=)368 b(2)9844 942 y Fn(0)10155 1424 y Fo(p)295 b Fq(+)g(2)p Fo(p)13713 942 y Fn(0)14393 1424 y Fq(=)368 b Fo(p)295 b Fq(+)g(2)p Fo(:)p 19473 1424 630 630 v 2751 3665 a Fq(Some)397 b(other)f(primes)h(also)h(can)f(b)36 b(e)397 b(obtained)f(as)i(a)f(deriv)-72 b(ativ)-36 b(e)398 b(of)f(a)h(natural)e(n)-36 b(um)g(b)36 b(er)396 b(\(e.g.)566 b(7\),)405 b(but)800 5270 y(it)460 b(is)g(more)g(in)-36 b(teresting)460 b(whic)-36 b(h)460 b(of)g(n)-36 b(um)g(b)36 b(ers)459 b(cannot.)657 b(Here)460 b(is)g(a)g(list)h(of)g(all)f Fo(a)414 b Fk(\267)g Fq(1000)461 b(for)g(whic)-36 b(h)460 b(the)800 6875 y(equation)434 b Fo(n)6887 6393 y Fn(0)7567 6875 y Fq(=)369 b Fo(a)433 b Fq(has)g(no)h(solutions)g(\(obtained)f (using)g(Maple)h(and)f(\()p 0 1 0 0 TeXcolorcmyk(3)p (#equation.3) [[395 655 401 667] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\)\):)5544 9708 y(2)p Fo(;)221 b Fq(3)p Fo(;)g Fq(11)p Fo(;)g Fq(17)p Fo(;)g Fq(23)p Fo(;)g Fq(29)p Fo(;)g Fq(35)p Fo(;)g Fq(37)p Fo(;)g Fq(47)p Fo(;)g Fq(53)p Fo(;)g Fq(57)p Fo(;)g Fq(65)p Fo(;)g Fq(67)p Fo(;)g Fq(79)p Fo(;)g Fq(83)p Fo(;)g Fq(89)p Fo(;)g Fq(93)p Fo(;)g Fq(97)p Fo(;)g Fq(107)p Fo(;)g Fq(117)p Fo(;)h Fq(125)q Fo(;)f Fq(1)q(27)p Fo(;)5374 12541 y Fq(137)p Fo(;)g Fq(145)p Fo(;)g Fq(149)p Fo(;)g Fq(157)p Fo(;)g Fq(163)p Fo(;)g Fq(173)p Fo(;)g Fq(177)p Fo(;)g Fq(179)p Fo(;)g Fq(189)p Fo(;)g Fq(197)p Fo(;)g Fq(205)p Fo(;)g Fq(207)p Fo(;)g Fq(209)p Fo(;)g Fq(217)p Fo(;)g Fq(219)p Fo(;)g Fq(223)q Fo(;)g Fq(2)q(33)p Fo(;)5374 14823 y Fq(237)p Fo(;)g Fq(245)p Fo(;)g Fq(257)p Fo(;)g Fq(261)p Fo(;)g Fq(277)p Fo(;)g Fq(289)p Fo(;)g Fq(303)p Fo(;)g Fq(305)p Fo(;)g Fq(307)p Fo(;)g Fq(317)p Fo(;)g Fq(323)p Fo(;)g Fq(325)p Fo(;)g Fq(337)p Fo(;)g Fq(345)p Fo(;)g Fq(353)p Fo(;)g Fq(367)q Fo(;)g Fq(3)q(73)p Fo(;)5374 17105 y Fq(377)p Fo(;)g Fq(379)p Fo(;)g Fq(387)p Fo(;)g Fq(389)p Fo(;)g Fq(393)p Fo(;)g Fq(397)p Fo(;)g Fq(409)p Fo(;)g Fq(413)p Fo(;)g Fq(415)p Fo(;)g Fq(427)p Fo(;)g Fq(429)p Fo(;)g Fq(443)p Fo(;)g Fq(449)p Fo(;)g Fq(453)p Fo(;)g Fq(457)p Fo(;)g Fq(473)q Fo(;)g Fq(4)q(77)p Fo(;)5374 19386 y Fq(485)p Fo(;)g Fq(497)p Fo(;)g Fq(499)p Fo(;)g Fq(509)p Fo(;)g Fq(513)p Fo(;)g Fq(515)p Fo(;)g Fq(517)p Fo(;)g Fq(529)p Fo(;)g Fq(531)p Fo(;)g Fq(533)p Fo(;)g Fq(537)p Fo(;)g Fq(547)p Fo(;)g Fq(553)p Fo(;)g Fq(561)p Fo(;)g Fq(569)p Fo(;)g Fq(577)q Fo(;)g Fq(5)q(93)p Fo(;)5374 21668 y Fq(597)p Fo(;)g Fq(605)p Fo(;)g Fq(613)p Fo(;)g Fq(625)p Fo(;)g Fq(629)p Fo(;)g Fq(639)p Fo(;)g Fq(657)p Fo(;)g Fq(659)p Fo(;)g Fq(665)p Fo(;)g Fq(673)p Fo(;)g Fq(677)p Fo(;)g Fq(681)p Fo(;)g Fq(683)p Fo(;)g Fq(697)p Fo(;)g Fq(699)p Fo(;)g Fq(709)q Fo(;)g Fq(7)q(13)p Fo(;)5374 23950 y Fq(715)p Fo(;)g Fq(733)p Fo(;)g Fq(747)p Fo(;)g Fq(749)p Fo(;)g Fq(757)p Fo(;)g Fq(765)p Fo(;)g Fq(769)p Fo(;)g Fq(777)p Fo(;)g Fq(781)p Fo(;)g Fq(783)p Fo(;)g Fq(785)p Fo(;)g Fq(787)p Fo(;)g Fq(793)p Fo(;)g Fq(797)p Fo(;)g Fq(805)p Fo(;)g Fq(809)q Fo(;)g Fq(8)q(17)p Fo(;)5374 26231 y Fq(819)p Fo(;)g Fq(827)p Fo(;)g Fq(833)p Fo(;)g Fq(835)p Fo(;)g Fq(845)p Fo(;)g Fq(847)p Fo(;)g Fq(849)p Fo(;)g Fq(853)p Fo(;)g Fq(857)p Fo(;)g Fq(869)p Fo(;)g Fq(873)p Fo(;)g Fq(877)p Fo(;)g Fq(881)p Fo(;)g Fq(891)p Fo(;)g Fq(895)p Fo(;)g Fq(897)q Fo(;)g Fq(9)q(07)p Fo(;)11709 28513 y Fq(917)p Fo(;)g Fq(925)p Fo(;)g Fq(933)p Fo(;)g Fq(937)p Fo(;)g Fq(947)p Fo(;)g Fq(953)p Fo(;)g Fq(963)p Fo(;)g Fq(965)p Fo(;)g Fq(967)p Fo(;)g Fq(981)p Fo(;)g Fq(989)p Fo(;)g Fq(997)p Fo(:)800 30795 y Fq(Note)498 b(that)g(a)g(large)h(p)36 b(ortion)498 b(of)h(them)f(\(69)g(from)h (153\))g(are)f(primes,)514 b(one)498 b(of)h(them)f(\(529)479 b(=)g(23)49247 30313 y Fl(2)49773 30795 y Fq(\))498 b(is)h(a)800 32400 y(square,)392 b(and)381 b(some)h(of)f(them)g(\(e.g.)561 b(765)370 b(=)f(3)22961 31918 y Fl(2)23675 32400 y Fk(\242)188 b Fq(5)g Fk(\242)g Fq(17\))383 b(ha)-36 b(v)g(e)381 b(at)h(least)g(4)f (prime)g(factors.)562 b(In)381 b(general)g(it)h(is)800 34005 y(in)-36 b(teresting)370 b(to)g(in)-36 b(v)g(estigate)372 b(the)d(b)36 b(eha)-36 b(vior)371 b(of)g(the)f(\\in)-36 b(tegrating")371 b(function)f Fo(I)104 b Fq(\()p Fo(a)p Fq(\))370 b(whic)-36 b(h)370 b(calculates)h(for)800 35610 y(ev)-36 b(ery)393 b Fo(a)f Fq(the)g(set)h(of)g(solutions)g(of)g(the)f (equation)g Fo(n)25972 35128 y Fn(0)26652 35610 y Fq(=)369 b Fo(a)392 b Fq(and)g(its)h(w)-36 b(eak)g(er)393 b(v)-72 b(arian)-36 b(t)393 b Fo(i)p Fq(\()p Fo(a)p Fq(\))e(that)h(calculates) 800 37215 y(the)372 b(n)-36 b(um)g(b)36 b(er)371 b(of)j(suc)-36 b(h)371 b(solutions.)559 b(As)372 b(w)-36 b(e)373 b(ha)-36 b(v)g(e)373 b(seen)f(ab)36 b(o)-36 b(v)g(e)374 b Fo(I)104 b Fq(\(0\))369 b(=)f Fk(f)p Fq(0)p Fo(;)221 b Fq(1)p Fk(g)p Fo(;)375 b(I)104 b Fq(\(1\))373 b(consist)f(of)i(all)f(primes) 800 38820 y(and)437 b Fo(i)p Fq(\(2\))375 b(=)h Fo(i)p Fq(\(3\))f(=)g Fo(i)p Fq(\(11\))h(=)f Fk(\242)221 b(\242)g(\242)377 b Fq(=)e Fo(i)p Fq(\(997\))h(=)f(0)p Fo(:)438 b Fq(Here)g(is)g(a)f (list)h(of)h(the)e(those)g(n)-36 b(um)g(b)36 b(ers)436 b Fo(a)375 b Fk(\267)h Fq(100)438 b(that)800 40425 y(ha)-36 b(v)g(e)468 b(more)f(than)f(one)i(\\in)-36 b(tegral")468 b(\(i.e.)680 b Fo(i)p Fq(\()p Fo(a)p Fq(\))426 b Fk(\270)g Fq(2\))p Fo(:)468 b Fq(F)-108 b(or)467 b(example)h(10)f(has)h(t)-36 b(w)g(o)467 b(\\in)-36 b(tegrals")469 b(\(namely)800 42030 y Fo(I)104 b Fq(\(10\))369 b(=)g Fk(f)p Fq(21)p Fo(;)221 b Fq(25)p Fk(g)p Fq(\))436 b(and)d(100)h(has)g(six)g(\()p Fo(I)104 b Fq(\(100\))370 b(=)e Fk(f)p Fq(291)p Fo(;)221 b Fq(979)p Fo(;)g Fq(1411)p Fo(;)g Fq(2059)p Fo(;)g Fq(2419)p Fo(;)g Fq(2491)p Fk(g)p Fq(\).)p Black Black 2167 44960 a Fc([10,)682 b(2],)g([12,)g(2],)g([14,)g(2],)g([16,)g(3],)g([18,)g (2],)g([20,)g(2],)2167 46565 y([21,)g(2],)g([22,)g(3],)g([24,)g(4],)g ([26,)g(3],)g([28,)g(2],)g([30,)g(3],)2167 48170 y([31,)g(2],)g([32,)g (4],)g([34,)g(4],)g([36,)g(4],)g([38,)g(2],)g([39,)g(2],)2167 49775 y([40,)g(3],)g([42,)g(4],)g([44,)g(4],)g([45,)g(2],)g([46,)g(4],) g([48,)g(6],)2167 51380 y([50,)g(4],)g([52,)g(3],)g([54,)g(5],)g([55,)g (2],)g([56,)g(4],)g([58,)g(4],)2167 52985 y([60,)g(7],)g([61,)g(2],)g ([62,)g(3],)g([64,)g(5],)g([66,)g(6],)g([68,)g(3],)2167 54591 y([70,)g(5],)g([71,)g(2],)g([72,)g(7],)g([74,)g(5],)g([75,)g(3],) g([76,)g(5],)2167 56196 y([78,)g(7],)g([80,)g(6],)g([81,)g(2],)g([82,)g (5],)g([84,)g(8],)g([86,)g(5],)2167 57801 y([87,)g(2],)g([88,)g(4],)g ([90,)g(9],)g([91,)g(3],)g([92,)g(6],)g([94,)g(5],)2167 59406 y([96,)g(8],)g([98,)g(3],)g([100,)f(6].)800 63940 y Fq(Note)454 b(that)f(only)h(three)f(of)h(them)f(are)h(primes.)637 b(T)-108 b(o)454 b(complete)g(the)f(picture)g(it)g(remains)h(to)f(list) h(the)f(set)800 65546 y(of)434 b(those)g Fo(a)369 b(<)p Fq(=)f(100)434 b(for)g(whic)-36 b(h)434 b Fo(i)p Fq(\()p Fo(a)p Fq(\))368 b(=)g(1)p Fo(:)16620 68756 y Fq(4)p Fo(;)221 b Fq(5)p Fo(;)g Fq(6)p Fo(;)g Fq(7)p Fo(;)g Fq(8)p Fo(;)g Fq(9)p Fo(;)g Fq(13)p Fo(;)g Fq(15)p Fo(;)g Fq(19)p Fo(;)g Fq(25)p Fo(;)g Fq(27)p Fo(;)g Fq(33)p Fo(;)g Fq(41)p Fo(;)16552 71037 y Fq(43)p Fo(;)g Fq(49)p Fo(;)g Fq(51)p Fo(;)g Fq(59)p Fo(;)g Fq(63)p Fo(;)g Fq(69)p Fo(;)g Fq(73)p Fo(;)g Fq(77)p Fo(;)g Fq(85)p Fo(;)g Fq(95)p Fo(;)g Fq(99)p Fo(:)p Black 26475 74617 a Fq(7)p Black eop %%Page: 8 8 8 7 bop Black 0 TeXcolorgray Black Black Black 800 1424 a Fr(Theorem)499 b(11)p Black 651 w Fj(The)465 b(function)e Fo(i)p Fq(\()p Fo(n)p Fq(\))i Fj(is)f(unb)-66 b(ounde)g(d)465 b(for)f Fo(n)370 b(>)e Fq(1)p Fo(:)800 4468 y Fj(Pr)-66 b(o)g(of.)597 b Fq(Supp)36 b(ose)432 b(that)h Fo(i)p Fq(\()p Fo(n)p Fq(\))369 b Fo(<)g(C)528 b Fq(for)434 b(all)h Fo(n)369 b(>)g Fq(1)434 b(for)g(some)g(constan)-36 b(t)433 b Fo(C)23 b(:)433 b Fq(Then)22979 6943 y Fl(2)p Fm(n)22541 7341 y Fi(X)22642 10170 y Fm(k)24 b Fl(=2)24681 8603 y Fo(i)p Fq(\()p Fo(k)45 b Fq(\))368 b Fo(<)h Fq(2)p Fo(C)95 b(n)800 12738 y Fq(for)434 b(an)-36 b(y)434 b Fo(n:)g Fq(But)f(for)h(an)-36 b(y)434 b(t)-36 b(w)g(o)434 b(primes)f Fo(p;)221 b(q)482 b Fq(the)433 b(pro)36 b(duct)432 b Fo(pq)481 b Fq(b)36 b(elongs)434 b(to)g Fo(I)104 b Fq(\()p Fo(p)294 b Fq(+)h Fo(q)48 b Fq(\))433 b(th)-36 b(us)14294 15213 y Fl(2)p Fm(n)13855 15612 y Fi(X)13957 18441 y Fm(k)24 b Fl(=2)15996 16874 y Fo(i)p Fq(\()p Fo(k)45 b Fq(\))368 b Fo(>)20442 15612 y Fi(X)19922 18418 y Fm(p)p Fn(\267)p Fm(q)32 b Fn(\267)p Fm(n)22882 15829 y Fn(0)23192 16874 y Fq(1)370 b(=)25725 15975 y Fo(\274)48 b Fq(\()p Fo(n)p Fq(\)\()p Fo(\274)g Fq(\()p Fo(n)p Fq(\))294 b(+)h(1\))p 25725 16568 8411 54 v 29605 17785 a(2)34638 16874 y Fo(>)36151 15975 y(\274)48 b Fq(\()p Fo(n)p Fq(\))38725 15493 y Fl(2)p 36151 16568 3100 54 v 37376 17785 a Fq(2)39383 16874 y Fo(;)800 21299 y Fq(where)4584 20303 y Fi(P)5986 20654 y Fn(0)6756 21299 y Fq(means)459 b(that)h(the)f(sum)g(runs)f(o) -36 b(v)g(er)460 b(the)f(primes,)466 b(and)460 b Fo(\274)48 b Fq(\()p Fo(n)p Fq(\))459 b(is)h(the)f(n)-36 b(um)g(b)36 b(er)458 b(of)i(primes)f(not)800 22905 y(exceeding)434 b Fo(n:)g Fq(This)g(leads)g(to)g(the)f(inequalit)-36 b(y)17861 26553 y(2)p Fo(C)95 b(n)370 b(>)22196 25655 y(\274)48 b Fq(\()p Fo(n)p Fq(\))24770 25173 y Fl(2)p 22196 26248 V 23421 27464 a Fq(2)25797 26553 y Fk(\))370 b Fo(\274)48 b Fq(\()p Fo(n)p Fq(\))368 b Fo(<)h Fq(2)32468 25364 y Fk(p)p 33575 25364 1803 54 v 1189 x Fo(C)95 b(n)q(;)800 29900 y Fq(whic)-36 b(h)433 b(con)-36 b(tradicts)434 b(the)f(kno)-36 b(wn)433 b(asymptotic)i(b)36 b(eha)-36 b(vior)434 b Fo(\274)48 b Fq(\()p Fo(n)p Fq(\))368 b Fk(\274)34522 29377 y Fm(n)p 34051 29594 1512 54 v 34051 30358 a Fl(ln)157 b Fm(n)35696 29900 y Fo(:)p 36491 29900 630 630 v 2751 32169 a Fq(It)434 b(w)-36 b(ould)433 b(b)36 b(e)434 b(in)-36 b(teresting)433 b(to)g(pro)-36 b(v)g(e)434 b(a)g(stronger)f(result.)p Black 800 34881 a Fr(Conjecture)499 b(7)p Black 651 w Fj(F)-100 b(or)465 b(any)g(nonne)-66 b(gative)462 b Fo(m)i Fj(ther)-66 b(e)464 b(exists)h(in\257nitely)d (many)i Fo(a)h Fj(such)h(that)e Fo(i)p Fq(\()p Fo(a)p Fq(\))368 b(=)h Fo(m:)2751 37593 y Fq(Another)433 b(related)g (conjecture)h(is)g(the)f(follo)-36 b(wing:)p Black 800 40305 a Fr(Conjecture)499 b(8)p Black 651 w Fj(Ther)-66 b(e)535 b(exists)h(an)g(in\257nite)e(se)-66 b(quenc)g(e)534 b Fo(a)29811 40504 y Fm(n)30973 40305 y Fj(of)i(di\256er)-66 b(ent)535 b(natur)-66 b(al)536 b(numb)-66 b(ers)535 b(such)i(that)800 41910 y Fo(a)1483 42109 y Fl(1)2378 41910 y Fq(=)368 b(1)p Fo(;)221 b Fq(\()p Fo(a)6179 42109 y Fm(n)6806 41910 y Fq(\))7312 41428 y Fn(0)7991 41910 y Fq(=)369 b Fo(a)10055 42109 y Fm(n)p Fn(\241)p Fl(1)12348 41910 y Fj(for)465 b Fo(n)369 b Fq(=)g(2)p Fo(;)221 b Fq(3)g Fo(:)g(:)g(:)2751 44622 y Fq(Here)434 b(is)f(an)h(example)g(of)h(p)36 b(ossible)433 b(b)36 b(eginning)434 b(of)g(suc)-36 b(h)433 b(a)h(sequence:)9517 47556 y(1)369 b Fk(\303)h Fq(7)f Fk(\303)g Fq(10)h Fk(\303)f Fq(25)h Fk(\303)f Fq(46)h Fk(\303)f Fq(129)h Fk(\303)g Fq(170)g Fk(\303)f Fq(501)h Fk(\303)f Fq(414)h Fk(\303)f Fq(2045)p Fo(:)2751 50489 y Fq(The)598 b(follo)-36 b(wing)600 b(table)d(sho)-36 b(ws)598 b(the)g(maxim)-36 b(um)597 b(of)i Fo(i)p Fq(\()p Fo(n)p Fq(\))e(dep)36 b(ending)597 b(of)h(the)g(n)-36 b(um)g(b)36 b(er)596 b Fo(m)h Fq(of)h(\(not)800 52094 y(necessary)434 b(di\256eren)-36 b(t\))432 b(prime)i(factors)g(in)f (the)g(factorization)i(of)g Fo(n)f Fq(for)g Fo(n)369 b Fk(\267)g Fq(1000)p Fo(:)p 14622 53910 24357 45 v 14600 55515 45 1606 v 15722 55033 a(m)p 17938 55515 V 1654 w Fq(1)1432 b(2)1758 b(3)f(4)h(5)g(6)f(7)h(8)g(9)p 38956 55515 V 14622 55559 24357 45 v 14600 57164 45 1606 v 15175 56683 a Fo(i)p Fq(\()p Fo(n)p Fq(\))p 17938 57164 V 1107 w(8)1107 b(22)h(35)f(46)h(52)g(52)f(40)h(47)f(32)p 38956 57164 V 14622 57209 24357 45 v 2751 59644 a(The)464 b(next)g(more)g(detailed)g(picture)f(sho)-36 b(ws)464 b(the)f(distribution)g(of)h Fo(i)p Fq(\()p Fo(n)p Fq(\))g(dep)36 b(ending)463 b(of)h(the)g(n)-36 b(um)g(b)36 b(er)800 61249 y Fo(m)513 b Fq(for)h Fo(i)p Fq(\()p Fo(n)p Fq(\))504 b Fo(<)h Fq(33)p Fo(:)514 b Fq(Note)f(that)g(maxim)-36 b(um)514 b(p)36 b(ossible)513 b Fo(i)p Fq(\()p Fo(n)p Fq(\))g(is)h(equal)g(52)p Fo(;)g Fq(so)g(w)-36 b(e)513 b(ha)-36 b(v)g(e)514 b(only)g(part)f(of)h(a)800 62854 y(p)36 b(ossible)434 b(table.)578 b(W)-108 b(e)434 b(lea)-36 b(v)g(e)435 b(to)e(the)g(reader)h(the)f(pleasure)g(of)h(making)h(some)f (natural)f(conjectures.)p Black 26475 74617 a(8)p Black eop %%Page: 9 9 9 8 bop Black 0 TeXcolorgray Black Black 15346 2428 22908 45 v 15324 4033 45 1606 v 15900 3551 a Fo(i)p Fq(\()p Fo(n)p Fq(\))p Fk(n)p Fo(m)p 20465 4033 V 1432 w Fq(1)1757 b(2)h(3)1432 b(4)1108 b(5)f(6)g(7)h(8)f(9)p 38232 4033 V 15346 4077 22908 45 v 15324 5682 45 1606 v 17592 5201 a(0)p 20465 5682 V 2799 w(69)g(49)h(28)f(6)h(1)f(0)g(0)h(0)f(0)p 38232 5682 V 15324 7287 V 17592 6806 a(1)p 20465 7287 V 2799 w(46)g(89)h(35)f(8)h(3)f(1)g(0)h(0)f(0)p 38232 7287 V 15324 8892 V 17592 8411 a(2)p 20465 8892 V 2799 w(25)g(44)h(18)f(7)h(1)f(0)g(0)h(0)f(0)p 38232 8892 V 15324 10497 V 17592 10016 a(3)p 20465 10497 V 2799 w(13)g(16)h(17)f(7)h (0)f(0)g(0)h(0)f(0)p 38232 10497 V 15324 12103 V 17592 11621 a(4)p 20465 12103 V 3124 w(9)1432 b(12)h(8)f(5)1108 b(2)f(0)g(1)h(0)f(0)p 38232 12103 V 15324 13708 V 17592 13226 a(5)p 20465 13708 V 3124 w(2)1757 b(6)h(3)1432 b(4)1108 b(0)f(1)g(0)h(0)f(0)p 38232 13708 V 15324 15313 V 17592 14831 a(6)p 20465 15313 V 3124 w(1)1757 b(7)h(8)1432 b(1)1108 b(2)f(0)g(0)h(0)f(0)p 38232 15313 V 15324 16918 V 17592 16436 a(7)p 20465 16918 V 3124 w(1)1432 b(10)h(4)f(3)1108 b(2)f(1)g(0)h(0)f(0)p 38232 16918 V 15324 18523 V 17592 18041 a(8)p 20465 18523 V 3124 w(2)1757 b(3)h(8)1432 b(3)1108 b(2)f(2)g(0)h(1)f(0)p 38232 18523 V 15324 20128 V 17592 19647 a(9)p 20465 20128 V 3124 w(0)1757 b(8)h(6)1432 b(7)1108 b(4)f(0)g(0)h(0)f(0)p 38232 20128 V 15324 21733 V 17266 21252 a(10)p 20465 21733 V 2800 w(0)1757 b(3)h(7)1432 b(5)1108 b(1)f(1)g(0)h(0)f(0)p 38232 21733 V 15324 23338 V 17266 22857 a(11)p 20465 23338 V 2800 w(0)1757 b(8)1433 b(13)1107 b(2)h(1)f(2)g(0)h(0)f(0)p 38232 23338 V 15324 24943 V 17266 24462 a(12)p 20465 24943 V 2800 w(0)1757 b(4)h(4)1432 b(5)1108 b(2)f(0)g(1)h(0)f(1)p 38232 24943 V 15324 26548 V 17266 26067 a(13)p 20465 26548 V 2800 w(0)1757 b(3)1433 b(10)1107 b(5)h(2)f(2)g(1)h(0)f(0)p 38232 26548 V 15324 28153 V 17266 27672 a(14)p 20465 28153 V 2800 w(0)1757 b(7)h(7)1432 b(5)1108 b(4)f(1)g(1)h(0)f(0)p 38232 28153 V 15324 29759 V 17266 29277 a(15)p 20465 29759 V 2800 w(0)1757 b(8)h(8)1432 b(3)1108 b(3)f(1)g(0)h(0)f(0)p 38232 29759 V 15324 31364 V 17266 30882 a(16)p 20465 31364 V 2800 w(0)1757 b(1)1433 b(15)1107 b(6)h(5)f(1)g(0)h(0)f(0)p 38232 31364 V 15324 32969 V 17266 32487 a(17)p 20465 32969 V 2800 w(0)1432 b(10)h(4)f(8)1108 b(2)f(0)g(0)h(0)f(0)p 38232 32969 V 15324 34574 V 17266 34092 a(18)p 20465 34574 V 2800 w(0)1757 b(3)h(4)1432 b(5)1108 b(2)f(1)g(1)h(0)f(0)p 38232 34574 V 15324 36179 V 17266 35697 a(19)p 20465 36179 V 2800 w(0)1757 b(4)h(5)1432 b(9)1108 b(4)f(2)g(1)h(1)f(0)p 38232 36179 V 15324 37784 V 17266 37303 a(20)p 20465 37784 V 2800 w(0)1757 b(3)h(7)1432 b(1)1108 b(0)f(1)g(0)h(1)f(0)p 38232 37784 V 15324 39389 V 17266 38908 a(21)p 20465 39389 V 2800 w(0)1757 b(0)h(5)1432 b(2)1108 b(4)f(3)g(0)h(1)f(0)p 38232 39389 V 15324 40994 V 17266 40513 a(22)p 20465 40994 V 2800 w(0)1757 b(1)h(2)1432 b(5)1108 b(1)f(0)g(1)h(0)f(0)p 38232 40994 V 15324 42599 V 17266 42118 a(23)p 20465 42599 V 2800 w(0)1757 b(0)h(4)1432 b(1)1108 b(1)f(1)g(2)h(0)f(0)p 38232 42599 V 15324 44204 V 17266 43723 a(24)p 20465 44204 V 2800 w(0)1757 b(0)h(1)1432 b(6)1108 b(3)f(1)g(0)h(0)f(0)p 38232 44204 V 15324 45809 V 17266 45328 a(25)p 20465 45809 V 2800 w(0)1757 b(0)h(3)1432 b(2)1108 b(1)f(1)g(0)h(0)f(0)p 38232 45809 V 15324 47415 V 17266 46933 a(26)p 20465 47415 V 2800 w(0)1757 b(0)h(1)1432 b(2)1108 b(4)f(1)g(0)h(1)f(0)p 38232 47415 V 15324 49020 V 17266 48538 a(27)p 20465 49020 V 2800 w(0)1757 b(0)h(2)1432 b(1)1108 b(2)f(1)g(0)h(0)f(0)p 38232 49020 V 15324 50625 V 17266 50143 a(28)p 20465 50625 V 2800 w(0)1757 b(0)h(1)1432 b(1)1108 b(0)f(1)g(1)h(0)f(0)p 38232 50625 V 15324 52230 V 17266 51748 a(29)p 20465 52230 V 2800 w(0)1757 b(0)h(2)1432 b(2)1108 b(1)f(0)g(1)h(0)f(0)p 38232 52230 V 15324 53835 V 17266 53353 a(30)p 20465 53835 V 2800 w(0)1757 b(0)h(1)1432 b(1)1108 b(1)f(0)g(0)h(0)f(0)p 38232 53835 V 15324 55440 V 17266 54958 a(31)p 20465 55440 V 2800 w(0)1757 b(0)h(1)1432 b(4)1108 b(3)f(0)g(0)h(0)f(0)p 38232 55440 V 15324 57045 V 17266 56564 a(32)p 20465 57045 V 2800 w(0)1757 b(0)h(0)1432 b(6)1108 b(1)f(1)g(1)h(0)f(1)p 38232 57045 V 15346 57089 22908 45 v 800 60563 a Fp(4)2152 b(The)716 b(equation)g Fg(n)18320 59869 y Fk(00)19637 60563 y Ff(=)531 b(1)800 63484 y Fq(The)434 b(main)f(conjecture)h(for)g (the)f(second-order)f(equations)i(is)g(the)f(follo)-36 b(wing:)p Black 800 65785 a Fr(Conjecture)499 b(9)p Black 651 w Fj(The)620 b(di\256er)-66 b(ential)618 b(e)-66 b(quation)618 b Fo(n)25496 65303 y Fn(00)26719 65785 y Fq(=)656 b(1)621 b Fj(has)f(in\257nitely)e(many)i(solutions)g(in)f (natur)-66 b(al)800 67390 y(numb)g(ers.)800 69691 y Fq(Theorem)p 0 1 0 0 TeXcolorcmyk 596 w(10)p (#Thm.10) [[130 90 142 102] [1 1 1 [3 3]] [0 0 1]] pdfm Black 596 w(sho)-36 b(ws)596 b(that)f(2)p Fo(p)h Fq(is)g(a)g(solution)g (if)h Fo(p;)221 b(p)405 b Fq(+)h(2)596 b(are)f(primes.)1065 b(So)595 b(the)h(follo)-36 b(wing)597 b(famous)800 71296 y(conjecture)433 b(w)-36 b(ould)434 b(b)36 b(e)433 b(su\261cien)-36 b(t)433 b(to)h(pro)-36 b(v)g(e.)p Black 26475 74617 a(9)p Black eop %%Page: 10 10 10 9 bop Black 0 TeXcolorgray Black Black Black 800 1424 a Fr(Conjecture)499 b(10)p Black 651 w Fj(\()p Fr(prime)369 b(t)-42 b(wins)p Fj(\))363 b(Ther)-66 b(e)360 b(exists)g(in\257nitely)f (many)h(p)-66 b(airs)361 b Fo(p;)221 b(p)64 b Fq(+)g(2)361 b Fj(of)f(prime)g(numb)-66 b(ers.)800 4136 y Fq(The)434 b(follo)-36 b(wing)435 b(problem)e(is)h(another)f(alternativ)-36 b(e)435 b(whic)-36 b(h)433 b(w)-36 b(ould)434 b(b)36 b(e)433 b(su\261cien)-36 b(t:)p Black 800 7180 a Fr(Conjecture)499 b(11)p Black 651 w Fj(\()p Fr(prime)530 b(triples)p Fj(\))492 b(Ther)-66 b(e)489 b(exists)g(in\257nitely)e(many)j(triples)f Fo(p;)221 b(q)48 b(;)221 b(r)527 b Fj(of)489 b(prime)g(num-)800 8785 y(b)-66 b(ers)464 b(such)i(that)f Fo(P)550 b Fq(=)368 b Fo(pq)343 b Fq(+)294 b Fo(pr)332 b Fq(+)295 b Fo(q)48 b(r)500 b Fj(is)465 b(a)g(prime.)800 11830 y Fq(Suc)-36 b(h)526 b(a)i(triple)g(giv)-36 b(es)529 b(a)f(solution)g Fo(n)h Fq(=)g Fo(pq)48 b(r)563 b Fq(to)528 b(our)g(equation,)552 b(b)36 b(ecause)527 b Fo(n)39088 11347 y Fn(0)39928 11830 y Fq(=)i Fo(P)36 b(:)529 b Fq(In)e(realit)-36 b(y)529 b(all)g(the)800 13435 y(solutions)434 b(can)f(b)36 b(e)434 b(describ)36 b(ed)433 b(as)h(follo)-36 b(ws.)p Black 800 16147 a Fr(Theorem)499 b(12)p Black 651 w Fj(A)437 b(numb)-66 b(er)435 b Fo(n)j Fj(is)e(a)i(solution)f(of)f(the)h (di\256er)-66 b(ential)435 b(e)-66 b(quation)436 b Fo(n)39945 15665 y Fn(00)40880 16147 y Fq(=)368 b(1)438 b Fj(if)e(and)h(only)f(if) g(the)800 17752 y(thr)-66 b(e)g(e)464 b(fol)66 b(lowing)464 b(c)-66 b(onditions)463 b(ar)-66 b(e)465 b(valid:)p Black 2338 20464 a(1.)p Black 651 w(The)f(numb)-66 b(er)464 b Fo(n)h Fj(is)g(a)g(pr)-66 b(o)g(duct)465 b(of)f(di\256er)-66 b(ent)463 b(primes:)597 b Fo(n)370 b Fq(=)33722 19468 y Fi(Q)34977 19818 y Fm(k)34977 20851 y(i)p Fl(=1)36776 20464 y Fo(p)37429 20663 y Fm(i)37805 20464 y Fo(:)p Black 2338 23330 a Fj(2.)p Black 4052 22333 a Fi(P)5454 22684 y Fm(k)5454 23717 y(i)p Fl(=1)7253 23330 y Fq(1)p Fo(=p)9206 23529 y Fm(i)9951 23330 y Fq(=)11513 22746 y Fm(p)p 11465 23024 571 54 v 11465 23788 a(n)12169 23330 y Fo(;)465 b Fj(wher)-66 b(e)464 b Fo(p)h Fj(is)g(a)g(prime.)p Black 2338 26042 a(3.)p Black 651 w(If)f Fo(k)510 b Fj(is)465 b(even,)e(then)h(the)h(smal)66 b(lest)465 b(prime)f(of)h Fo(p)27363 26241 y Fm(i)28203 26042 y Fj(should)i(b)-66 b(e)464 b(e)-66 b(qual)464 b(to)h Fq(2)p Fo(:)800 28754 y Fj(Pr)-66 b(o)g(of.)730 b Fq(If)482 b Fo(n)452 b Fq(=)e Fo(p)9644 28272 y Fl(2)10170 28754 y Fo(m)482 b Fq(for)g(some)g(prime)g Fo(p)f Fq(then)g Fo(n)25787 28272 y Fn(0)26549 28754 y Fq(=)451 b Fo(p)p Fq(\(2)p Fo(m)328 b Fq(+)f Fo(pm)34417 28272 y Fn(0)34727 28754 y Fq(\))482 b(is)g(not)g(prime)f(and)h (according)g(to)800 30359 y(Theorem)p 0 1 0 0 TeXcolorcmyk 413 w(8)p (#Thm.8) [[128 444 134 456] [1 1 1 [3 3]] [0 0 1]] pdfm Black 413 w(the)413 b(n)-36 b(um)g(b)36 b(er)411 b Fo(n)j Fq(cannot)e(b)36 b(e)413 b(a)g(solution.)572 b(So,)417 b(it)c(is)h(a)f(pro)36 b(duct)412 b(of)h(di\256eren)-36 b(t)412 b(primes.)571 b(Then)800 31964 y(the)352 b(second)g(condition)g (means)g(that)g Fo(n)20116 31482 y Fn(0)20780 31964 y Fq(is)g(a)h(prime)f(and)g(b)-36 b(y)352 b(Theorem)p 0 1 0 0 TeXcolorcmyk 352 w(8)p (#Thm.8) [[398 429 404 441] [1 1 1 [3 3]] [0 0 1]] pdfm Black 353 w(it)g(is)h(necessary)g(and)f(su\261cien)-36 b(t)800 33569 y(to)486 b(b)36 b(e)486 b(a)h(solution.)736 b(As)487 b(to)f(the)g(n)-36 b(um)g(b)36 b(er)484 b Fo(k)532 b Fq(of)487 b(factors)g(it)f(cannot)g(b)36 b(e)486 b(ev)-36 b(en)486 b(if)h(all)g(primes)f Fo(p)47225 33768 y Fm(i)48087 33569 y Fq(are)g(o)36 b(dd,)800 35174 y(b)g(ecause)434 b Fo(n)6389 34692 y Fn(0)7133 35174 y Fq(in)f(this)h(case)g(is)g(\(as)f (the)h(sum)f(of)h Fo(k)478 b Fq(o)36 b(dd)434 b(n)-36 b(um)g(b)36 b(ers\))432 b(ev)-36 b(en)433 b(and)g(larger)h(than)f(t)-36 b(w)g(o.)p 48471 35174 630 630 v 800 40275 a Fp(5)2152 b(Deriv)-120 b(ativ)-60 b(e)717 b(for)g(in)-60 b(tegers)800 43196 y Fq(It)434 b(is)g(time)f(to)h(extend)f(our)g(de\257nition)g(to)h (in)-36 b(tegers.)p Black 800 46240 a Fr(Theorem)499 b(13)p Black 651 w Fj(A)464 b(derivative)g(is)g(uniquely)h(de\257ne)-66 b(d)463 b(over)i(the)g(inte)-66 b(gers)462 b(by)j(the)g(rule)23156 49173 y Fq(\()p Fk(\241)p Fo(x)p Fq(\))25940 48625 y Fn(0)26619 49173 y Fq(=)369 b Fk(\241)p Fo(x)29772 48625 y Fn(0)30083 49173 y Fo(:)800 52218 y Fj(Pr)-66 b(o)g(of.)725 b Fq(Because)480 b(\()p Fk(\241)p Fq(1\))12664 51736 y Fl(2)13638 52218 y Fq(=)447 b(1)481 b(w)-36 b(e)480 b(should)g(ha)-36 b(v)g(e)480 b(\(according)g(to)g(the)g(Leibnitz)f (rule\))h(2\()p Fk(\241)p Fq(1\))47356 51736 y Fn(0)48115 52218 y Fq(=)447 b(0)481 b(and)800 53823 y(\()p Fk(\241)p Fq(1\))3495 53341 y Fn(0)4204 53823 y Fq(=)398 b(0)451 b(is)g(the)f(only)i(c)-36 b(hoice.)630 b(After)451 b(that)g(\()p Fk(\241)p Fo(x)p Fq(\))26625 53341 y Fn(0)27333 53823 y Fq(=)398 b(\(\()p Fk(\241)p Fq(1\))307 b Fk(\242)g Fo(x)p Fq(\))34172 53341 y Fn(0)34881 53823 y Fq(=)398 b(0)307 b Fk(\242)g Fo(x)38663 53341 y Fn(0)39280 53823 y Fq(+)g(\()p Fk(\241)p Fq(1\))g Fk(\242)g Fo(x)45016 53341 y Fn(0)45725 53823 y Fq(=)398 b Fk(\241)p Fo(x)48907 53341 y Fn(0)49668 53823 y Fq(is)452 b(the)800 55428 y(only)483 b(c)-36 b(hoice)483 b(for)h(negativ)-36 b(e)483 b Fk(\241)p Fo(x)g Fq(and)f(as)h(a)g(result)g(is)g(true)e(for)j(p)36 b(ositiv)-36 b(e)483 b(in)-36 b(tegers)483 b(also.)727 b(It)482 b(remains)h(to)800 57033 y(c)-36 b(hec)g(k)443 b(that)f(the)g(Leibnitz)g(rule)h(is)g(still)g(v)-72 b(alid.)606 b(It)443 b(is)g(su\261cien)-36 b(t)442 b(to)h(c)-36 b(hec)g(k)442 b(that)h(it)f(is)h(v)-72 b(alid)444 b(for)f Fk(\241)p Fo(a)g Fq(and)800 58638 y Fo(b)434 b Fq(if)g(it)g(w)-36 b(as)434 b(v)-72 b(alid)434 b(for)g Fo(a)f Fq(and)h Fo(b:)f Fq(It)h(follo)-36 b(ws)435 b(directly:)3538 61571 y(\(\()p Fk(\241)p Fo(a)p Fq(\))294 b Fk(\242)h Fo(b)p Fq(\))8789 61023 y Fn(0)9469 61571 y Fq(=)368 b Fk(\241)p Fq(\()p Fo(a)295 b Fk(\242)g Fo(b)p Fq(\))15089 61023 y Fn(0)15769 61571 y Fq(=)368 b Fk(\241)p Fq(\()p Fo(a)19371 61023 y Fn(0)19977 61571 y Fk(\242)295 b Fo(b)g Fq(+)g Fo(a)g Fk(\242)g Fo(b)24991 61023 y Fn(0)25302 61571 y Fq(\))369 b(=)f Fk(\241)p Fo(a)29273 61023 y Fn(0)29879 61571 y Fk(\242)295 b Fo(b)g Fq(+)g(\()p Fk(\241)p Fo(a)p Fq(\))g Fk(\242)g Fo(b)36938 61023 y Fn(0)37618 61571 y Fq(=)368 b(\()p Fk(\241)p Fo(a)p Fq(\))41726 61023 y Fn(0)42331 61571 y Fk(\242)296 b Fo(b)f Fq(+)g(\()p Fk(\241)p Fo(a)p Fq(\))f Fk(\242)i Fo(b)49391 61023 y Fn(0)49701 61571 y Fo(:)p 800 64505 V Black 26150 74617 a Fq(10)p Black eop %%Page: 11 11 11 10 bop Black 0 TeXcolorgray Black Black 800 1424 a Fp(6)2152 b(Deriv)-120 b(ativ)-60 b(e)717 b(for)g(rational)f(n)-60 b(um)g(b)60 b(ers)800 4345 y Fq(The)534 b(next)f(step)g(is)h(to)g (di\256eren)-36 b(tiate)533 b(a)g(rational)i(n)-36 b(um)g(b)36 b(er.)877 b(W)-108 b(e)533 b(start)g(from)h(the)f(p)36 b(ositiv)-36 b(e)535 b(rationals.)800 5950 y(The)i(shortest)f(w)-36 b(a)g(y)538 b(is)f(to)g(use)g(\()p 0 1 0 0 TeXcolorcmyk(1)p (#equation.1) [[228 663 234 675] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\).)888 b(Namely)-108 b(,)563 b(if)538 b Fo(x)545 b Fq(=)29001 4953 y Fi(Q)30255 5304 y Fm(k)30255 6337 y(i)p Fl(=1)32055 5950 y Fo(p)32708 5357 y Fm(x)33238 5492 y Fh(i)32708 6290 y Fm(i)34181 5950 y Fq(is)537 b(a)g(a)g(factorization)i(of)e(a)g(rational)800 7555 y(n)-36 b(um)g(b)36 b(er)432 b Fo(x)i Fq(in)f(prime)g(p)36 b(o)-36 b(w)g(ers,)434 b(\(where)g(some)f Fo(x)24837 7754 y Fm(i)25647 7555 y Fq(ma)-36 b(y)434 b(b)36 b(e)433 b(negativ)-36 b(e\))435 b(then)d(w)-36 b(e)434 b(put)23160 11669 y Fo(x)23899 11121 y Fn(0)24579 11669 y Fq(=)368 b Fo(x)27622 10009 y Fm(k)26920 10407 y Fi(X)27118 13207 y Fm(i)p Fl(=1)29193 10771 y Fo(x)29932 10970 y Fm(i)p 29193 11364 1115 54 v 29236 12580 a Fo(p)29889 12779 y Fm(i)p Black 51138 11669 a Fq(\(5\))p Black 800 15636 a(and)518 b(the)f(same)i(pro)36 b(of)519 b(as)f(in)g(Theorem)p 0 1 0 0 TeXcolorcmyk 518 w(1)p (#Thm.1) [[265 576 271 588] [1 1 1 [3 3]] [0 0 1]] pdfm Black 519 w(sho)-36 b(ws)518 b(that)g(this)g(de\257nition)f(is)i(still) g(consisten)-36 b(t)517 b(with)i(the)800 17241 y(Leibnitz)433 b(rule.)2751 18846 y(Here)h(is)f(a)h(table)g(of)g(deriv)-72 b(ativ)-36 b(es)435 b(of)f Fo(i=j)509 b Fq(for)434 b(small)g Fo(i;)221 b(j)r(:)p 9136 20606 35328 45 v 9114 22211 45 1606 v 9690 21730 a(i=j)p 11928 22211 V 1832 w Fq(1)1756 b(2)2407 b(3)f(4)g(5)2731 b(6)g(7)h(8)3056 b(9)2731 b(10)p 44442 22211 V 9136 22256 35328 45 v 9114 23861 45 1606 v 11928 23861 V 44442 23861 V 9114 27241 45 3381 v 10746 25615 a(1)p 11928 27241 V 1758 w(0)15044 24717 y Fk(\241)p Fq(1)p 15044 25310 1684 54 v 15560 26527 a(4)18100 24717 y Fk(\241)p Fq(1)p 18100 25310 V 18617 26527 a(9)21156 24717 y Fk(\241)p Fq(1)p 21156 25310 V 21673 26527 a(4)24212 24717 y Fk(\241)p Fq(1)p 24212 25310 V 24403 26527 a(25)27594 24717 y Fk(\241)p Fq(5)p 27594 25310 V 27785 26527 a(36)30975 24717 y Fk(\241)p Fq(1)p 30975 25310 V 31166 26527 a(49)34356 24717 y Fk(\241)p Fq(3)p 34356 25310 V 34547 26527 a(16)38063 24717 y Fk(\241)p Fq(2)p 38063 25310 V 38254 26527 a(27)41769 24717 y Fk(\241)p Fq(7)p 41635 25310 1951 54 v 41635 26527 a(100)p 44442 27241 45 3381 v 9114 30621 V 10746 28996 a(2)p 11928 30621 V 1758 w(1)1756 b(0)18617 28097 y(1)p 18617 28690 651 54 v 18617 29907 a(9)21156 28097 y Fk(\241)p Fq(1)p 21156 28690 1684 54 v 21673 29907 a(4)24729 28097 y(3)p 24404 28690 1301 54 v 24404 29907 a(25)27594 28097 y Fk(\241)p Fq(1)p 27594 28690 1684 54 v 28110 29907 a(9)31491 28097 y(5)p 31166 28690 1301 54 v 31166 29907 a(49)34356 28097 y Fk(\241)p Fq(1)p 34356 28690 1684 54 v 34873 29907 a(4)38063 28097 y Fk(\241)p Fq(1)p 38063 28690 V 38254 29907 a(27)41769 28097 y Fk(\241)p Fq(1)p 41769 28690 V 41960 29907 a(25)p 44442 30621 45 3381 v 9114 34001 V 10746 32376 a(3)p 11928 34001 V 1758 w(1)15044 31477 y Fk(\241)p Fq(1)p 15044 32070 1684 54 v 15560 33287 a(4)18617 32376 y(0)21156 31477 y Fk(\241)p Fq(1)p 21156 32070 V 21673 33287 a(2)24729 31477 y(2)p 24404 32070 1301 54 v 24404 33287 a(25)27594 31477 y Fk(\241)p Fq(1)p 27594 32070 1684 54 v 28110 33287 a(4)31491 31477 y(4)p 31166 32070 1301 54 v 31166 33287 a(49)34356 31477 y Fk(\241)p Fq(7)p 34356 32070 1684 54 v 34547 33287 a(16)38063 31477 y Fk(\241)p Fq(1)p 38063 32070 V 38579 33287 a(9)41444 31477 y Fk(\241)p Fq(11)p 41444 32070 2334 54 v 41635 33287 a(100)p 44442 34001 45 3381 v 9114 37381 V 10746 35756 a(4)p 11928 37381 V 1758 w(4)g(1)18617 34857 y(8)p 18617 35450 651 54 v 18617 36667 a(9)21673 35756 y(0)24404 34857 y(16)p 24404 35450 1301 54 v 24404 36667 a(25)28110 34857 y(1)p 28110 35450 651 54 v 28110 36667 a(9)31166 34857 y(24)p 31166 35450 1301 54 v 31166 36667 a(49)34356 34857 y Fk(\241)p Fq(1)p 34356 35450 1684 54 v 34873 36667 a(4)38579 34857 y(4)p 38254 35450 1301 54 v 38254 36667 a(27)42286 34857 y(3)p 41960 35450 V 41960 36667 a(25)p 44442 37381 45 3381 v 9114 40761 V 10746 39136 a(5)p 11928 40761 V 1758 w(1)15044 38237 y Fk(\241)p Fq(3)p 15044 38830 1684 54 v 15560 40047 a(4)18100 38237 y Fk(\241)p Fq(2)p 18100 38830 V 18617 40047 a(9)21156 39136 y Fk(\241)p Fq(1)1890 b(0)27268 38237 y Fk(\241)p Fq(19)p 27268 38830 2334 54 v 27785 40047 a(36)31491 38237 y(2)p 31166 38830 1301 54 v 31166 40047 a(49)34031 38237 y Fk(\241)p Fq(13)p 34031 38830 2334 54 v 34548 40047 a(16)38063 38237 y Fk(\241)p Fq(7)p 38063 38830 1684 54 v 38254 40047 a(27)41769 38237 y Fk(\241)p Fq(1)p 41769 38830 V 42286 40047 a(4)p 44442 40761 45 3381 v 9114 44141 V 10746 42516 a(6)p 11928 44141 V 1758 w(5)1756 b(1)2407 b(1)21156 41617 y Fk(\241)p Fq(1)p 21156 42210 1684 54 v 21673 43427 a(4)24404 41617 y(19)p 24404 42210 1301 54 v 24404 43427 a(25)28110 42516 y(0)31166 41617 y(29)p 31166 42210 V 31166 43427 a(49)34356 41617 y Fk(\241)p Fq(1)p 34356 42210 1684 54 v 34873 43427 a(2)38579 41617 y(1)p 38579 42210 651 54 v 38579 43427 a(9)42286 41617 y(2)p 41960 42210 1301 54 v 41960 43427 a(25)p 44442 44141 45 3381 v 9114 47521 V 10746 45896 a(7)p 11928 47521 V 1758 w(1)15044 44997 y Fk(\241)p Fq(5)p 15044 45590 1684 54 v 15560 46807 a(4)18100 44997 y Fk(\241)p Fq(4)p 18100 45590 V 18617 46807 a(9)21156 44997 y Fk(\241)p Fq(3)p 21156 45590 V 21673 46807 a(2)24212 44997 y Fk(\241)p Fq(2)p 24212 45590 V 24403 46807 a(25)27268 44997 y Fk(\241)p Fq(29)p 27268 45590 2334 54 v 27785 46807 a(36)31491 45896 y(0)34031 44997 y Fk(\241)p Fq(19)p 34031 45590 V 34548 46807 a(16)37737 44997 y Fk(\241)p Fq(11)p 37737 45590 V 38254 46807 a(27)41444 44997 y Fk(\241)p Fq(39)p 41444 45590 V 41635 46807 a(100)p 44442 47521 45 3381 v 9114 50901 V 10746 49276 a(8)p 11928 50901 V 1107 w(12)1757 b(4)18291 48377 y(28)p 18291 48970 1301 54 v 18617 50187 a(9)21673 49276 y(1)24404 48377 y(52)p 24404 48970 V 24404 50187 a(25)28110 48377 y(8)p 28110 48970 651 54 v 28110 50187 a(9)31166 48377 y(76)p 31166 48970 1301 54 v 31166 50187 a(49)34873 49276 y(0)38254 48377 y(20)p 38254 48970 V 38254 50187 a(27)41960 48377 y(16)p 41960 48970 V 41960 50187 a(25)p 44442 50901 45 3381 v 9114 54281 V 10746 52656 a(9)p 11928 54281 V 1758 w(6)15560 51757 y(3)p 15560 52351 651 54 v 15560 53567 a(4)18617 52656 y(1)21156 51757 y Fk(\241)p Fq(3)p 21156 52351 1684 54 v 21673 53567 a(4)24404 51757 y(21)p 24404 52351 1301 54 v 24404 53567 a(25)27594 51757 y Fk(\241)p Fq(1)p 27594 52351 1684 54 v 28110 53567 a(4)31166 51757 y(33)p 31166 52351 1301 54 v 31166 53567 a(49)34031 51757 y Fk(\241)p Fq(15)p 34031 52351 2334 54 v 34548 53567 a(16)38579 52656 y(0)41769 51757 y Fk(\241)p Fq(3)p 41635 52351 1951 54 v 41635 53567 a(100)p 44442 54281 45 3381 v 9114 56947 45 2666 v 10096 56036 a(10)p 11928 56947 V 1758 w(7)f(1)18291 55137 y(11)p 18291 55731 1301 54 v 18617 56947 a(9)21156 55137 y Fk(\241)p Fq(3)p 21156 55731 1684 54 v 21673 56947 a(4)24729 56036 y(1)27594 55137 y Fk(\241)p Fq(2)p 27594 55731 V 28110 56947 a(9)31166 55137 y(39)p 31166 55731 1301 54 v 31166 56947 a(49)34356 56036 y Fk(\241)p Fq(1)38579 55137 y(1)p 38254 55731 V 38254 56947 a(27)42286 56036 y(0)p 44442 56947 45 2666 v 9114 58552 45 1606 v 11928 58552 V 44442 58552 V 9136 58597 35328 45 v 800 60322 a(A)434 b(natural)f(prop)36 b(ert)-36 b(y)433 b(is)h(the)f(follo)-36 b(wing:)p Black 800 62978 a Fr(Theorem)499 b(14)p Black 651 w Fj(F)-100 b(or)466 b(any)e(two)h(r)-66 b(ationals)465 b Fo(a;)221 b(b)465 b Fj(we)g(have)21831 65008 y Fi(\263)22757 65584 y Fo(a)p 22757 66177 683 54 v 22822 67394 a(b)23573 65008 y Fi(\264)24366 65306 y Fn(0)25045 66483 y Fq(=)26559 65584 y Fo(a)27242 65102 y Fn(0)27552 65584 y Fo(b)295 b Fk(\241)h Fo(ab)30965 65102 y Fn(0)p 26559 66177 4717 54 v 28378 67394 a Fo(b)28931 67010 y Fl(2)31408 66483 y Fo(:)800 69691 y Fj(A)455 b(derivative)e(c)-66 b(an)454 b(b)-66 b(e)454 b(wel)66 b(l)456 b(de\257ne)-66 b(d)454 b(for)h(r)-66 b(ational)454 b(numb)-66 b(ers)454 b(using)h(this)f (formula)i(and)f(this)f(is)h(the)g(only)800 71296 y(way)466 b(to)e(de\257ne)g(a)h(derivative)e(over)i(r)-66 b(ationals)464 b(that)h(pr)-66 b(eserves)464 b(the)h(L)-66 b(eibnitz)462 b(rule.)p Black 26150 74617 a Fq(11)p Black eop %%Page: 12 12 12 11 bop Black 0 TeXcolorgray Black Black 800 1424 a Fj(Pr)-66 b(o)g(of.)597 b Fq(If)434 b Fo(a)369 b Fq(=)8552 428 y Fi(Q)9807 778 y Fm(k)9807 1812 y(i)p Fl(=1)11606 1424 y Fo(p)12259 832 y Fm(a)12759 967 y Fh(i)12259 1765 y Fm(i)13165 1424 y Fo(;)221 b(b)370 b Fq(=)16051 428 y Fi(Q)17305 778 y Fm(k)17305 1812 y(i)p Fl(=1)19105 1424 y Fo(p)19758 832 y Fm(a)20258 967 y Fh(i)19758 1765 y Fm(i)21097 1424 y Fq(then)433 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)13688 4184 y Fi(\263)14614 4760 y Fo(a)p 14614 5353 683 54 v 14679 6570 a(b)15430 4184 y Fi(\264)16223 4482 y Fn(0)16902 5659 y Fq(=)369 b(\()19381 3998 y Fm(k)18789 4397 y Fi(Y)18876 7196 y Fm(i)p Fl(=1)20708 5659 y Fo(p)21361 5066 y Fm(a)21861 5201 y Fh(i)22212 5066 y Fn(\241)p Fm(b)23347 5201 y Fh(i)21361 5999 y Fm(i)23753 5659 y Fq(\))24259 5110 y Fn(0)24938 5659 y Fq(=)f(\()27416 3998 y Fm(k)26824 4397 y Fi(Y)26911 7196 y Fm(i)p Fl(=1)28743 5659 y Fo(p)29396 5066 y Fm(a)29896 5201 y Fh(i)30247 5066 y Fn(\241)p Fm(b)31382 5201 y Fh(i)29396 5999 y Fm(i)31788 5659 y Fq(\))33218 3998 y Fm(k)32515 4397 y Fi(X)32713 7196 y Fm(i)p Fl(=1)34788 4760 y Fo(a)35471 4959 y Fm(i)36142 4760 y Fk(\241)295 b Fo(b)38023 4959 y Fm(i)p 34788 5353 3611 54 v 36079 6570 a Fo(p)36732 6769 y Fm(i)38901 5659 y Fq(=)12553 9702 y Fi(\263)13479 10278 y Fo(a)p 13479 10871 683 54 v 13544 12088 a(b)14295 9702 y Fi(\264)16012 9516 y Fm(k)15310 9914 y Fi(X)15508 12714 y Fm(i)p Fl(=1)17583 10278 y Fo(a)18266 10477 y Fm(i)p 17583 10871 1059 54 v 17598 12088 a Fo(p)18251 12287 y Fm(i)19069 11176 y Fk(\241)20397 9303 y Fi(\265)21508 10278 y Fo(ab)p 21508 10871 1236 54 v 21586 12088 a(b)22139 11704 y Fl(2)22877 9303 y Fi(\266)24778 9516 y Fm(k)24076 9914 y Fi(X)24274 12714 y Fm(i)p Fl(=1)26399 10278 y Fo(b)26952 10477 y Fm(i)p 26349 10871 1029 54 v 26349 12088 a Fo(p)27002 12287 y Fm(i)27879 11176 y Fq(=)29393 10278 y Fo(a)30076 9796 y Fn(0)p 29393 10871 994 54 v 29613 12088 a Fo(b)30814 11176 y Fk(\241)32275 10278 y Fo(ab)33511 9796 y Fn(0)p 32275 10871 1547 54 v 32509 12088 a Fo(b)33062 11704 y Fl(2)34323 11176 y Fq(=)35837 10278 y Fo(a)36520 9796 y Fn(0)36830 10278 y Fo(b)g Fk(\241)h Fo(ab)40243 9796 y Fn(0)p 35837 10871 4717 54 v 37655 12088 a Fo(b)38208 11704 y Fl(2)40686 11176 y Fo(:)800 14771 y Fq(Let)433 b(us)g(c)-36 b(hec)g(k)434 b(uniqueness.)578 b(If)434 b Fo(n)g Fq(is)g(an)f(in)-36 b(teger)434 b(then)e Fo(n)296 b Fk(\242)30169 14248 y Fl(1)p 30119 14465 571 54 v 30119 15229 a Fm(n)31192 14771 y Fq(=)368 b(1)434 b(and)f(the)g(Leibnitz)g(rule)g(demands)16059 18735 y Fo(n)16835 18187 y Fn(0)17441 18735 y Fk(\242)18301 17837 y Fq(1)p 18238 18430 777 54 v 18238 19647 a Fo(n)19442 18735 y Fq(+)295 b Fo(n)21746 16862 y Fi(\265)22920 17837 y Fq(1)p 22857 18430 V 22857 19647 a Fo(n)23767 16862 y Fi(\266)24744 17160 y Fn(0)25424 18735 y Fq(=)369 b(0)g Fk(\))29521 16862 y Fi(\265)30695 17837 y Fq(1)p 30632 18430 V 30632 19647 a Fo(n)31541 16862 y Fi(\266)32519 17160 y Fn(0)33199 18735 y Fq(=)f Fk(\241)35853 17837 y Fo(n)36629 17355 y Fn(0)p 35745 18430 1303 54 v 35745 19647 a Fo(n)36521 19263 y Fl(2)37180 18735 y Fo(:)800 22359 y Fq(After)434 b(that)8415 24391 y Fi(\263)9341 24967 y Fo(a)p 9341 25560 683 54 v 9406 26777 a(b)10157 24391 y Fi(\264)10950 24688 y Fn(0)11630 25865 y Fq(=)13010 23992 y Fi(\265)13988 25865 y Fo(a)295 b Fk(\242)15763 24967 y Fq(1)p 15763 25560 651 54 v 15812 26777 a Fo(b)16546 23992 y Fi(\266)17524 24290 y Fn(0)18204 25865 y Fq(=)368 b Fo(a)20267 25317 y Fn(0)20873 25865 y Fk(\242)21670 24967 y Fq(1)p 21670 25560 V 21719 26777 a Fo(b)22748 25865 y Fq(+)295 b Fo(a)g Fk(\242)25697 23992 y Fi(\265)26808 24967 y Fq(1)p 26808 25560 V 26857 26777 a Fo(b)27591 23992 y Fi(\266)28569 24290 y Fn(0)29248 25865 y Fq(=)30762 24967 y Fo(a)31445 24485 y Fn(0)p 30762 25560 994 54 v 30982 26777 a Fo(b)32183 25865 y Fk(\241)g Fo(a)g Fk(\242)35153 23992 y Fi(\265)36372 24967 y Fo(b)36925 24485 y Fn(0)p 36264 25560 1079 54 v 36264 26777 a Fo(b)36817 26393 y Fl(2)37476 23992 y Fi(\266)38823 25865 y Fq(=)40336 24967 y Fo(a)41019 24485 y Fn(0)41329 24967 y Fo(b)h Fk(\241)f Fo(ab)44742 24485 y Fn(0)p 40336 25560 4717 54 v 42155 26777 a Fo(b)42708 26393 y Fl(2)800 29489 y Fq(is)537 b(the)e(only)i(c)-36 b(hoice)537 b(that)f(satis\257es)g (the)g(Leibnitz)g(rule.)886 b(This)536 b(pro)-36 b(v)g(es)537 b(uniqueness.)886 b(T)-108 b(o)536 b(pro)-36 b(v)g(e)536 b(that)800 31094 y(suc)-36 b(h)433 b(a)h(de\257nition)e(is)i(w)-36 b(ell-de\257ned,)433 b(it)h(is)g(su\261cien)-36 b(t)432 b(to)i(see)g(that)8300 33212 y Fi(\263)9226 33788 y Fo(ac)p 9226 34381 1243 54 v 9291 35597 a(bc)10601 33212 y Fi(\264)11395 33509 y Fn(0)12074 34686 y Fq(=)13588 33788 y(\()p Fo(ac)p Fq(\))15843 33306 y Fn(0)16152 33788 y Fq(\()p Fo(bc)p Fq(\))295 b Fk(\241)g Fq(\()p Fo(ac)p Fq(\)\()p Fo(bc)p Fq(\))24280 33306 y Fn(0)p 13588 34381 11002 54 v 17763 35597 a Fq(\()p Fo(bc)p Fq(\))19888 35214 y Fl(2)25091 34686 y Fq(=)26605 33788 y(\()p Fo(a)27794 33306 y Fn(0)28104 33788 y Fo(c)g Fq(+)g Fo(ac)31509 33306 y Fn(0)31819 33788 y Fq(\)\()p Fo(bc)p Fq(\))f Fk(\241)h Fq(\()p Fo(ac)p Fq(\)\()p Fo(b)39386 33306 y Fn(0)39696 33788 y Fo(c)g Fq(+)g Fo(bc)42971 33306 y Fn(0)43281 33788 y Fq(\))p 26605 34381 17183 54 v 34114 35597 a Fo(b)34667 35214 y Fl(2)35192 35597 y Fo(c)35752 35214 y Fl(2)44289 34686 y Fq(=)14711 38421 y(\()p Fo(a)15900 37939 y Fn(0)16210 38421 y Fo(bc)17323 37939 y Fl(2)18144 38421 y Fq(+)g Fo(abc)21247 37939 y Fn(0)21557 38421 y Fo(c)p Fq(\))g Fk(\241)g Fq(\()p Fo(ab)25988 37939 y Fn(0)26298 38421 y Fo(c)26858 37939 y Fl(2)27679 38421 y Fq(+)g Fo(abcc)31342 37939 y Fn(0)31652 38421 y Fq(\))p 14711 39014 17447 54 v 22352 40231 a Fo(b)22905 39847 y Fl(2)23431 40231 y Fo(c)23991 39847 y Fl(2)32659 39320 y Fq(=)34173 38421 y Fo(a)34856 37939 y Fn(0)35166 38421 y Fo(b)g Fk(\241)h Fo(ab)38579 37939 y Fn(0)p 34173 39014 4717 54 v 35991 40231 a Fo(b)36544 39847 y Fl(2)800 41984 y Fq(has)434 b(the)f(same)g(v)-72 b(alue.)p 12484 41984 630 630 v 2751 44253 a(F)-108 b(or)433 b(negativ)-36 b(e)435 b(rationals)f(w)-36 b(e)434 b(can)f(pro)36 b(ceed)434 b(as)g(ab)36 b(o)-36 b(v)g(e)434 b(and)f(put)f(\()p Fk(\241)p Fo(x)p Fq(\))37650 43771 y Fn(0)38330 44253 y Fq(=)368 b Fk(\241)p Fo(x)41482 43771 y Fn(0)41793 44253 y Fo(:)800 48690 y Fp(7)2152 b(Rational)716 b(solutions)g(of)g(the)h(equation)f Fg(x)37766 47996 y Fk(0)38718 48690 y Ff(=)531 b Fg(a:)800 51611 y Fq(Unexp)36 b(ectedly)493 b(the)g(equation)h Fo(x)17434 51129 y Fn(0)18215 51611 y Fq(=)470 b(0)493 b(has)g(non)-36 b(trivial)494 b(rational)g(solutions,)508 b(for)494 b(instance)f Fo(x)470 b Fq(=)g(4)p Fo(=)p Fq(27)p Fo(:)800 53216 y Fq(W)-108 b(e)434 b(can)f(describ)36 b(e)433 b(all)i(of)f(them.)p Black 800 55928 a Fr(Theorem)499 b(15)p Black 651 w Fj(L)-66 b(et)600 b Fo(k)646 b Fj(b)-66 b(e)600 b(some)h(natur)-66 b(al)601 b(numb)-66 b(er,)634 b Fk(f)p Fo(p)29289 56127 y Fm(i)29664 55928 y Fo(;)221 b(i)370 b Fq(=)e(1)p Fo(;)221 b(:)g(:)g(:)j(k)45 b Fk(g)602 b Fj(b)-66 b(e)600 b(a)h(set)g(of)g (di\256er)-66 b(ent)599 b(prime)800 57533 y(numb)-66 b(ers)464 b(and)h Fk(f)p Fo(\256)9979 57732 y Fm(i)10355 57533 y Fo(;)221 b(i)370 b Fq(=)e(1)p Fo(;)221 b(:)g(:)g(:)j(k)45 b Fk(g)465 b Fj(b)-66 b(e)464 b(a)h(set)g(of)g(inte)-66 b(gers)463 b(such)i(that)34683 56537 y Fi(P)36085 56887 y Fm(k)36085 57920 y(i)p Fl(=1)37885 57533 y Fo(\256)38712 57732 y Fm(i)39457 57533 y Fq(=)368 b(0)p Fo(:)466 b Fj(Then)22725 61768 y Fo(x)369 b Fq(=)f Fk(\247)27060 60107 y Fm(k)26468 60506 y Fi(Y)26555 63305 y Fm(i)p Fl(=1)28387 61768 y Fo(p)29040 61138 y Fm(\256)29644 61273 y Fh(i)29995 61138 y Fm(p)30469 61273 y Fh(i)29040 62108 y Fm(i)800 65851 y Fj(ar)-66 b(e)334 b(solutions)h(of)f(the)g (di\256er)-66 b(ential)332 b(e)-66 b(quation)333 b Fo(x)23836 65369 y Fn(0)24516 65851 y Fq(=)368 b(0)335 b Fj(and)f(any)g(other)h (nonzer)-66 b(o)333 b(solution)i(c)-66 b(an)333 b(b)-66 b(e)334 b(obtaine)-66 b(d)800 67456 y(in)464 b(this)h(manner.)p Black 26150 74617 a Fq(12)p Black eop %%Page: 13 13 13 12 bop Black 0 TeXcolorgray Black Black 800 1424 a Fj(Pr)-66 b(o)g(of.)989 b Fq(Because)576 b(\()p Fk(\241)p Fo(x)p Fq(\))13113 942 y Fn(0)14035 1424 y Fq(=)611 b Fk(\241)p Fo(x)17430 942 y Fn(0)18317 1424 y Fq(it)576 b(is)g(su\261cien)-36 b(t)576 b(to)g(consider)f(p)36 b(ositiv)-36 b(e)577 b(solutions)g(only)-108 b(.)1005 b(Let)576 b Fo(x)611 b Fq(=)800 2033 y Fi(Q)2055 2383 y Fm(k)2055 3417 y(i)p Fl(=1)3854 3029 y Fo(p)4507 2437 y Fm(a)5007 2572 y Fh(i)4507 3370 y Fm(i)5847 3029 y Fq(Then)433 b(from)h(\()p 0 1 0 0 TeXcolorcmyk(5)p (#equation.5) [[187 690 193 702] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))19375 4275 y Fm(k)18672 4674 y Fi(X)18870 7473 y Fm(i)p Fl(=1)20945 5037 y Fo(a)21628 5236 y Fm(i)p 20945 5630 1059 54 v 20960 6847 a Fo(p)21613 7046 y Fm(i)22506 5936 y Fq(=)368 b(0)h Fk(\))27305 4275 y Fm(k)26603 4674 y Fi(X)26801 7473 y Fm(i)p Fl(=1)28743 5936 y Fo(a)29426 6135 y Fm(i)30097 5936 y Fk(\242)295 b Fo(Q)31791 6135 y Fm(i)32536 5936 y Fq(=)368 b(0)p Fo(;)800 9839 y Fq(where)333 b Fo(Q)5487 10038 y Fm(i)6232 9839 y Fq(=)7613 8365 y Fi(\263)8406 8843 y(Q)9661 9194 y Fm(k)9661 10227 y(j)51 b Fl(=1)11571 9839 y Fo(p)12224 10038 y Fm(j)12711 8365 y Fi(\264)13726 9839 y Fo(=p)15029 10038 y Fm(i)15738 9839 y Fq(is)334 b(not)f(divisible)h(b)-36 b(y)333 b Fo(p)26518 10038 y Fm(i)26894 9839 y Fo(:)g Fq(Th)-36 b(us)333 b Fo(a)31466 10038 y Fm(i)32175 9839 y Fq(should)g(b)36 b(e)333 b(divisible)h(b)-36 b(y)334 b Fo(p)45232 10038 y Fm(i)45941 9839 y Fq(and)e Fo(\256)49196 10038 y Fm(i)49941 9839 y Fq(=)51455 9293 y Fm(a)51955 9428 y Fh(i)p 51455 9534 852 54 v 51468 10297 a Fm(p)51942 10432 y Fh(i)52439 9839 y Fo(:)p 800 11444 630 630 v 2751 13631 a Fq(Other)433 b(equations)h(are)f(more)h(di\261cult.)p Black 800 16094 a Fr(Conjecture)499 b(12)p Black 651 w Fj(The)465 b(e)-66 b(quation)463 b Fo(x)19182 15612 y Fn(0)19862 16094 y Fq(=)368 b(1)466 b Fj(has)f(only)g(primes)g(as)g(p)-66 b(ositive)463 b(r)-66 b(ational)464 b(solutions.)800 18557 y Fq(Note)549 b(that)f(there)g(exists)h(a)g(negativ)-36 b(e)550 b(solution,)578 b(namely)549 b Fo(x)565 b Fq(=)f Fk(\241)34684 18034 y Fl(5)p 34684 18252 471 54 v 34684 19015 a(4)35287 18557 y Fo(:)549 b Fq(One)f(p)36 b(ossible)549 b(solution)g(of)g(this)800 20331 y(equation)395 b(w)-36 b(ould)395 b(b)36 b(e)394 b Fo(x)369 b Fq(=)14372 19808 y Fm(n)p 14180 20025 956 54 v 14180 20789 a(p)14654 20537 y Fh(p)15663 20331 y Fq(for)395 b(some)g(natural)g Fo(n)g Fq(and)f(prime)g Fo(p:)h Fq(Because)g Fo(x)39736 19849 y Fn(0)40416 20331 y Fq(=)41929 19808 y Fm(n)42500 19495 y Fe(0)42798 19808 y Fn(\241)p Fm(n)p 41929 20025 2172 54 v 42537 20789 a(p)43011 20537 y Fh(p)44629 20331 y Fq(in)f(this)h(case)g(w)-36 b(e)800 21936 y(can)434 b(reform)-36 b(ulate)433 b(the)h(conjecture)f(as)p Black 800 24399 a Fr(Conjecture)499 b(13)p Black 651 w Fj(L)-66 b(et)440 b Fo(p)h Fj(b)-66 b(e)440 b(a)i(prime.)588 b(The)441 b(e)-66 b(quation)440 b Fo(n)29420 23917 y Fn(0)30100 24399 y Fq(=)369 b Fo(n)243 b Fq(+)f Fo(p)34407 23917 y Fm(p)35377 24399 y Fj(has)442 b(no)f(natur)-66 b(al)441 b(solutions)h(exc)-66 b(ept)800 26004 y Fo(n)369 b Fq(=)g Fo(q)48 b(p)4604 25522 y Fm(p)5132 26004 y Fo(;)465 b Fj(wher)-66 b(e)465 b Fo(q)512 b Fj(is)465 b(a)g(prime.)800 28467 y Fq(Note,)453 b(that)c(according)g(to)g(Theorem)p 0 1 0 0 TeXcolorcmyk 449 w(5)p (#Thm.5) [[253 461 259 473] [1 1 1 [3 3]] [0 0 1]] pdfm Black 449 w(if)h(a)f(solution)g Fo(n)g Fq(is)g(divisible)h(b)-36 b(y)449 b Fo(p)f Fq(it)h(should)g(b)36 b(e)448 b(divisible)i(b)-36 b(y)800 30072 y Fo(p)1453 29590 y Fm(p)1982 30072 y Fo(:)391 b Fq(Therefore)h Fo(n)369 b Fq(=)g Fo(mp)12898 29590 y Fm(p)13818 30072 y Fq(and)390 b Fo(p)16957 29590 y Fm(p)17486 30072 y Fq(\()p Fo(m)19130 29590 y Fn(0)19649 30072 y Fq(+)209 b Fo(m)p Fq(\))368 b(=)g Fo(p)24915 29590 y Fm(p)25444 30072 y Fq(\()p Fo(m)209 b Fq(+)g(1\))390 b(b)-36 b(y)391 b(Theorem)p 0 1 0 0 TeXcolorcmyk 391 w(4)p (#Thm.4) [[407 446 413 458] [1 1 1 [3 3]] [0 0 1]] pdfm Black 391 w(and)g Fo(m)g Fq(should)g(b)36 b(e)391 b(a)g(prime.)800 31678 y(Th)-36 b(us)433 b(it)h(is)g(su\261cien)-36 b(t)432 b(to)i(pro)-36 b(v)g(e)434 b(that)f(an)-36 b(y)434 b(solution)f(is)h (divisible)h(b)-36 b(y)433 b Fo(p:)2751 33283 y Fq(W)-108 b(e)509 b(do)f(not)h(exp)36 b(ect)508 b(that)h(it)f(is)h(p)36 b(ossible)509 b(to)g(in)-36 b(tegrate)509 b(ev)-36 b(ery)509 b(rational)g(n)-36 b(um)g(b)36 b(er,)526 b(though)508 b(w)-36 b(e)509 b(do)800 34888 y(not)433 b(kno)-36 b(w)435 b(a)e(coun)-36 b(terexample.)p Black 800 37600 a Fr(Conjecture)499 b(14)p Black 651 w Fj(Ther)-66 b(e)464 b(exists)g Fo(a)h Fj(such)h(that)e(the)h(e)-66 b(quation)464 b Fo(x)32734 37118 y Fn(0)33413 37600 y Fq(=)369 b Fo(a)464 b Fj(has)i(no)f(r)-66 b(ational)464 b(solutions.)800 40313 y Fq(The)434 b(\257rst)e(natural)i (candidates)f(do)h(not)f(v)-36 b(erify)435 b(the)e(conjecture:)16421 43474 y(\()p Fk(\241)18093 42576 y Fq(21)p 18093 43169 1301 54 v 18093 44386 a(16)19527 43474 y(\))20033 42926 y Fn(0)20712 43474 y Fq(=)369 b(2;)221 b(\()p Fk(\241)24997 42576 y Fq(13)p 24997 43169 V 25323 44386 a(4)26431 43474 y(\))26937 42926 y Fn(0)27616 43474 y Fq(=)369 b(3;)221 b(\()p Fk(\241)31901 42576 y Fq(22)p 31901 43169 V 31901 44386 a(27)33335 43474 y(\))33841 42926 y Fn(0)34521 43474 y Fq(=)36034 42576 y(1)p 36034 43169 651 54 v 36034 44386 a(3)36817 43474 y Fo(:)800 47869 y Fp(8)2152 b(Logarithmic)717 b(deriv)-120 b(ativ)-60 b(e)800 50789 y Fq(One)421 b(thing)h(that)f(is) h(still)h(absen)-36 b(t)421 b(in)g(our)h(picture)f(is)h(the)f(analogue) i(of)g(the)e(logarithm)h({)g(the)g(primitiv)-36 b(e)800 52394 y(of)2399 51871 y Fl(1)p 2349 52089 571 54 v 2349 52852 a Fm(n)3053 52394 y Fo(:)369 b Fq(Because)g(our)f(deriv)-72 b(ativ)-36 b(e)369 b(is)g(not)f(linear)h(w)-36 b(e)369 b(cannot)g(exp)36 b(ect)368 b(that)g(the)h(logarithm)g(of)g(the)f(pro) 36 b(duct)800 53999 y(is)535 b(equal)g(to)g(the)f(sum)g(of)h (logarithms.)882 b(Instead)534 b(this)g(is)h(true)f(for)h(its)g(deriv) -72 b(ativ)-36 b(e.)882 b(So)534 b(let)h(us)f(de\257ne)800 55604 y(a)513 b(logarithmic)g(deriv)-72 b(ativ)-36 b(e)513 b(ld\()p Fo(x)p Fq(\))f(as)h(follo)-36 b(ws.)817 b(If)513 b Fo(x)503 b Fq(=)29118 54608 y Fi(Q)30373 54959 y Fm(k)30373 55992 y(i)p Fl(=1)32172 55604 y Fo(p)32825 55012 y Fm(x)33355 55147 y Fh(i)32825 55945 y Fm(i)34274 55604 y Fq(for)513 b(di\256eren)-36 b(t)511 b(primes)h Fo(p)46479 55803 y Fm(i)47367 55604 y Fq(and)g(some)800 57210 y(in)-36 b(tegers)434 b Fo(x)6352 57409 y Fm(i)6727 57210 y Fo(;)g Fq(then)14413 59987 y(ld\()p Fo(x)p Fq(\))368 b(=)19700 58326 y Fm(k)18997 58725 y Fi(X)19195 61524 y Fm(i)p Fl(=1)21270 59088 y Fo(x)22009 59287 y Fm(i)p 21270 59681 1115 54 v 21313 60898 a Fo(p)21966 61097 y Fm(i)22518 59987 y Fo(;)655 b Fq(ld\()p Fk(\241)p Fo(x)p Fq(\))369 b(=)f(ld\()p Fo(x)p Fq(\))p Fo(;)655 b Fq(ld\(0\))369 b(=)g Fk(1)p Fo(:)800 63285 y Fq(In)433 b(other)h(w)-36 b(ords)23670 65269 y(ld)o(\()p Fo(x)p Fq(\))369 b(=)28387 64370 y Fo(x)29126 63888 y Fn(0)p 28387 64963 1050 54 v 28542 66180 a Fo(x)29569 65269 y(:)p Black 800 68320 a Fr(Theorem)499 b(16)p Black 651 w Fj(F)-100 b(or)466 b(any)e(r)-66 b(ational)464 b(numb)-66 b(ers)20511 70964 y(ld)p Fq(\()p Fo(xy)48 b Fq(\))369 b(=)f Fj(ld)q Fq(\()p Fo(x)p Fq(\))294 b(+)h Fj(ld)q Fq(\()p Fo(y)48 b Fq(\))p Fo(:)p Black 26150 74617 a Fq(13)p Black eop %%Page: 14 14 14 13 bop Black 0 TeXcolorgray Black Black 800 1424 a Fj(Pr)-66 b(o)g(of.)11497 3688 y Fq(ld\()p Fo(xy)48 b Fq(\))369 b(=)16896 2789 y(\()p Fo(xy)48 b Fq(\))19329 2307 y Fn(0)p 16896 3382 2744 54 v 17557 4599 a Fo(xy)20141 3688 y Fq(=)21655 2789 y Fo(x)22394 2307 y Fn(0)22704 2789 y Fo(y)344 b Fq(+)294 b Fo(xy)26409 2307 y Fn(0)p 21655 3382 5066 54 v 23477 4599 a Fo(xy)27222 3688 y Fq(=)28735 2789 y Fo(x)29474 2307 y Fn(0)p 28735 3382 1050 54 v 28890 4599 a Fo(x)30213 3688 y Fq(+)31652 2789 y Fo(y)32334 2307 y Fn(0)p 31652 3382 993 54 v 31808 4599 a Fo(y)33147 3688 y Fq(=)368 b(ld\()p Fo(x)p Fq(\))295 b(+)g(ld\()p Fo(y)48 b Fq(\))p Fo(:)p 800 6251 630 630 v 2751 8476 a Fq(It)527 b(is)g(useful)g(to)g(divide)g(ev)-36 b(ery)527 b(in)-36 b(teger)527 b(n)-36 b(um)g(b)36 b(er)525 b(in)-36 b(to)526 b(large)i(and)e(small)i(parts.)857 b(Let)527 b(sign\()p Fo(x)p Fq(\))p Fo(x)g Fq(=)800 10081 y Fk(j)p Fo(x)p Fk(j)369 b Fq(=)4027 9085 y Fi(Q)5281 9436 y Fm(k)5281 10469 y(i)p Fl(=1)7081 10081 y Fo(p)7734 9489 y Fm(x)8264 9624 y Fh(i)7734 10422 y Fm(i)9103 10081 y Fq(and)433 b Fo(x)12371 10280 y Fm(i)13116 10081 y Fq(=)369 b Fo(a)15180 10280 y Fm(i)15555 10081 y Fo(p)16208 10280 y Fm(i)16879 10081 y Fq(+)295 b Fo(r)18772 10280 y Fm(i)19148 10081 y Fo(;)433 b Fq(where)h(0)369 b Fk(\267)g Fo(r)26707 10280 y Fm(i)27452 10081 y Fo(<)g(p)29486 10280 y Fm(i)29861 10081 y Fo(:)434 b Fq(W)-108 b(e)434 b(de\257ne)11727 14161 y Fo(P)181 b Fq(\()p Fo(x)p Fq(\))369 b(=)f(sign\()p Fo(x)p Fq(\))21058 12501 y Fm(k)20466 12900 y Fi(Y)20553 15699 y Fm(i)p Fl(=1)22384 14161 y Fo(p)23037 13532 y Fm(a)23537 13667 y Fh(i)23888 13532 y Fm(p)24362 13667 y Fh(i)23037 14502 y Fm(i)24769 14161 y Fo(;)221 b(R)11 b Fq(\()p Fo(x)p Fq(\))369 b(=)30444 12501 y Fm(k)29852 12900 y Fi(Y)29940 15699 y Fm(i)p Fl(=1)31771 14161 y Fo(p)32424 13569 y Fm(r)32849 13704 y Fh(i)32424 14502 y Fm(i)33255 14161 y Fo(;)221 b(A)p Fq(\()p Fo(x)p Fq(\))370 b(=)39016 12501 y Fm(k)38313 12900 y Fi(X)38511 15699 y Fm(i)p Fl(=1)40453 14161 y Fo(a)41136 14360 y Fm(i)41512 14161 y Fo(:)p Black 800 19696 a Fr(Theorem)499 b(17)p Black 651 w Fj(The)465 b(fol)66 b(lowing)464 b(pr)-66 b(op)g(erties)464 b(hold)p Black 2737 22275 a Fk(\262)p Black 651 w Fj(ld)p Fq(\()p Fo(x)p Fq(\))369 b(=)f Fo(A)p Fq(\()p Fo(x)p Fq(\))295 b(+)g Fj(ld)p Fq(\()p Fo(R)11 b Fq(\()p Fo(x)p Fq(\)\))p Fo(:)p Black 2737 24943 a Fk(\262)p Black 651 w Fo(x)4791 24461 y Fn(0)5470 24943 y Fq(=)369 b Fo(A)p Fq(\()p Fo(x)p Fq(\))p Fo(x)295 b Fq(+)g Fo(P)181 b Fq(\()p Fo(x)p Fq(\)\()p Fo(R)11 b Fq(\()p Fo(x)p Fq(\)\))18452 24461 y Fn(0)19131 24943 y Fq(=)368 b Fo(x)p Fq(\()p Fo(A)p Fq(\()p Fo(x)p Fq(\))295 b(+)g Fj(ld)p Fq(\()p Fo(R)11 b Fq(\()p Fo(x)p Fq(\)\)\))p Fo(:)p Black 2737 27611 a Fk(\262)p Black 651 w Fj(If)464 b Fo(x)h Fj(is)g(a)g(nonzer)-66 b(o)464 b(inte)-66 b(ger,)462 b(then)19882 30389 y Fo(x)p Fk(j)p Fo(x)21729 29841 y Fn(0)22409 30389 y Fk(,)369 b Fj(ld)q Fq(\()p Fo(x)p Fq(\))f Fk(2)h Fr(Z)g Fk(,)h Fo(R)11 b Fq(\()p Fo(x)p Fq(\))368 b(=)h(1)p Fo(:)p Black 2737 33699 a Fk(\262)p Black 651 w Fj(if)5314 32623 y Fi(\241)6055 33176 y Fm(a)p 6055 33394 500 54 v 6104 34158 a(b)6688 32623 y Fi(\242)7297 32921 y Fn(0)8072 33699 y Fj(is)465 b(an)g(inte)-66 b(ger,)462 b(and)j Fq(gcd\()p Fo(a;)221 b(b)p Fq(\))369 b(=)f(1)466 b Fj(then)e Fo(R)11 b Fq(\()p Fo(b)p Fq(\))368 b(=)h(1)p Fo(:)800 36279 y Fj(Pr)-66 b(o)g(of.)597 b Fq(First)433 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)8524 39058 y(ld\()p Fo(x)p Fq(\))369 b(=)g(ld\()p Fo(P)181 b Fq(\()p Fo(x)p Fq(\))p Fo(R)11 b Fq(\()p Fo(x)p Fq(\)\))368 b(=)g(ld\()p Fo(P)181 b Fq(\()p Fo(x)p Fq(\)\))295 b(+)g(ld)o(\()p Fo(R)11 b Fq(\()p Fo(x)p Fq(\)\))369 b(=)f Fo(A)p Fq(\()p Fo(x)p Fq(\))295 b(+)g(ld\()p Fo(R)11 b Fq(\()p Fo(x)p Fq(\)\))p Fo(:)800 41836 y Fq(Using)434 b(this)f(w)-36 b(e)434 b(get)10384 44615 y Fo(x)11123 44067 y Fn(0)11803 44615 y Fq(=)368 b Fo(x)p Fq(ld\()p Fo(x)p Fq(\))h(=)g Fo(x)p Fq(\()p Fo(A)p Fq(\()p Fo(x)p Fq(\))295 b(+)f(ld\()p Fo(R)11 b Fq(\()p Fo(x)p Fq(\)\)\))368 b(=)h Fo(xA)p Fq(\()p Fo(x)p Fq(\))295 b(+)g Fo(x)p Fq(ld\()p Fo(R)11 b Fq(\()p Fo(x)p Fq(\)\))368 b(=)12072 47394 y Fo(xA)p Fq(\()p Fo(x)p Fq(\))295 b(+)f Fo(P)181 b Fq(\()p Fo(x)p Fq(\))p Fo(R)11 b Fq(\()p Fo(x)p Fq(\)ld\()p Fo(R)g Fq(\()p Fo(x)p Fq(\)\))368 b(=)h Fo(A)p Fq(\()p Fo(x)p Fq(\))p Fo(x)295 b Fq(+)g Fo(P)181 b Fq(\()p Fo(x)p Fq(\)\()p Fo(R)11 b Fq(\()p Fo(x)p Fq(\)\))40858 46845 y Fn(0)41167 47394 y Fo(:)800 49652 y Fq(If)434 b Fo(R)11 b Fq(\()p Fo(x)p Fq(\))369 b Fk(6)p Fq(=)g(1)433 b(then)g(the)g(sum)21785 52503 y(ld)o(\()p Fo(R)11 b Fq(\()p Fo(x)p Fq(\)\))369 b(=)29084 50843 y Fm(k)28381 51241 y Fi(X)28579 54041 y Fm(i)p Fl(=1)30688 51605 y Fo(r)31274 51804 y Fm(i)p 30654 52198 1029 54 v 30654 53414 a Fo(p)31307 53613 y Fm(i)800 55860 y Fq(cannot)433 b(b)36 b(e)434 b(an)f(in)-36 b(teger.)578 b(Otherwise)19483 59811 y(ld\()p Fo(R)11 b Fq(\()p Fo(x)p Fq(\)\))25143 58150 y Fm(k)24551 58549 y Fi(Y)24639 61348 y Fm(i)p Fl(=1)26470 59811 y Fo(p)27123 60010 y Fm(i)27868 59811 y Fq(=)29951 58150 y Fm(k)29248 58549 y Fi(X)29446 61348 y Fm(i)p Fl(=1)31388 59811 y Fo(r)31974 60010 y Fm(i)32350 59811 y Fo(Q)33380 60010 y Fm(i)33756 59811 y Fo(;)800 63688 y Fq(and)471 b(if)h(0)434 b Fo(<)f(r)7712 63887 y Fm(j)8633 63688 y Fo(<)g(p)10731 63887 y Fm(j)11689 63688 y Fq(then)470 b(an)i(in)-36 b(teger)471 b(on)h(the)e(left)j(hand)d(side)i(is)f(divisible)i(b)-36 b(y)471 b Fo(p)42226 63887 y Fm(j)42713 63688 y Fo(;)h Fq(but)e(on)h(the)g(righ)-36 b(t)800 65710 y(hand)533 b(side)h(is)g(not)g(b)36 b(ecause)534 b Fo(Q)16625 65909 y Fm(j)17652 65710 y Fq(=)19336 64357 y Fb(Q)20225 64631 y Fh(k)20225 65298 y(i)p Fd(=1)21760 65022 y Fm(p)22234 65157 y Fh(i)p 19336 65405 3249 54 v 20507 66168 a Fm(p)20981 66303 y Fh(j)23251 65710 y Fq(and)g(all)h(primes)e Fo(p)32739 65909 y Fm(i)33649 65710 y Fq(are)h(di\256eren)-36 b(t.)878 b(The)534 b(last)h(statemen)-36 b(t)800 67466 y(follo)g(ws)436 b(from)e(Theorem)p 0 1 0 0 TeXcolorcmyk 433 w(14)p (#Thm.14) [[195 110 206 122] [1 1 1 [3 3]] [0 0 1]] pdfm Black(.)p 15873 67466 630 630 v 2751 69691 a(No)-36 b(w)410 b(w)-36 b(e)409 b(are)h(able)g(to)f(solv)-36 b(e)410 b(the)f(equation)h Fo(x)25414 69209 y Fn(0)26093 69691 y Fq(=)369 b Fo(\256)8 b(x)410 b Fq(with)f(rational)i Fo(\256)418 b Fq(in)409 b(the)f(rationals.)572 b(W)-108 b(e)409 b(ha)-36 b(v)g(e)800 71296 y(already)435 b(solv)-36 b(ed)434 b(this)f(equation)h(in)g(the)f(case)h Fo(\256)377 b Fq(=)369 b(0)p Fo(;)434 b Fq(so)g(let)g Fo(\256)378 b Fk(6)p Fq(=)368 b(0)p Fo(:)p Black 26150 74617 a Fq(14)p Black eop %%Page: 15 15 15 14 bop Black 0 TeXcolorgray Black Black Black 800 1424 a Fr(Theorem)499 b(18)p Black 651 w Fj(L)-66 b(et)464 b Fo(\256)378 b Fq(=)14240 901 y Fm(a)p 14240 1119 500 54 v 14289 1882 a(b)15338 1424 y Fj(b)-66 b(e)464 b(a)h(r)-66 b(ational)464 b(numb)-66 b(er)463 b(with)i Fq(gcd\()p Fo(a;)221 b(b)p Fq(\))369 b(=)g(1)p Fo(;)221 b(b)370 b(>)e Fq(0)p Fo(:)466 b Fj(Then)p Black 2737 4136 a Fk(\262)p Black 651 w Fj(The)e(e)-66 b(quation)26239 5741 y Fo(x)26978 5193 y Fn(0)27657 5741 y Fq(=)369 b Fo(\256)8 b(x)p Black 20526 w Fq(\(6\))p Black 4052 8066 a Fj(has)423 b(nonzer)-66 b(o)421 b(r)-66 b(ational)422 b(solutions)g(if)g(and)g(only)g(if)f Fo(b)i Fj(is)f(a)g(pr)-66 b(o)g(duct)422 b(of)g(di\256er)-66 b(ent)421 b(primes)g(or)i Fo(b)369 b Fq(=)f(1)p Fo(:)p Black 2737 10778 a Fk(\262)p Black 651 w Fj(If)433 b Fo(x)6124 10977 y Fl(0)7084 10778 y Fj(is)h(a)h(nonzer)-66 b(o)433 b(p)-66 b(articular)434 b(solution)g(\()p 0 1 0 0 TeXcolorcmyk(6)p (#equation.6) [[303 620 309 632] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))h(and)g Fo(y)482 b Fj(is)434 b(any)g(r)-66 b(ational)433 b(solution)h(of)g(the)g(e)-66 b(quation)4052 12383 y Fo(y)4734 11901 y Fn(0)5490 12383 y Fq(=)445 b(0)507 b Fj(then)f Fo(x)445 b Fq(=)h Fo(x)14416 12582 y Fl(0)14942 12383 y Fo(y)554 b Fj(is)506 b(also)h(a)g(solution)f(of)g(\()p 0 1 0 0 TeXcolorcmyk(6)p (#equation.6) [[328 606 334 618] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))h(and)g(any)f(solution)g(of)g(\()p 0 1 0 0 TeXcolorcmyk(6)p (#equation.6) [[453 606 459 618] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))i(c)-66 b(an)505 b(b)-66 b(e)506 b(obtaine)-66 b(d)4052 13988 y(in)464 b(this)g(manner.)p Black 2737 16700 a Fk(\262)p Black 651 w Fj(T)-100 b(o)472 b(obtain)f(a)i(p)-66 b(articular)471 b(solution)h(of)g(the)g(e)-66 b(quation)471 b(\()p 0 1 0 0 TeXcolorcmyk(6)p (#equation.6) [[355 567 361 579] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))i(it)e(is)h(su\261cient)f(to)h(de)-66 b(c)g(omp)g(ose)471 b Fo(\256)481 b Fj(into)4052 18305 y(the)464 b(elementary)g(fr)-66 b(actions:)22077 21082 y Fo(\256)378 b Fq(=)24795 20184 y Fo(a)p 24795 20777 683 54 v 24860 21994 a(b)25980 21082 y Fq(=)369 b Fk(b)p Fo(\256)8 b Fk(c)296 b Fq(+)31682 19422 y Fm(k)30979 19821 y Fi(X)31177 22620 y Fm(i)p Fl(=1)33298 20184 y Fo(c)33858 20383 y Fm(i)p 33252 20777 1029 54 v 33252 21994 a Fo(p)33905 22193 y Fm(i)34413 21082 y Fo(;)4052 24844 y Fj(wher)-66 b(e)464 b Fo(b)369 b Fq(=)10024 23847 y Fi(Q)11278 24198 y Fm(k)11278 25231 y(i)p Fl(=1)13078 24844 y Fo(p)13731 25043 y Fm(i)14106 24844 y Fo(;)221 b Fq(1)370 b Fk(\267)f(j)p Fo(c)18039 25043 y Fm(i)18415 24844 y Fk(j)g Fo(<)g(p)21187 25043 y Fm(i)21562 24844 y Fo(:)465 b Fj(Then)24042 29078 y Fo(x)24781 29277 y Fl(0)25676 29078 y Fq(=)368 b(4)27706 28530 y Fn(b)p Fm(\256)p Fn(c)30016 27418 y Fm(k)29424 27817 y Fi(Y)29512 30616 y Fm(i)p Fl(=1)31343 29078 y Fo(p)31996 28486 y Fm(c)32404 28621 y Fh(i)31996 29419 y Fm(i)4052 33184 y Fj(is)487 b(a)h(p)-66 b(articular)487 b(solution.)665 b(\(Of)488 b(c)-66 b(ourse)487 b(the)g(numb)-66 b(er)487 b Fq(4)g Fj(c)-66 b(an)487 b(b)-66 b(e)487 b(r)-66 b(eplac)g(e)g(d)486 b(by)h Fo(p)43901 32702 y Fm(p)44917 33184 y Fj(for)h(any)f(prime)4052 34789 y Fo(p)p Fj(\).)800 37501 y(Pr)-66 b(o)g(of.)597 b Fq(The)433 b(equation)h(\()p 0 1 0 0 TeXcolorcmyk(6)p (#equation.6) [[192 379 198 391] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))g(is)g(equiv)-72 b(alen)-36 b(t)434 b(to)f(the)h(equation) 15647 40743 y(ld\()p Fo(x)p Fq(\))369 b(=)f Fo(\256)378 b Fk(,)369 b Fo(A)p Fq(\()p Fo(x)p Fq(\))295 b(+)g(ld\()p Fo(R)11 b Fq(\()p Fo(x)p Fq(\)\))368 b(=)h Fo(\256)378 b Fq(=)36776 39844 y Fo(a)p 36776 40437 683 54 v 36841 41654 a(b)37592 40743 y(:)800 44354 y Fq(Because)485 b Fo(A)p Fq(\()p Fo(x)p Fq(\))f(is)g(an)g(in)-36 b(teger)485 b(and)e(ld\()p Fo(R)11 b Fq(\()p Fo(x)p Fq(\)\))455 b(=)25951 43358 y Fi(P)27353 43708 y Fm(k)27353 44741 y(i)p Fl(=1)29310 43807 y Fm(r)29735 43942 y Fh(i)p 29285 44048 825 54 v 29285 44812 a Fm(p)29759 44947 y Fh(i)30243 44354 y Fo(;)485 b Fq(the)e(natural)h(n)-36 b(um)g(b)36 b(er)483 b Fo(b)h Fq(should)g(b)36 b(e)484 b(equal)800 45959 y(to)392 b(the)f(pro)36 b(duct)391 b(of)h(the)f(di\256eren)-36 b(t)391 b(primes)g(or)h(should)f(b)36 b(e)391 b(equal)i(to)e(1)p Fo(:)i Fq(Supp)36 b(ose)390 b(that)h Fo(b)h Fq(is)g(of)g(this)g(t)-36 b(yp)36 b(e.)800 47564 y(Then)16525 50083 y(ld)17830 47811 y Fi(\303)18882 50083 y Fq(4)19532 49534 y Fn(b)p Fm(\256)p Fn(c)21842 48423 y Fm(k)21250 48821 y Fi(Y)21337 51620 y Fm(i)p Fl(=1)23169 50083 y Fo(p)23822 49490 y Fm(c)24230 49625 y Fh(i)23822 50423 y Fm(i)24636 47811 y Fi(!)26056 50083 y Fq(=)369 b Fk(b)p Fo(\256)8 b Fk(c)296 b Fq(+)31758 48423 y Fm(k)31055 48821 y Fi(X)31253 51620 y Fm(i)p Fl(=1)33375 49184 y Fo(c)33935 49383 y Fm(i)p 33328 49777 1029 54 v 33328 50994 a Fo(p)33981 51193 y Fm(i)34859 50083 y Fq(=)368 b Fo(\256)800 53557 y Fq(and)433 b(w)-36 b(e)434 b(obtain)f(a)h(desired)f(particular)h(solution.)579 b(If)434 b Fo(y)28196 53075 y Fn(0)28875 53557 y Fq(=)369 b(0)434 b(and)f Fo(x)34608 53756 y Fl(0)35567 53557 y Fq(an)-36 b(y)434 b(particular)g(solution)f(then)18941 56491 y(\()p Fo(x)20186 56690 y Fl(0)20712 56491 y Fo(y)48 b Fq(\))21900 55942 y Fn(0)22579 56491 y Fq(=)369 b Fo(x)24699 55942 y Fn(0)24699 56819 y Fl(0)25224 56491 y Fo(y)344 b Fq(+)294 b Fo(x)28247 56690 y Fl(0)28773 56491 y Fo(y)29455 55942 y Fn(0)30135 56491 y Fq(=)368 b Fo(\256)8 b(x)33089 56690 y Fl(0)33616 56491 y Fo(y)48 b(;)800 59424 y Fq(also)435 b(satis\257es)e(\()p 0 1 0 0 TeXcolorcmyk(6)p (#equation.6) [[150 182 156 194] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\).)579 b(Finally)-108 b(,)434 b(if)h Fo(x)17411 58942 y Fn(0)18090 59424 y Fq(=)369 b Fo(\256)8 b(x)434 b Fq(and)f Fo(y)417 b Fq(=)26803 58901 y Fm(x)p 26573 59119 991 54 v 26573 59882 a(x)27103 60005 y Fd(0)28130 59424 y Fq(then)19427 62358 y(ld\()p Fo(y)48 b Fq(\))369 b(=)f(ld\()p Fo(x)p Fq(\))295 b Fk(\241)g Fq(ld\()p Fo(x)30741 62557 y Fl(0)31267 62358 y Fq(\))369 b(=)f(0)800 65291 y(means)434 b(that)f Fo(y)481 b Fq(is)434 b(a)g(solution)g(of)g(the)f (equation)h Fo(y)25742 64809 y Fn(0)26421 65291 y Fq(=)369 b(0)p Fo(:)p 29247 65291 630 630 v 2751 67561 a Fq(F)-108 b(or)537 b(instance)f(the)h(equation)g Fo(x)18850 67078 y Fn(0)19706 67561 y Fq(=)21395 67038 y Fm(x)p 21395 67255 530 54 v 21425 68019 a Fl(4)22595 67561 y Fq(has)g(no)g (solutions,)563 b Fo(x)33591 67760 y Fl(0)34662 67561 y Fq(=)36351 67038 y Fl(2)p 36351 67255 471 54 v 36351 68019 a(3)37492 67561 y Fq(is)537 b(a)g(partial)h(solution)f(of)h(the) 800 69166 y(equation)430 b Fo(x)6846 68684 y Fn(0)7526 69166 y Fq(=)9039 68643 y Fm(x)p 9039 68860 530 54 v 9069 69624 a Fl(6)10132 69166 y Fq(and)f(to)h(obtain)f(all)i(nonzero)e (solutions)h(w)-36 b(e)430 b(need)f(to)h(m)-36 b(ultiply)430 b Fo(x)42821 69365 y Fl(0)43777 69166 y Fq(with)f(an)-36 b(y)430 b Fo(y)478 b Fq(suc)-36 b(h)800 70771 y(that)433 b Fo(R)11 b Fq(\()p Fo(y)48 b Fq(\))369 b(=)f(1)p Fo(;)221 b(A)p Fq(\()p Fo(y)48 b Fq(\))370 b(=)e(0)p Fo(:)p Black 26150 74617 a Fq(15)p Black eop %%Page: 16 16 16 15 bop Black 0 TeXcolorgray Black Black 800 1424 a Fp(9)2152 b(Ho)-60 b(w)717 b(to)g(di\256eren)-60 b(tiate)716 b(irrational)g(n)-60 b(um)g(b)60 b(ers)800 4345 y Fq(The)580 b(next)f(step)g(is)h(to)g(try)g(to)f(generalize)i(our)e(de\257nition)g (to)h(irrational)g(n)-36 b(um)g(b)36 b(ers.)1015 b(The)580 b(equation)800 5950 y(\()p 0 1 0 0 TeXcolorcmyk(1)p (#equation.1) [[84 663 90 675] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))542 b(can)g(still)g(b)36 b(e)542 b(used)f(in)h(the)f(more)h (general)h(situation.)903 b(But)542 b(\257rst)f(w)-36 b(e)542 b(need)f(to)h(think)g(ab)36 b(out)542 b(the)800 7555 y(correctness)433 b(of)i(the)e(de\257nition.)p Black 800 9903 a Fr(Lemma)499 b(1)p Black 651 w Fj(L)-66 b(et)499 b Fk(f)p Fo(p)11088 10102 y Fl(1)11614 9903 y Fo(;)221 b(:)g(:)g(:)j(;)d(p)15180 10102 y Fm(k)15749 9903 y Fk(g)501 b Fj(b)-66 b(e)500 b(a)h(set)g(of)f(di\256er)-66 b(ent)499 b(primes)i(and)f Fk(f)p Fo(x)36800 10102 y Fl(1)37327 9903 y Fo(;)221 b(:)g(:)g(:)i(;)e(x)40978 10102 y Fm(k)41548 9903 y Fk(g)501 b Fj(a)g(set)g(of)f(r)-66 b(ationals.)800 11508 y(Then)14730 14027 y Fo(P)550 b Fq(=)18090 12367 y Fm(k)17499 12766 y Fi(Y)17586 15565 y Fm(i)p Fl(=1)19417 14027 y Fo(p)20070 13435 y Fm(x)20600 13570 y Fh(i)20070 14368 y Fm(i)21375 14027 y Fq(=)369 b(1)g Fk(,)h Fo(x)26212 14226 y Fl(1)27107 14027 y Fq(=)e Fo(x)29226 14226 y Fl(2)30121 14027 y Fq(=)h Fk(\242)221 b(\242)g(\242)369 b Fq(=)g Fo(x)35540 14226 y Fm(k)36478 14027 y Fq(=)g(0)p Fo(:)800 17474 y Fj(Pr)-66 b(o)g(of.)569 b Fq(It)343 b(is)h(eviden)-36 b(t)343 b(if)g(all)i Fo(x)15347 17673 y Fm(i)16066 17474 y Fq(are)e(in)-36 b(tegers,)361 b(b)36 b(ecause)344 b(the)e(primes)h(are)g(di\256eren)-36 b(t.)548 b(If)343 b(they)h(are)f(rational,)800 19079 y(let)455 b(us)g(c)-36 b(ho)36 b(ose)455 b(a)g(natural)g Fo(m)g Fq(suc)-36 b(h)454 b(that)g(all)i Fo(y)24066 19278 y Fm(i)24847 19079 y Fq(=)405 b Fo(mx)28141 19278 y Fm(i)28972 19079 y Fq(are)455 b(in)-36 b(teger.)642 b(Then)454 b Fo(P)40466 18597 y Fm(m)41759 19079 y Fq(=)405 b(1)456 b(to)36 b(o)455 b(and)g(w)-36 b(e)455 b(get)800 20685 y Fo(y)1434 20884 y Fm(i)2179 20685 y Fq(=)369 b(0)g Fk(\))g Fo(x)7015 20884 y Fm(i)7760 20685 y Fq(=)g(0)p Fo(:)p 10586 20685 630 630 v 2751 22833 a Fq(No)-36 b(w)395 b(w)-36 b(e)395 b(can)f(extend)h(our)f(de\257nition)f(to)i(an)-36 b(y)395 b(real)g(n)-36 b(um)g(b)36 b(er)393 b Fo(x)h Fq(that)g(can)h(b)36 b(e)394 b(written)g(as)h(a)g(pro)36 b(duct)800 24438 y Fo(x)566 b Fq(=)3683 23441 y Fi(Q)4937 23792 y Fm(k)4937 24825 y(i)p Fl(=1)6737 24438 y Fo(p)7390 23845 y Fm(x)7920 23980 y Fh(i)7390 24778 y Fm(i)8875 24438 y Fq(for)550 b(di\256eren)-36 b(t)548 b(primes)h Fo(p)21191 24637 y Fm(i)22116 24438 y Fq(and)g(some)g(nonzero)g (rationals)h Fo(x)39301 24637 y Fm(i)39677 24438 y Fq(.)926 b(The)549 b(previous)h(lemma)800 26043 y(sho)-36 b(ws)434 b(that)f(this)g(form)h(is)g(unique)f(and)g(as)h(ab)36 b(o)-36 b(v)g(e)435 b(w)-36 b(e)433 b(can)h(de\257ne)22979 29724 y Fo(x)23718 29176 y Fn(0)24398 29724 y Fq(=)368 b Fo(x)27442 28064 y Fm(k)26739 28462 y Fi(X)26937 31262 y Fm(i)p Fl(=1)29012 28826 y Fo(x)29751 29025 y Fm(i)p 29012 29419 1115 54 v 29055 30636 a Fo(p)29708 30835 y Fm(i)30260 29724 y Fo(:)800 33332 y Fq(The)434 b(pro)36 b(of)434 b(for)g(the)f(Leibnitz)g(rule)g(is)h(still)g(v)-72 b(alid)435 b(to)36 b(o)434 b(and)f(w)-36 b(e)434 b(skip)g(it.)578 b(F)-108 b(or)433 b(example)i(w)-36 b(e)434 b(ha)-36 b(v)g(e)17557 36709 y(\()18063 35545 y Fk(p)p 19170 35545 651 54 v 1164 x Fq(3\))20326 36161 y Fn(0)21006 36709 y Fq(=)368 b(\(3)23542 36161 y Fl(1)p Fm(=)p Fl(2)25009 36709 y Fq(\))25515 36161 y Fn(0)26195 36709 y Fq(=)g(3)28225 36161 y Fl(1)p Fm(=)p Fl(2)29825 35810 y Fq(1)p Fo(=)p Fq(2)p 29825 36404 1951 54 v 30475 37620 a(3)32278 36709 y(=)33791 34712 y Fk(p)p 34898 34712 651 54 v 1098 x Fq(3)p 33791 36404 1758 54 v 34345 37620 a(6)35682 36709 y Fo(:)800 39558 y Fq(More)434 b(generally)g(w)-36 b(e)434 b(ha)-36 b(v)g(e)434 b(the)f(follo)-36 b(wing)436 b(con)-36 b(v)g(enien)g(t)433 b(form)-36 b(ula:)p Black 800 41906 a Fr(Theorem)499 b(19)p Black 651 w Fj(L)-66 b(et)464 b Fo(x;)221 b(y)513 b Fj(b)-66 b(e)465 b(r)-66 b(ationals)464 b(and)h Fo(x)g Fj(b)-66 b(e)464 b(p)-66 b(ositive.)596 b(Then)16571 45052 y Fq(\()p Fo(x)17816 44504 y Fm(y)18368 45052 y Fq(\))18874 44504 y Fn(0)19554 45052 y Fq(=)368 b Fo(y)48 b(x)22355 44504 y Fm(y)32 b Fn(\241)p Fl(1)24110 45052 y Fo(x)24849 44504 y Fn(0)25529 45052 y Fq(=)27042 44154 y Fo(y)48 b(x)28463 43672 y Fn(0)p 27042 44747 1732 54 v 27538 45964 a Fo(x)28906 45052 y(x)29645 44504 y Fm(y)30567 45052 y Fq(=)368 b Fo(y)48 b(x)33368 44504 y Fm(y)33920 45052 y Fj(ld)q Fq(\()p Fo(x)p Fq(\))p Fo(:)p Black 14109 w Fq(\(7\))p Black 800 48239 a Fj(Pr)-66 b(o)g(of.)597 b Fq(If)434 b Fo(x)369 b Fq(=)8608 47243 y Fi(Q)9863 47593 y Fm(k)9863 48627 y(i)p Fl(=1)11662 48239 y Fo(p)12315 47647 y Fm(x)12845 47782 y Fh(i)12315 48580 y Fm(i)13252 48239 y Fo(;)433 b Fq(then)10352 52363 y(\()p Fo(x)11597 51815 y Fm(y)12149 52363 y Fq(\))12655 51815 y Fn(0)13334 52363 y Fq(=)369 b(\()15812 50703 y Fm(k)15221 51101 y Fi(Y)15308 53901 y Fm(i)p Fl(=1)17139 52363 y Fo(p)17792 51734 y Fm(y)32 b(x)18819 51869 y Fh(i)17792 52704 y Fm(i)19225 52363 y Fq(\))19731 51815 y Fn(0)20411 52363 y Fq(=)368 b Fo(x)22530 51815 y Fm(y)24006 50703 y(k)23304 51101 y Fi(X)23502 53901 y Fm(i)p Fl(=1)25577 51465 y Fo(y)48 b(x)26998 51664 y Fm(i)p 25577 52058 1797 54 v 25961 53274 a Fo(p)26614 53473 y Fm(i)27875 52363 y Fq(=)369 b Fo(y)48 b(x)30677 51815 y Fm(y)32 b Fn(\241)p Fl(1)32431 52363 y Fo(x)34094 50703 y Fm(k)33392 51101 y Fi(X)33590 53901 y Fm(i)p Fl(=1)35665 51465 y Fo(x)36404 51664 y Fm(i)p 35665 52058 1115 54 v 35708 53274 a Fo(p)36361 53473 y Fm(i)37281 52363 y Fq(=)369 b Fo(y)48 b(x)40083 51815 y Fm(y)32 b Fn(\241)p Fl(1)41838 52363 y Fo(x)42577 51815 y Fn(0)42887 52363 y Fo(:)p 800 55312 630 630 v 2751 57460 a Fq(An)433 b(in)-36 b(teresting)433 b(corollary)j(is)p Black 800 60020 a Fr(Corollary)500 b(4)p Black 650 w Fj(L)-66 b(et)464 b Fo(a;)221 b(b;)g(c;)g(d)467 b Fj(b)-66 b(e)464 b(r)-66 b(ationals)464 b(such)i(that)e Fo(a)29015 59538 y Fm(b)29842 60020 y Fq(=)369 b Fo(c)31783 59538 y Fm(d)32787 60020 y Fj(\()p Fo(a;)221 b(c)465 b Fj(b)-66 b(eing)463 b(p)-66 b(ositive\).)596 b(Then)21722 62529 y Fo(b)295 b Fk(\242)h Fj(ld)p Fq(\()p Fo(a)p Fq(\))368 b(=)h Fo(d)295 b Fk(\242)g Fj(ld)p Fq(\()p Fo(c)p Fq(\))800 65038 y Fj(and)23577 66643 y Fo(a)24260 66095 y Fn(0)24570 66643 y Fo(bc)369 b Fq(=)g Fo(c)27993 66095 y Fn(0)28303 66643 y Fo(ad:)800 68786 y Fj(In)464 b(p)-66 b(articular,)464 b(for)h(the)f(c)-66 b(ase)465 b Fo(a)369 b Fq(=)f Fo(b;)221 b(c)370 b Fq(=)e Fo(d)p Fj(,)465 b(we)g(have)21775 71296 y Fo(a)22458 70747 y Fm(a)23381 71296 y Fq(=)369 b Fo(c)25322 70747 y Fm(c)26154 71296 y Fk(\))g Fo(a)28534 70747 y Fn(0)29213 71296 y Fq(=)g Fo(c)31154 70747 y Fn(0)31464 71296 y Fo(:)p Black 26150 74617 a Fq(16)p Black eop %%Page: 17 17 17 16 bop Black 0 TeXcolorgray Black Black 800 1424 630 630 v 2751 3635 a Fq(As)434 b(an)f(example)h(w)-36 b(e)434 b(can)g(c)-36 b(hec)g(k)433 b(directly)h(that)f Fo(x)27667 3153 y Fm(y)28589 3635 y Fq(=)368 b Fo(y)30651 3153 y Fm(x)31670 3635 y Fq(has)433 b(the)g(solutions)16546 7275 y Fo(x)370 b Fq(=)19035 5402 y Fi(\265)20146 6376 y Fo(m)295 b Fq(+)g(1)p 20146 6969 3391 54 v 21272 8186 a Fo(m)23669 5402 y Fi(\266)24647 5699 y Fm(m)25756 7275 y Fq(;)221 b Fo(y)417 b Fq(=)28770 5402 y Fi(\265)29880 6376 y Fo(m)295 b Fq(+)g(1)p 29880 6969 V 31006 8186 a Fo(m)33403 5402 y Fi(\266)34381 5699 y Fm(m)p Fl(+1)36692 7275 y Fo(;)800 10694 y Fq(th)-36 b(us)24483 11779 y Fo(x)25222 11297 y Fn(0)p 24375 12372 1265 54 v 24375 13589 a Fo(x)25114 13205 y Fl(2)26142 12677 y Fq(=)27763 11779 y Fo(y)28445 11297 y Fn(0)p 27656 12372 1208 54 v 27656 13589 a Fo(y)28338 13205 y Fl(2)28996 12677 y Fo(;)800 15853 y Fq(so)434 b(the)f(equation)h Fo(x)10687 15371 y Fn(0)11367 15853 y Fq(=)12880 15330 y Fm(x)13410 15018 y Fd(2)p 12880 15548 991 54 v 13140 16311 a Fl(4)14437 15853 y Fq(has)g(at)g(least)g(t)-36 b(w)g(o)434 b(solutions)f(obtained) h(from)g Fo(m)368 b Fq(=)h(1.)2751 17458 y(Another)433 b(example)h(is)13176 21019 y(\(1)p Fo(=)p Fq(2\))16138 20471 y Fl(1)p Fm(=)p Fl(2)17975 21019 y Fq(=)368 b(\(1)p Fo(=)p Fq(4\))22317 20471 y Fl(1)p Fm(=)p Fl(4)24154 21019 y Fk(\))25851 19146 y Fi(\265)26962 20120 y Fq(1)p 26962 20714 651 54 v 26962 21930 a(2)27745 19146 y Fi(\266)28723 19444 y Fn(0)29402 21019 y Fq(=)30783 19146 y Fi(\265)31894 20120 y Fq(1)p 31894 20714 V 31894 21930 a(4)32677 19146 y Fi(\266)33655 19444 y Fn(0)34334 21019 y Fq(=)h Fk(\241)36969 19146 y Fi(\265)38080 20120 y Fq(1)p 38080 20714 V 38080 21930 a(4)38863 19146 y Fi(\266)40062 21019 y Fo(:)800 24439 y Fq(In)425 b(general)h(it)g(is)g(not)f(di\261cult)g(to)h(pro)-36 b(v)g(e)426 b(that)f(all)h(rational)g(solutions)g(of)h(the)e(equation)h Fo(x)46241 23956 y Fm(x)47195 24439 y Fq(=)368 b Fo(y)49257 23956 y Fm(y)50235 24439 y Fq(ha)-36 b(v)g(e)800 26044 y(the)433 b(form)16838 28301 y Fo(x)369 b Fq(=)19327 26428 y Fi(\265)21563 27402 y Fo(m)p 20437 27995 3391 54 v 20437 29212 a(m)295 b Fq(+)g(1)23960 26428 y Fi(\266)24938 26725 y Fm(m)26047 28301 y Fo(;)221 b(y)417 b Fq(=)29061 26428 y Fi(\265)31298 27402 y Fo(m)p 30172 27995 V 30172 29212 a(m)295 b Fq(+)f(1)33695 26428 y Fi(\266)34672 26725 y Fm(m)p Fl(+1)800 31228 y Fq(for)434 b(some)g(natural)f Fo(m:)h Fq(Direct)g(calculations)h(giv)-36 b(e)434 b(the)f(same)h (result)f(as)h(ab)36 b(o)-36 b(v)g(e:)6920 34867 y Fo(x)7659 34319 y Fn(0)8339 34867 y Fq(=)368 b Fo(m)11078 32994 y Fi(\265)13315 33969 y Fo(m)p 12189 34562 V 12189 35779 a(m)295 b Fq(+)g(1)15712 32994 y Fi(\266)16690 33292 y Fm(m)p Fn(\241)p Fl(1)19001 32994 y Fi(\265)21238 33969 y Fo(m)p 20112 34562 V 20112 35779 a(m)g Fq(+)g(1)23635 32994 y Fi(\266)24612 33292 y Fn(0)25292 34867 y Fq(=)369 b(\()o Fo(m)295 b Fq(+)g(1\))31296 32994 y Fi(\265)33533 33969 y Fo(m)p 32406 34562 V 32406 35779 a(m)g Fq(+)g(1)35929 32994 y Fi(\266)36907 33292 y Fm(m)38016 32994 y Fi(\265)40253 33969 y Fo(m)p 39127 34562 V 39127 35779 a(m)g Fq(+)f(1)42650 32994 y Fi(\266)43627 33292 y Fn(0)44307 34867 y Fq(=)369 b Fo(y)46370 34319 y Fn(0)800 38287 y Fq(and)433 b(sho)-36 b(ws)434 b(that)f(this)g(w)-36 b(orks)435 b(ev)-36 b(en)433 b(for)i(rational)f Fo(m:)2751 39892 y Fq(It)611 b(w)-36 b(ould)611 b(b)36 b(e)611 b(natural)g(to)g(extend)g(our)f(de\257nition) g(to)i(in\257nite)e(pro)36 b(ducts:)932 b(if)612 b Fo(x)671 b Fq(=)46672 38896 y Fi(Q)47926 39246 y Fn(1)47926 40279 y Fm(i)p Fl(=1)49726 39892 y Fo(p)50379 39299 y Fm(x)50909 39434 y Fh(i)50379 40232 y Fm(i)51926 39892 y Fq(is)800 41497 y(con)-36 b(v)g(ergen)g(t)455 b(then)e(it)i(is)h(easy)f(to)g(sho) -36 b(w)455 b(that)g(the)f(sum)g Fo(x)29532 40501 y Fi(P)30935 40851 y Fn(1)30935 41884 y Fm(i)p Fl(=1)32867 40950 y Fm(x)33397 41085 y Fh(i)p 32867 41191 881 54 v 32895 41955 a Fm(p)33369 42090 y Fh(i)34336 41497 y Fq(is)h(also)h(con)-36 b(v)g(ergen)g(t.)641 b(Ho)-36 b(w)g(ev)g(er,)462 b(the)800 43102 y(problem)325 b(is)h(that)f(the)g(sum)g(is)h(not)f(necessary)i (con)-36 b(v)g(ergen)g(t)325 b(to)h(zero,)347 b(when)325 b Fo(x)370 b Fq(=)e(1)p Fo(:)326 b Fq(This)g(is)g(a)g(reason)g(wh)-36 b(y)800 44707 y(suc)g(h)531 b(a)i(natural)e(generalization)j(of)e(the)g (deriv)-72 b(ativ)-36 b(e)533 b(is)f(not)g(w)-36 b(ell-de\257ned.)873 b(Ma)-36 b(yb)36 b(e)532 b(a)g(more)g(natural)800 46312 y(approac)-36 b(h)560 b(is)g(to)g(restrict)g(p)36 b(ossible)560 b(pro)36 b(ducts,)591 b(but)559 b(w)-36 b(e)560 b(still)h(do)f(not)g (kno)-36 b(w)560 b(a)h(nice)f(solution)g(of)h(the)800 47917 y(problems)433 b(that)g(arise.)579 b(But)433 b(there)g(is)h (another)f(w)-36 b(a)g(y)-108 b(,)435 b(whic)-36 b(h)433 b(w)-36 b(e)434 b(consider)f(in)h(Section)p 0 1 0 0 TeXcolorcmyk 433 w(11)p (#section.11) [[471 286 483 298] [1 1 1 [3 3]] [0 0 1]] pdfm Black(.)800 52324 y Fp(10)2152 b(Arithmetic)716 b(Deriv)-120 b(ativ)-60 b(e)717 b(for)g(UFD)800 55245 y Fq(The)437 b(de\257nition)f(of)i(the)f(deriv)-72 b(ativ)-36 b(e)438 b(and)e(most)i(of)f(the)g(pro)36 b(ofs)438 b(are)f(based)g(only)h(on)f (the)f(fact)i(that)e(that)800 56850 y(ev)-36 b(ery)385 b(natural)f(n)-36 b(um)g(b)36 b(er)383 b(has)h(a)h(unique)f (factorization)i(in)-36 b(to)384 b(primes.)561 b(So)385 b(it)f(is)h(not)f(di\261cult)g(to)g(transfer)800 58455 y(it)498 b(to)h(an)f(arbitrary)g(UFD)g(\(unique)g(factorization)h (domain\))f Fo(R)509 b Fq(using)498 b(the)g(same)g(de\257nition:)707 b Fo(p)49870 57973 y Fn(0)50659 58455 y Fq(=)479 b(1)800 60060 y(for)447 b(ev)-36 b(ery)447 b(\\canonical")i(prime)d (\(irreducible\))f(elemen)-36 b(t,)450 b(the)c(Leibnitz)g(rule)g(and)g (additionally)i Fo(u)50045 59578 y Fn(0)50747 60060 y Fq(=)391 b(0)800 61665 y(for)411 b(all)g(units)e(\(in)-36 b(v)g(ertible)410 b(elemen)-36 b(ts\))410 b(in)f Fo(R)11 b(:)411 b Fq(F)-108 b(or)409 b(example)i(w)-36 b(e)411 b(can)f(do)g(it)g(for)g(a)h(p)36 b(olynomial)411 b(ring)f Fo(K)95 b Fq([)p Fo(x)p Fq(])800 63270 y(or)489 b(for)h(the)e(Gaussian) h(n)-36 b(um)g(b)36 b(ers)487 b Fo(a)333 b Fq(+)f Fo(bi:)489 b Fq(In)g(the)f(\257rst)g(case)i(the)e(canonical)i(irreducible)e(p)36 b(olynomials)800 64875 y(are)381 b(monic,)393 b(in)381 b(the)f(second)h(the)g(canonical)h(primes)f(are)g(\\p)36 b(ositiv)-36 b(e")383 b(primes)e([)p 0 1 0 0 TeXcolorcmyk(3)p (#cite.Con) [[428 133 434 145] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(].)561 b(This)381 b(leads)h(to)f(a)h(w)-36 b(ell-)800 66480 y(de\257ned)486 b(deriv)-72 b(ativ)-36 b(e)488 b(for)g(the)f(\257eld)g(of)h(fractions.)741 b(Note)487 b(also,)502 b(that)487 b(ev)-36 b(en)488 b(the)f(condition)g(UFD)h(is)f (not)800 68086 y(necessary)407 b({)g(w)-36 b(e)406 b(only)h(need)f(to)g (ha)-36 b(v)g(e)407 b(a)g(w)-36 b(ell-de\257ned)405 b(deriv)-72 b(ativ)-36 b(e,)413 b(i.e.)570 b(indep)36 b(enden)-36 b(t)404 b(of)j(factorization.)800 69691 y(W)-108 b(e)451 b(do)g(not)g(plan)g(to)g(dev)-36 b(elop)451 b(the)g(theory)g(in)g(this) g(more)g(abstract)g(direction)g(and)g(restrict)f(ourselv)-36 b(es)800 71296 y(b)g(y)434 b(the)f(follo)-36 b(wing)435 b(trivial)g(\(but)d(in)-36 b(teresting\))433 b(result.)p Black 26150 74617 a(17)p Black eop %%Page: 18 18 18 17 bop Black 0 TeXcolorgray Black Black Black 800 1424 a Fr(Theorem)499 b(20)p Black 651 w Fj(L)-66 b(et)536 b Fo(K)632 b Fj(b)-66 b(e)537 b(a)g(\257eld)g(of)g(char)-66 b(acteristic)535 b(zer)-66 b(o)537 b(and)g(with)g(the)g(derivative)e Fo(f)46155 942 y Fn(0)47003 1424 y Fj(in)h Fo(K)95 b Fq([)p Fo(x)p Fq(])538 b Fj(is)800 3164 y(de\257ne)-66 b(d)352 b(as)h(ab)-66 b(ove.)559 b(L)-66 b(et)13163 2641 y Fm(d)p 12898 2859 1014 54 v 12898 3622 a(dx)14398 3164 y Fj(b)g(e)352 b(a)h(usual)h(derivative.)558 b(Then)353 b Fo(f)30686 2682 y Fn(0)30996 3164 y Fq(\()p Fo(x)p Fq(\))369 b(=)34629 2531 y Fm(d)-157 b(f)98 b Fl(\()p Fm(x)p Fl(\))p 34629 2859 2139 54 v 35192 3622 a Fm(dx)37254 3164 y Fj(if)351 b(and)i(only)g(if)f(the)h(p)-66 b(olynomial)800 4769 y Fo(f)142 b Fq(\()p Fo(x)p Fq(\))465 b Fj(is)f(a)i(pr)-66 b(o)g(duct)464 b(of)h(line)-66 b(ar)463 b(factors.)800 7199 y(Pr)-66 b(o)g(of.)1169 b Fq(Because)641 b(b)36 b(oth)640 b(deriv)-72 b(ativ)-36 b(es)642 b(are)f(equal)h(to)f(zero)g (on)g(constan)-36 b(ts)641 b(they)f(coincide)i(on)f(linear)800 8804 y(p)36 b(olynomials.)973 b(If)566 b Fo(f)142 b Fq(\()p Fo(x)p Fq(\))564 b(has)h(no)f(linear)h(irreducible)f(factors)i(then)d Fo(f)36398 8322 y Fn(0)36709 8804 y Fq(\()p Fo(x)p Fq(\))h(has)h (smaller)g(degree)g(then)933 9911 y Fm(d)-157 b(f)98 b Fl(\()p Fm(x)p Fl(\))p 933 10238 V 1495 11002 a Fm(dx)3204 10544 y Fo(:)434 b Fq(Otherwise)f Fo(f)142 b Fq(\()p Fo(x)p Fq(\))369 b(=)g Fo(l)29 b Fq(\()p Fo(x)p Fq(\))p Fo(g)48 b Fq(\()p Fo(x)p Fq(\))433 b(for)h(some)g(linear)g(p)36 b(olynomial)435 b Fo(l)29 b Fq(\()p Fo(x)p Fq(\))434 b(and)8518 14006 y Fo(f)9301 13457 y Fn(0)9611 14006 y Fq(\()p Fo(x)p Fq(\))295 b Fk(\241)13119 13107 y Fo(d)-217 b(f)142 b Fq(\()p Fo(x)p Fq(\))p 13119 13700 2993 54 v 13908 14917 a Fo(dx)16613 14006 y Fq(=)369 b Fo(l)18410 13457 y Fn(0)18721 14006 y Fq(\()p Fo(x)p Fq(\))p Fo(g)48 b Fq(\()p Fo(x)p Fq(\))294 b(+)h Fo(l)29 b Fq(\()p Fo(x)p Fq(\))p Fo(g)27333 13457 y Fn(0)27643 14006 y Fq(\()p Fo(x)p Fq(\))295 b Fk(\241)31151 13107 y Fo(dl)29 b Fq(\()p Fo(x)p Fq(\))p 31151 13700 2844 54 v 31865 14917 a Fo(dx)34127 14006 y(g)48 b Fq(\()p Fo(x)p Fq(\))294 b Fk(\241)h Fo(l)29 b Fq(\()p Fo(x)p Fq(\))40471 13107 y Fo(dg)48 b Fq(\()p Fo(x)p Fq(\))p 40471 13700 3098 54 v 41313 14917 a Fo(dx)44070 14006 y Fq(=)20769 17952 y Fo(l)29 b Fq(\()p Fo(x)p Fq(\))23157 16079 y Fi(\265)24135 17952 y Fo(g)24806 17404 y Fn(0)25116 17952 y Fq(\()p Fo(x)p Fq(\))295 b Fk(\241)28624 17054 y Fo(dg)48 b Fq(\()p Fo(x)p Fq(\))p 28624 17647 V 29465 18864 a Fo(dx)31853 16079 y Fi(\266)800 20826 y Fq(and)433 b(w)-36 b(e)434 b(can)g(use)f(induction.)p 16162 20826 630 630 v 2751 23001 a(So,)h(for)g(the)e(complex)i(p)36 b(olynomials)435 b(b)36 b(oth)433 b(de\257nitions)g(coincide.)578 b(On)433 b(the)f(other)h(hand)g(\()p Fo(x)48628 22519 y Fl(2)49448 23001 y Fq(+)295 b Fo(x)f Fq(+)800 24606 y(1\))1956 24124 y Fn(0)2825 24606 y Fq(=)4793 24083 y Fm(d)p 4528 24300 1014 54 v 4528 25064 a(dx)5675 24606 y Fq(\()p Fo(x)6920 24124 y Fl(2)7817 24606 y Fq(+)371 b Fo(x)g Fq(+)f(1\))559 b(=)f(1)546 b(in)e Fr(Z)18714 24805 y Fa(2)19312 24606 y Fq([)p Fo(x)p Fq(])p Fo(;)i Fq(though)f(\()p Fo(x)27445 24124 y Fl(2)28341 24606 y Fq(+)371 b Fo(x)g Fq(+)g(1\))545 b(is)g(irreducible,)573 b(th)-36 b(us)544 b(c)-36 b(haracteristic)800 26211 y(restrictions)434 b(are)f(essen)-36 b(tial.)2751 27816 y(Let)352 b(us)g(no)-36 b(w)352 b(lo)36 b(ok)354 b(at)f(the)f(Gaussian)g(n)-36 b(um)g(b)36 b(ers.)550 b(W)-108 b(e)352 b(lea)-36 b(v)g(e)353 b(to)g(the)f(reader)g(the)f(pleasure)i(of)g(creating)800 29421 y(similar)565 b(conjectures)f(as)g(for)h(in)-36 b(tegers,)596 b(for)565 b(example)g(the)e(analogs)i(of)g(Goldbac)-36 b(h)563 b(and)g(prime)h(t)-36 b(wins)800 31026 y(conjectures)506 b(\(t)-36 b(wins)507 b(seem)f(to)h(b)36 b(e)506 b(pairs)g(with)h (distance)29853 29927 y Fk(p)p 30960 29927 651 54 v 1099 x Fq(2)g(b)36 b(et)-36 b(w)g(een)506 b(t)-36 b(w)g(o)506 b(elemen)-36 b(ts;)543 b(more)507 b(history)800 32631 y(and)433 b(v)-72 b(arian)-36 b(ts)434 b(can)g(b)36 b(e)433 b(found)g(in)g(\\The)h(Gaussian)g(zo)36 b(o")434 b([)p 0 1 0 0 TeXcolorcmyk(5)p (#cite.ref:Zoo) [[341 423 346 435] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(]\).)578 b(W)-108 b(e)434 b(go)g(in)-36 b(to)434 b(another)f(direction.)2751 34236 y(Note,)402 b(that)392 b(b)36 b(ecause)393 b(2)212 b(+)g Fo(i)393 b Fq(and)g(2)212 b Fk(\241)g Fo(i)393 b Fq(are)g(\\p)36 b(ositiv)-36 b(e")395 b(primes)e(and)f(5)369 b(=)g(\(2)212 b(+)g Fo(i)p Fq(\)\(2)g Fk(\241)g Fo(i)p Fq(\))p Fo(;)393 b Fq(w)-36 b(e)394 b(should)800 35841 y(ha)-36 b(v)g(e)465 b(5)4480 35359 y Fn(0)5212 35841 y Fq(=)421 b(\(2)316 b(+)g Fo(i)p Fq(\))g(+)g(\(2)g Fk(\241)g Fo(i)p Fq(\))421 b(=)g(4)p Fo(;)465 b Fq(but)f(this)g(do)36 b(es)464 b(not)h(coincide)f(with)h(the)f(earlier)h(de\257nition.)670 b(So)464 b(it)800 37446 y(ma)-36 b(y)434 b(b)36 b(e)434 b(is)f(time)h(to)g(c)-36 b(hange)433 b(our)g(p)36 b(oin)-36 b(t)434 b(of)g(view)h(radically)-108 b(.)800 41835 y Fp(11)2152 b(Generalized)716 b(deriv)-120 b(ativ)-60 b(es)800 44755 y Fq(Our)491 b(de\257nition)f(is)i(based)f(on)g(t)-36 b(w)g(o)492 b(k)-36 b(ey)492 b(p)36 b(oin)-36 b(ts)491 b({)h(the)f(Leibnitz)g(rule)g(and)g Fo(p)39639 44273 y Fn(0)40417 44755 y Fq(=)467 b(1)492 b(for)g(primes.)752 b(If)492 b(w)-36 b(e)800 46360 y(skip)406 b(the)f(second)g(one)h(and)f (use)g(the)g(Leibnitz)g(rule)g(only)i(w)-36 b(e)406 b(get)f(a)h(more)g (general)g(de\257nition)f(of)h Fo(D)36 b Fq(\()p Fo(x)p Fq(\))p Fo(:)800 47966 y Fq(No)-36 b(w,)435 b(if)f Fo(x)369 b Fq(=)7804 46969 y Fi(Q)9059 47320 y Fm(k)9059 48353 y(i)p Fl(=1)10858 47966 y Fo(p)11511 47373 y Fm(x)12041 47508 y Fh(i)11511 48306 y Fm(i)12448 47966 y Fo(;)433 b Fq(then)20493 50872 y Fo(D)36 b Fq(\()p Fo(x)p Fq(\))368 b(=)h Fo(x)26772 49212 y Fm(k)26069 49610 y Fi(X)26268 52409 y Fm(i)p Fl(=1)28342 49973 y Fo(x)29081 50172 y Fm(i)29457 49973 y Fo(D)36 b Fq(\()p Fo(p)31732 50172 y Fm(i)32108 49973 y Fq(\))p 28342 50566 4271 54 v 29964 51783 a Fo(p)30617 51982 y Fm(i)32746 50872 y Fo(;)800 54154 y Fq(and)581 b(w)-36 b(e)583 b(can)e(again)i(rep)36 b(eat)582 b(most)g(of)g(the)f(pro)36 b(ofs)583 b(ab)36 b(o)-36 b(v)g(e.)1024 b(But)581 b(it)h(is)g(m)-36 b(uc)g(h)581 b(more)h(natural)f(to)h(use)800 55759 y(another)433 b(approac)-36 b(h.)p Black 800 58188 a Fr(Theorem)499 b(21)p Black 651 w Fj(L)-66 b(et)488 b Fo(R)500 b Fj(b)-66 b(e)488 b(a)h(c)-66 b(ommutative)487 b(ring)h(without)i(zer)-66 b(o)488 b(divisors)h(and)g(let)g Fo(L)414 b Fq(:)g Fo(R)45901 57706 y Fn(\244)46840 58188 y Fk(\241)-221 b(!)414 b Fo(R)50395 57706 y Fl(+)51671 58188 y Fj(b)-66 b(e)800 59793 y(a)465 b(homomorphism)g(of)g(its)f(multiplic)-66 b(ative)463 b(semigr)-66 b(oup)464 b(to)h(the)f(additive)g(gr)-66 b(oup.)597 b(Then)465 b(a)g(map)16085 62397 y Fo(D)405 b Fq(:)369 b Fo(R)380 b Fk(\241)-221 b(!)369 b Fo(R)11 b(;)221 b(D)36 b Fq(\()p Fo(x)p Fq(\))370 b(=)f Fo(xL)p Fq(\()p Fo(x)p Fq(\))p Fo(;)221 b(D)36 b Fq(\(0\))369 b(=)g(0)800 65135 y Fj(satis\257es)560 b(the)g(L)-66 b(eibnitz)558 b(rule.)884 b(Conversely,)583 b(if)559 b Fo(D)36 b Fq(\()p Fo(xy)48 b Fq(\))546 b(=)f Fo(D)36 b Fq(\()p Fo(x)p Fq(\))p Fo(y)413 b Fq(+)366 b Fo(xD)36 b Fq(\()p Fo(y)48 b Fq(\))560 b Fj(then)f Fo(L)p Fq(\()p Fo(x)p Fq(\))546 b(=)47893 64502 y Fm(D)26 b Fl(\()p Fm(x)p Fl(\))p 47893 64830 2060 54 v 48658 65593 a Fm(x)50646 65135 y Fj(is)560 b(a)800 66740 y(homomorphism.)597 b(If)464 b Fo(R)476 b Fj(is)465 b(a)g(\257eld)g(then)f Fo(L)h Fj(is)f(a)i(gr)-66 b(oup)465 b(homomorphism)f(and)19102 70034 y Fo(D)20440 68161 y Fi(\265)21551 69135 y Fo(x)p 21551 69728 740 54 v 21580 70945 a(y)22422 68161 y Fi(\266)23769 70034 y Fq(=)25283 69135 y Fo(D)36 b Fq(\()p Fo(x)p Fq(\))p Fo(y)343 b Fk(\241)295 b Fo(xD)36 b Fq(\()p Fo(y)48 b Fq(\))p 25283 69728 8721 54 v 29039 70945 a Fo(y)29721 70561 y Fl(2)34136 70034 y Fo(:)p Black 26150 74617 a Fq(18)p Black eop %%Page: 19 19 19 18 bop Black 0 TeXcolorgray Black Black 800 1424 a Fj(Pr)-66 b(o)g(of.)2977 4194 y Fo(D)36 b Fq(\()p Fo(xy)48 b Fq(\))295 b Fk(\241)h Fo(D)36 b Fq(\()p Fo(x)p Fq(\))p Fo(y)342 b Fk(\241)296 b Fo(xD)36 b Fq(\()p Fo(y)48 b Fq(\))368 b(=)h Fo(xy)48 b(L)p Fq(\()p Fo(xy)g Fq(\))295 b Fk(\241)g Fo(xL)p Fq(\()p Fo(x)p Fq(\))p Fo(y)343 b Fk(\241)295 b Fo(xy)48 b(L)p Fq(\()p Fo(y)g Fq(\))369 b(=)f Fo(xy)48 b Fq(\()p Fo(L)p Fq(\()p Fo(xy)g Fq(\))295 b Fk(\241)g Fo(L)p Fq(\()p Fo(x)p Fq(\))g Fk(\241)g Fo(L)p Fq(\()p Fo(y)48 b Fq(\)\))800 6964 y(and)574 b(w)-36 b(e)575 b(see)f(that)g(the)g(Leibnitz)g(rule)g(is)h(equiv)-72 b(alen)-36 b(t)575 b(to)g(the)e(homomorphism)h(condition.)1001 b(If)575 b Fo(R)586 b Fq(is)800 8569 y(a)504 b(\257eld)g(then)f(the)h (semigroup)g(homomorphism)f(is)i(automatically)h(the)d(group)h (homomorphism)g(and)800 10174 y Fo(L)p Fq(\(1)p Fo(=x)p Fq(\))369 b(=)g Fk(\241)p Fo(L)p Fq(\()p Fo(x)p Fq(\))433 b(whic)-36 b(h)434 b(is)g(su\261cien)-36 b(t)432 b(to)i(get)16821 13708 y Fo(D)18159 11835 y Fi(\265)19285 12809 y Fq(1)p 19270 13402 682 54 v 19270 14619 a Fo(y)20084 11835 y Fi(\266)21431 13708 y Fq(=)22812 11835 y Fi(\265)23938 12809 y Fq(1)p 23922 13402 V 23922 14619 a Fo(y)24737 11835 y Fi(\266)25936 13708 y Fq(\()p Fk(\241)p Fo(L)p Fq(\()p Fo(y)48 b Fq(\)\))368 b(=)32442 12809 y Fk(\241)p Fo(D)36 b Fq(\()p Fo(y)48 b Fq(\))p 32442 13402 3843 54 v 33759 14619 a Fo(y)34441 14235 y Fl(2)36417 13708 y Fo(:)800 17168 y Fq(Then)433 b(it)h(remains)f(to)h(rep)36 b(eat)434 b(the)f(pro)36 b(of)434 b(of)g(Theorem)p 0 1 0 0 TeXcolorcmyk 434 w(14)p (#Thm.14) [[329 562 341 574] [1 1 1 [3 3]] [0 0 1]] pdfm Black(.)p 30795 17168 630 630 v Black 800 20642 a Fr(Corollary)500 b(5)p Black 650 w Fj(Ther)-66 b(e)398 b(exist)f(in\257nitely)e(many)j (p)-66 b(ossibilities)396 b(to)i(extend)f(the)h(derivative)e Fo(x)45370 20160 y Fn(0)45681 20642 y Fo(;)i Fj(c)-66 b(onstructe)g(d)800 22247 y(in)464 b(Se)-66 b(ction)p 0 1 0 0 TeXcolorcmyk 463 w(9)p (#section.9) [[134 517 140 529] [1 1 1 [3 3]] [0 0 1]] pdfm Black 465 w(on)465 b Fr(Q)g Fj(to)g(al)66 b(l)465 b(r)-66 b(e)g(al)464 b(numb)-66 b(ers)464 b(pr)-66 b(eserving)464 b(the)g(L)-66 b(eibnitz)463 b(rule.)800 25104 y(Pr)-66 b(o)g(of.)795 b Fq(W)-108 b(e)505 b(start)h(from)g(the)f(p)36 b(ositiv)-36 b(e)506 b(n)-36 b(um)g(b)36 b(ers.)793 b(It)505 b(is)h(su\261cien)-36 b(t)505 b(to)h(extend)f Fo(l)29 b(d)p Fq(\()p Fo(x)p Fq(\))p Fo(:)506 b Fq(Note)g(that)f(the)800 26709 y(m)-36 b(ultiplicativ)g(e)472 b(group)f(of)g(p)36 b(ositiv)-36 b(e)472 b(real)g(n)-36 b(um)g(b)36 b(ers)469 b(is)i(isomorphic)h(to)f(the)f(additiv)-36 b(e)471 b(group)g(and)g(b)36 b(oth)800 28315 y(of)393 b(them)f(are)g(v)-36 b(ector)393 b(spaces)g(o)-36 b(v)g(er)393 b(rationals.)565 b(In)392 b(Section)p 0 1 0 0 TeXcolorcmyk 392 w(9)p (#section.9) [[343 462 349 474] [1 1 1 [3 3]] [0 0 1]] pdfm Black 393 w(a)h(map)f Fo(l)29 b(d)p Fq(\()p Fo(x)p Fq(\))393 b(is)f(de\257ned)f(o)-36 b(v)g(er)393 b(a)g(subspace)800 29920 y(and)540 b(there)f(are)i(in\257nitely)f(man)-36 b(y)540 b(p)36 b(ossibilities)541 b(to)g(extend)f(a)g(linear)h(map)f (from)g(a)h(subspace)e(to)i(the)800 31525 y(whole)469 b(space.)682 b(Ob)-36 b(viously)468 b(it)g(w)-36 b(ould)468 b(b)36 b(e)468 b(a)h(group)e(homomorphism)h(and)f(this)h(giv)-36 b(es)469 b(a)g(deriv)-72 b(ativ)-36 b(e)468 b(for)800 33130 y(p)36 b(ositiv)-36 b(e)435 b(n)-36 b(um)g(b)36 b(ers.)576 b(F)-108 b(or)433 b(the)g(negativ)-36 b(e)435 b(n)-36 b(um)g(b)36 b(ers)432 b(w)-36 b(e)434 b(pro)36 b(ceed)433 b(as)h(in)f(Section)p 0 1 0 0 TeXcolorcmyk 434 w(5)p (#section.5) [[438 419 444 431] [1 1 1 [3 3]] [0 0 1]] pdfm Black(.)p 42287 33130 V 2751 35352 a(Note)329 b(that)g(the)f(Axiom)i (of)g(Choice)f(is)h(b)36 b(eing)329 b(used)f(here.)543 b(It)329 b(w)-36 b(ould)329 b(b)36 b(e)329 b(nice)g(to)g(\257nd)e(some) j(\\natural")800 36957 y(extension,)386 b(whic)-36 b(h)373 b(preserv)-36 b(es)374 b(condition)f(\()p 0 1 0 0 TeXcolorcmyk(7)p (#equation.7) [[272 384 278 396] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\),)386 b(but)373 b(note)g(that)g(no)g(suc)-36 b(h)373 b(extension)h(can)f(b)36 b(e)374 b(con)-36 b(tin)g(uous.)800 38563 y(T)-108 b(o)434 b(sho)-36 b(w)434 b(this)f(let)h(us)f(consider)g (a)h(sequence)19594 41858 y Fo(x)20333 42057 y Fm(n)21329 41858 y Fq(=)22842 40960 y(2)23492 40478 y Fm(a)23992 40589 y Fh(n)p 22842 41553 1773 54 v 23090 42769 a Fq(3)23740 42386 y Fm(n)24747 41858 y Fo(;)221 b(a)26012 42057 y Fm(n)27008 41858 y Fq(=)368 b Fk(b)p Fo(n)221 b Fq(log)31657 42172 y Fl(2)32404 41858 y Fq(3)p Fk(c)p Fo(:)800 45041 y Fq(It)434 b(is)g(b)36 b(ounded)432 b(and)h(has)g(a)h(con)-36 b(v)g(ergen)g(t)433 b(subsequence)g(\(ev)-36 b(en)434 b(con)-36 b(v)g(ergen)g(t)433 b(to)h(1)p Fo(:)p Fq(\))g(But)16915 48176 y(lim)16592 48973 y Fm(n)p Fn(!1)19044 48176 y Fq(\()p Fo(x)20289 48375 y Fm(n)20915 48176 y Fq(\))21421 47628 y Fn(0)22101 48176 y Fq(=)692 b(lim)23481 48973 y Fm(n)p Fn(!1)26156 48176 y Fo(x)26895 48375 y Fm(n)27742 46702 y Fi(\263)28669 47278 y Fo(a)29352 47477 y Fm(n)p 28669 47871 1310 54 v 28998 49087 a Fq(2)30406 48176 y Fk(\241)31867 47278 y Fo(n)p 31867 47871 777 54 v 31930 49087 a Fq(3)32776 46702 y Fi(\264)33938 48176 y Fq(=)369 b Fk(1)p Fo(:)2751 51378 y Fq(An)572 b(example)i(of)f(con)-36 b(tin)g(uous)572 b(generalized)h(deriv)-72 b(ativ)-36 b(e)574 b(giv)-36 b(es)574 b(us)e Fo(D)36 b Fq(\()p Fo(x)p Fq(\))606 b(=)f Fo(x)221 b Fq(ln)h Fo(x:)573 b Fq(It)g(is)g(easy)h(to) 800 52983 y(construct)398 b(a)i(surjectiv)-36 b(e)400 b(generalized)f(deriv)-72 b(ativ)-36 b(e)400 b(in)f(the)g(set)g(of)h (in)-36 b(tegers,)406 b(and)399 b(is)g(imp)36 b(ossible)400 b(to)f(mak)-36 b(e)800 54588 y(it)333 b(injectiv)-36 b(e)335 b(\(b)36 b(ecause)333 b Fo(D)36 b Fq(\(1\))369 b(=)f Fo(D)36 b Fq(\()p Fk(\241)p Fq(1\))370 b(=)e Fo(D)36 b Fq(\(0\))369 b(=)g(0\))p Fo(:)333 b Fq(But)g(probably)h(ev)-36 b(en)333 b(the)g(follo)-36 b(wing)335 b(conjecture)800 56193 y(is)434 b(true.)p Black 800 58765 a Fr(Conjecture)499 b(15)p Black 651 w Fj(Ther)-66 b(e)517 b(is)i(no)f(gener)-66 b(alize)g(d)517 b(derivative)f Fo(D)36 b Fq(\()p Fo(x)p Fq(\))518 b Fj(which)h(is)f(bije)-66 b(ction)516 b(b)-66 b(etwe)g(en)516 b(the)i(set)800 60370 y(of)465 b(natur)-66 b(al)464 b(numb)-66 b(ers)464 b(and)h(the)g(set)g(of)f(nonne)-66 b(gative)462 b(inte)-66 b(gers.)2751 62942 y Fq(W)-108 b(e)434 b(can)f(ev)-36 b(en)434 b(hop)36 b(e)433 b(for)h(a)g(stronger)f (v)-72 b(arian)-36 b(t:)p Black 800 65514 a Fr(Conjecture)499 b(16)p Black 651 w Fj(F)-100 b(or)530 b(any)e(gener)-66 b(alize)g(d)528 b(derivative)f Fo(D)36 b Fq(\()p Fo(x)p Fq(\))529 b Fj(on)g(the)g(set)f(of)h(inte)-66 b(gers)527 b(ther)-66 b(e)528 b(exist)h(two)800 67119 y(di\256er)-66 b(ent)463 b(p)-66 b(ositive)464 b(inte)-66 b(gers)462 b(which)j(have)g(the)g(same)g(derivative.)2751 69691 y Fq(Returning)413 b(to)h(the)g(generalized)g(deriv)-72 b(ativ)-36 b(es)415 b(in)f Fr(Q)g Fq(or)g Fr(R)g Fq(let)g(us)g(in)-36 b(v)g(estigate)414 b(their)g(structure)e(as)j(a)800 71296 y(set.)p Black 26150 74617 a(19)p Black eop %%Page: 20 20 20 19 bop Black 0 TeXcolorgray Black Black Black 800 1424 a Fr(Theorem)499 b(22)p Black 651 w Fj(If)446 b Fo(D)11705 1623 y Fl(1)12231 1424 y Fo(;)221 b(D)13893 1623 y Fl(2)14866 1424 y Fj(ar)-66 b(e)446 b(two)h(gener)-66 b(alize)g(d)445 b(derivatives)g(and)i Fo(a;)221 b(b)447 b Fj(ar)-66 b(e)446 b(some)h(r)-66 b(e)g(al)446 b(numb)-66 b(ers)446 b(then)800 3029 y Fo(aD)2563 3228 y Fl(1)3430 3029 y Fq(+)342 b Fo(bD)6417 3228 y Fl(2)7471 3029 y Fj(and)527 b Fq([)p Fo(D)11498 3228 y Fl(1)12024 3029 y Fo(;)221 b(D)13686 3228 y Fl(2)14213 3029 y Fq(])486 b(=)f Fo(D)17637 3228 y Fl(1)18163 3029 y Fo(D)19243 3228 y Fl(2)20110 3029 y Fk(\241)342 b Fo(D)22565 3228 y Fl(2)23091 3029 y Fo(D)24171 3228 y Fl(1)25225 3029 y Fj(ar)-66 b(e)527 b(gener)-66 b(alize)g(d)526 b(derivatives)h(to)-66 b(o.)785 b(Nevertheless)527 b(the)800 4634 y(set)465 b(of)f(al)66 b(l)466 b(gener)-66 b(alize)g(d)463 b(derivatives)h(is)g Fq(not)h Fj(a)g(Lie)f(algebr)-66 b(a.)800 7346 y(Pr)g(o)g(of.)597 b Fq(W)-108 b(e)433 b(ha)-36 b(v)g(e)2441 10280 y(\()p Fo(aD)4710 10479 y Fl(1)5530 10280 y Fq(+)295 b Fo(bD)8470 10479 y Fl(2)8996 10280 y Fq(\)\()p Fo(xy)48 b Fq(\))368 b(=)h Fo(aD)15447 10479 y Fl(1)15972 10280 y Fq(\()p Fo(xy)48 b Fq(\))295 b(+)g Fo(bD)21640 10479 y Fl(2)22166 10280 y Fq(\()p Fo(xy)48 b Fq(\))368 b(=)h Fo(aD)28111 10479 y Fl(1)28636 10280 y Fq(\()p Fo(x)p Fq(\))p Fo(y)343 b Fq(+)295 b Fo(axD)35173 10479 y Fl(1)35699 10280 y Fq(\()p Fo(y)48 b Fq(\))294 b(+)h Fo(bD)40627 10479 y Fl(2)41153 10280 y Fq(\()p Fo(x)p Fq(\))p Fo(y)343 b Fq(+)294 b Fo(bxD)47559 10479 y Fl(2)48085 10280 y Fq(\()p Fo(y)48 b Fq(\))369 b(=)3735 13213 y Fo(aD)5498 13412 y Fl(1)6023 13213 y Fq(\()p Fo(x)p Fq(\))p Fo(y)343 b Fq(+)295 b Fo(bD)11691 13412 y Fl(2)12217 13213 y Fq(\()p Fo(x)p Fq(\))p Fo(y)342 b Fq(+)295 b Fo(xaD)18753 13412 y Fl(1)19279 13213 y Fq(\()p Fo(y)48 b Fq(\))294 b(+)h Fo(xbD)24946 13412 y Fl(2)25472 13213 y Fq(\()p Fo(y)48 b Fq(\))369 b(=)f(\()p Fo(aD)31184 13412 y Fl(1)32005 13213 y Fq(+)294 b Fo(bD)34944 13412 y Fl(2)35470 13213 y Fq(\)\()p Fo(x)p Fq(\))p Fo(y)343 b Fq(+)295 b Fo(x)p Fq(\()p Fo(aD)43019 13412 y Fl(1)43839 13213 y Fq(+)g Fo(bD)46779 13412 y Fl(2)47305 13213 y Fq(\)\()p Fo(y)48 b Fq(\))p Fo(:)800 15538 y Fq(In)433 b(the)g(same)h(w)-36 b(a)g(y:)8604 18471 y([)p Fo(D)10045 18670 y Fl(1)10571 18471 y Fo(;)221 b(D)12233 18670 y Fl(2)12759 18471 y Fq(]\()p Fo(xy)48 b Fq(\))369 b(=)g(\()p Fo(D)18889 18670 y Fl(1)19414 18471 y Fo(D)36 b Fq(2)296 b Fk(\241)f Fo(D)23884 18670 y Fl(2)24410 18471 y Fo(D)25490 18670 y Fl(1)26016 18471 y Fq(\)\()p Fo(xy)48 b Fq(\))368 b(=)h Fo(D)31784 18670 y Fl(1)32310 18471 y Fo(D)33390 18670 y Fl(2)33915 18471 y Fq(\()p Fo(xy)48 b Fq(\))295 b Fk(\241)g Fo(D)39051 18670 y Fl(2)39577 18471 y Fo(D)40657 18670 y Fl(1)41183 18471 y Fq(\()p Fo(xy)48 b Fq(\))368 b(=)13002 21405 y Fo(D)14082 21604 y Fl(1)14608 21405 y Fq(\()p Fo(D)16194 21604 y Fl(2)16719 21405 y Fq(\()p Fo(x)p Fq(\))p Fo(y)343 b Fq(+)295 b Fo(xD)22573 21604 y Fl(2)23099 21405 y Fq(\()p Fo(y)48 b Fq(\)\))294 b Fk(\241)i Fo(D)28002 21604 y Fl(2)28527 21405 y Fq(\()p Fo(D)30113 21604 y Fl(1)30639 21405 y Fq(\()p Fo(x)p Fq(\))p Fo(y)343 b Fq(+)294 b Fo(xD)36492 21604 y Fl(1)37018 21405 y Fq(\()p Fo(y)48 b Fq(\)\))369 b(=)10569 23729 y Fo(D)11649 23928 y Fl(1)12175 23729 y Fq(\()p Fo(D)13761 23928 y Fl(2)14286 23729 y Fq(\()p Fo(x)p Fq(\)\))p Fo(y)343 b Fq(+)295 b Fo(D)19907 23928 y Fl(2)20433 23729 y Fq(\()p Fo(x)p Fq(\))p Fo(D)23264 23928 y Fl(1)23789 23729 y Fq(\()p Fo(y)48 b Fq(\))295 b(+)f Fo(D)28164 23928 y Fl(1)28690 23729 y Fq(\()p Fo(x)p Fq(\))p Fo(D)31521 23928 y Fl(2)32047 23729 y Fq(\()p Fo(y)48 b Fq(\))294 b(+)h Fo(xD)37161 23928 y Fl(1)37687 23729 y Fq(\()p Fo(D)39273 23928 y Fl(2)39799 23729 y Fq(\()p Fo(y)48 b Fq(\)\))p Fk(\241)9890 26054 y Fq(\()p Fo(D)11476 26253 y Fl(2)12001 26054 y Fq(\()p Fo(D)13587 26253 y Fl(1)14113 26054 y Fq(\()p Fo(x)p Fq(\)\))p Fo(y)342 b Fq(+)295 b Fo(D)19733 26253 y Fl(1)20259 26054 y Fq(\()p Fo(x)p Fq(\))p Fo(D)23090 26253 y Fl(2)23615 26054 y Fq(\()p Fo(y)48 b Fq(\))295 b(+)g Fo(D)27991 26253 y Fl(2)28516 26054 y Fq(\()p Fo(x)p Fq(\))p Fo(D)31347 26253 y Fl(1)31873 26054 y Fq(\()p Fo(y)48 b Fq(\))295 b(+)f Fo(xD)36987 26253 y Fl(2)37513 26054 y Fq(\()p Fo(D)39099 26253 y Fl(1)39625 26054 y Fq(\()p Fo(y)48 b Fq(\)\)\))368 b(=)10374 28379 y Fo(D)11454 28578 y Fl(1)11980 28379 y Fq(\()p Fo(D)13566 28578 y Fl(2)14091 28379 y Fq(\()p Fo(x)p Fq(\)\))p Fo(y)343 b Fq(+)295 b Fo(xD)20451 28578 y Fl(1)20976 28379 y Fq(\()p Fo(D)22562 28578 y Fl(2)23088 28379 y Fq(\()p Fo(y)48 b Fq(\)\))294 b Fk(\241)i Fo(D)27991 28578 y Fl(2)28516 28379 y Fq(\()p Fo(D)30102 28578 y Fl(1)30628 28379 y Fq(\()p Fo(x)p Fq(\)\))p Fo(y)343 b Fk(\241)295 b Fo(xD)37009 28578 y Fl(2)37535 28379 y Fq(\()p Fo(D)39121 28578 y Fl(1)39646 28379 y Fq(\()p Fo(y)48 b Fq(\)\))369 b(=)18869 30703 y([)p Fo(D)20310 30902 y Fl(1)20836 30703 y Fo(;)221 b(D)22498 30902 y Fl(2)23025 30703 y Fq(]\()p Fo(x)p Fq(\))p Fo(y)343 b Fq(+)294 b Fo(x)p Fq([)p Fo(D)29600 30902 y Fl(1)30127 30703 y Fo(;)221 b(D)31789 30902 y Fl(2)32315 30703 y Fq(]\()p Fo(y)48 b Fq(\))p Fo(:)800 33028 y Fq(But)433 b(the)g(comm)-36 b(utator)433 b(is)h(not)g (bilinear:)578 b(in)434 b(general)15509 35961 y([)p Fo(aD)17633 36160 y Fl(1)18454 35961 y Fq(+)295 b Fo(bD)21394 36160 y Fl(2)21920 35961 y Fo(;)221 b(D)23582 36160 y Fl(3)24108 35961 y Fq(])370 b Fk(6)p Fq(=)e Fo(a)p Fq([)p Fo(D)28343 36160 y Fl(1)28869 35961 y Fo(;)221 b(D)30531 36160 y Fl(3)31058 35961 y Fq(])295 b(+)g Fo(b)p Fq([)p Fo(D)35015 36160 y Fl(2)35541 35961 y Fo(;)221 b(D)37203 36160 y Fl(3)37729 35961 y Fq(])800 38895 y(so)434 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)434 b(no)f(Lie)h(algebra)g(structure.)p 21932 38895 630 630 v 2751 41164 a(Let)356 b(us)g(de\257ne)f Fo(D)11356 41371 y Fl(\()p Fm(p)12196 41506 y Fh(i)12547 41371 y Fl(\))13325 41164 y Fq(as)i(a)g(deriv)-72 b(ativ)-36 b(e)357 b(whic)-36 b(h)356 b(maps)g(a)h(prime)f Fo(p)33974 41363 y Fm(i)34706 41164 y Fq(to)h(1)g(and)f(other)g(primes)g Fo(p)47771 41363 y Fm(j)48614 41164 y Fq(to)g(zero.)800 42769 y(Then)571 b([)p Fo(D)5775 42976 y Fl(\()p Fm(p)6615 43111 y Fh(i)6966 42976 y Fl(\))7388 42769 y Fo(;)221 b(D)9050 42976 y Fl(\()p Fm(p)9890 43111 y Fh(j)10323 42976 y Fl(\))10745 42769 y Fq(])604 b(=)g(0)p Fo(;)572 b Fq(but)f(already)h([3)p Fo(D)24249 42976 y Fl(\(2\))25507 42769 y Fo(;)221 b(D)27169 42976 y Fl(\(3\))28428 42769 y Fq(])604 b(=)g Fk(\241)p Fo(D)33122 42976 y Fl(\(2\))34379 42769 y Fo(:)572 b Fq(Nev)-36 b(ertheless)572 b(ev)-36 b(ery)573 b(generalized)800 44374 y(deriv)-72 b(ativ)-36 b(e)434 b Fo(D)470 b Fq(can)434 b(b)36 b(e)433 b(uniquely)h(written)f (as)21192 48246 y Fo(D)405 b Fq(=)24547 46586 y Fn(1)24058 46984 y Fi(X)24256 49783 y Fm(i)p Fl(=1)26198 48246 y Fo(D)36 b Fq(\()p Fo(p)28473 48445 y Fm(i)28849 48246 y Fq(\))p Fo(D)30435 48453 y Fl(\()p Fm(p)31275 48588 y Fh(i)31625 48453 y Fl(\))32046 48246 y Fo(:)800 53357 y Fp(12)2152 b(The)716 b(generating)g(function)800 56453 y Fq(Let)447 b Fo(D)36 b Fq(\()p Fo(x)p Fq(\))447 b(b)36 b(e)447 b(a)h(generalized)g(deriv)-72 b(ativ)-36 b(e)448 b(o)-36 b(v)g(er)448 b(the)f(reals)h(and)f Fo(L)p Fq(\()p Fo(x)p Fq(\))392 b(=)37263 55819 y Fm(D)26 b Fl(\()p Fm(x)p Fl(\))p 37263 56147 2060 54 v 38028 56911 a Fm(x)39903 56453 y Fq(b)36 b(e)447 b(corresp)36 b(onding)447 b(loga-)800 58058 y(rithmic)433 b(deriv)-72 b(ativ)-36 b(e.)580 b(Let)15401 61671 y Fo(H)16479 61870 y Fm(D)17332 61671 y Fq(\()p Fo(t)p Fq(\))369 b(=)21052 60011 y Fn(1)20563 60409 y Fi(X)20636 63199 y Fm(n)p Fl(=0)22703 61671 y Fo(D)36 b Fq(\()p Fo(n)p Fq(\))p Fo(t)26077 61123 y Fm(n)26703 61671 y Fo(;)221 b(H)28363 61870 y Fm(L)29059 61671 y Fq(\()p Fo(t)p Fq(\))368 b(=)32779 60011 y Fn(1)32290 60409 y Fi(X)32363 63199 y Fm(n)p Fl(=1)34430 61671 y Fo(L)p Fq(\()p Fo(n)p Fq(\))p Fo(t)37573 61123 y Fm(n)800 65693 y Fq(b)36 b(e)433 b(their)h(generating)f(functions.)p Black 26150 74617 a(20)p Black eop %%Page: 21 21 21 20 bop Black 0 TeXcolorgray Black Black Black 800 1424 a Fr(Theorem)499 b(23)p Black 651 w Fj(The)465 b(gener)-66 b(ating)462 b(functions)h Fo(H)24952 1623 y Fm(D)25806 1424 y Fq(\()p Fo(t)p Fq(\))p Fo(;)221 b(H)28948 1623 y Fm(L)29642 1424 y Fq(\()p Fo(t)p Fq(\))465 b Fj(c)-66 b(an)464 b(b)-66 b(e)464 b(b)-66 b(e)464 b(c)-66 b(alculate)g(d)464 b(as)h(fol)66 b(lows:)20744 4961 y Fo(H)21822 5160 y Fm(D)22675 4961 y Fq(\()p Fo(t)p Fq(\))368 b(=)h Fo(t)26743 4062 y(d)p 26509 4655 1146 54 v 26509 5872 a(dt)28008 4961 y Fq(\()p Fo(H)29592 5160 y Fm(L)30287 4961 y Fq(\()p Fo(t)p Fq(\)\))220 b Fo(:)18561 8877 y(H)19639 9076 y Fm(L)20334 8877 y Fq(\()p Fo(t)p Fq(\))369 b(=)23565 7615 y Fi(X)24288 10405 y Fm(p)25484 7833 y Fn(0)25795 8877 y Fo(L)p Fq(\()p Fo(p)p Fq(\))29054 7217 y Fn(1)28565 7615 y Fi(X)28708 10414 y Fm(j)51 b Fl(=1)32192 7978 y Fo(t)32662 7496 y Fm(p)p 30838 8571 3707 54 v 30838 9807 a Fq(1)296 b Fk(\241)f Fo(t)33582 9423 y Fm(p)34056 9171 y Fh(j)34677 8877 y Fo(;)800 12472 y Fj(wher)-66 b(e)465 b(the)f(\257rst)h(sum)g(runs)g(over)g(al)66 b(l)466 b(primes.)800 15184 y(Pr)-66 b(o)g(of.)803 b Fq(The)508 b(\257rst)g(form)-36 b(ula)508 b(is)h(equiv)-72 b(alen)-36 b(t)509 b(to)f(the)g(condition)g Fo(D)36 b Fq(\()p Fo(n)p Fq(\))497 b(=)f Fo(n)346 b Fk(\242)g Fo(L)p Fq(\()p Fo(n)p Fq(\))p Fo(:)509 b Fq(As)f(to)h(the)e(second)800 16789 y(form)-36 b(ula)466 b(it)g(is)g(su\261cien)-36 b(t)465 b(to)h(pro)-36 b(v)g(e)466 b(it)g(for)h(the)e(sp)36 b(ecial)466 b(case)h(when)e Fo(L)p Fq(\()p Fo(p)p Fq(\))423 b(=)h(1)466 b(for)g(some)g(prime)g Fo(p)g Fq(and)800 18394 y Fo(L)p Fq(\()p Fo(q)48 b Fq(\))368 b(=)h(0)433 b(for)i(all)f(other)f(primes.) 578 b(Then)433 b(w)-36 b(e)434 b(need)f(to)h(pro)-36 b(v)g(e)434 b(that)20223 20680 y Fn(1)19734 21078 y Fi(X)19807 23868 y Fm(n)p Fl(=0)21874 22340 y Fo(L)p Fq(\()p Fo(n)p Fq(\))p Fo(t)25017 21792 y Fm(n)26012 22340 y Fq(=)27882 20680 y Fn(1)27393 21078 y Fi(X)27535 23877 y Fm(j)51 b Fl(=1)31019 21441 y Fo(t)31489 20959 y Fm(p)p 29666 22035 V 29666 23270 a Fq(1)295 b Fk(\241)h Fo(t)32410 22886 y Fm(p)32884 22634 y Fh(j)33505 22340 y Fo(:)800 26719 y Fq(If)434 b Fo(n)370 b Fq(=)e Fo(p)5279 26237 y Fm(k)5848 26719 y Fo(m)433 b Fq(and)g(gcd\()p Fo(p;)221 b(m)p Fq(\))369 b(=)g(1)433 b(then)g Fo(t)21550 26237 y Fm(n)22610 26719 y Fq(app)36 b(ears)433 b(exactly)i(in)f Fo(k)478 b Fq(sums)23480 29766 y Fo(t)23950 29284 y Fm(p)p 22126 30359 V 22126 31594 a Fq(1)295 b Fk(\241)h Fo(t)24870 31210 y Fm(p)25344 30958 y Fh(j)26334 30664 y Fq(=)28203 29004 y Fn(1)27714 29402 y Fi(X)27912 32202 y Fm(i)p Fl(=1)29854 30664 y Fo(t)30324 30116 y Fm(ip)31118 29803 y Fh(j)800 34770 y Fq(for)434 b Fo(j)444 b Fq(=)369 b(1)p Fo(;)221 b Fq(2)p Fo(;)g(:)g(:)g(:)k(;)c(k)45 b(:)434 b Fq(It)g(only)g(remains)g(to)f(note)h(that)f Fo(L)p Fq(\()p Fo(n)p Fq(\))369 b(=)f Fo(k)45 b(:)p 33810 34770 630 630 v Black 800 38478 a Fr(Corollary)500 b(6)p Black 15954 41022 a Fo(L)p Fq(\()p Fo(n)p Fq(!\))369 b(=)21412 39361 y Fm(n)20738 39760 y Fi(X)20936 42559 y Fm(i)p Fl(=1)22878 41022 y Fo(L)p Fq(\()p Fo(i)p Fq(\))f(=)26968 39760 y Fi(X)27039 42566 y Fm(p)p Fn(\267)p Fm(n)28887 39978 y Fn(0)29197 41022 y Fo(L)p Fq(\()p Fo(p)p Fq(\))32457 39361 y Fn(1)31968 39760 y Fi(X)32110 42559 y Fm(j)51 b Fl(=1)34108 39149 y Fi(\271)35198 40123 y Fo(n)p 35016 40716 1140 54 v 35016 41933 a(p)35669 41549 y Fm(j)36288 39149 y Fi(\272)37285 41022 y Fo(:)800 46151 y Fj(Pr)-66 b(o)g(of.)573 b Fq(If)356 b(w)-36 b(e)356 b(replace)h(ev)-36 b(ery)16491 45628 y Fm(t)16831 45315 y Fh(p)p 15673 45845 2458 54 v 15673 46791 a Fl(1)p Fn(\241)p Fm(t)17215 46539 y Fh(p)17642 46338 y(j)18620 46151 y Fq(b)g(y)20348 45154 y Fi(P)21750 45085 y Fn(b)22476 44723 y Fh(n)p 22301 44884 861 40 v 22301 45564 a(p)22728 45363 y(j)23295 45085 y Fn(c)21750 46491 y Fm(i)p Fl(=1)23990 46151 y Fo(t)24460 45668 y Fm(ip)25254 45356 y Fh(j)26098 46151 y Fq(w)g(e)356 b(do)g(not)g(c)-36 b(hange)355 b(the)h(co)36 b(e\261cien)-36 b(ts)356 b(in)g Fo(t)46698 45668 y Fm(k)47622 46151 y Fq(for)g Fo(k)415 b Fk(\267)369 b Fo(n)800 47990 y Fq(and)382 b(mak)-36 b(e)383 b(them)f(equal)i(to)f(zero)g(for)g Fo(k)414 b(>)368 b(n:)384 b Fq(So)e(it)h(is)g(su\261cien)-36 b(t)382 b(to)h(put)e Fo(t)369 b Fq(=)g(1)382 b(to)h(get)g(the)f (desired)g Fk(b)51338 47467 y Fm(n)p 51169 47685 908 54 v 51169 48448 a(p)51643 48196 y Fh(j)52210 47990 y Fk(c)800 49595 y Fq(in)433 b(ev)-36 b(ery)435 b(summand.)p 12188 49595 630 630 v 2751 51865 a(If)379 b(w)-36 b(e)379 b(use)f(the)g(same)h Fo(L)p Fq(\()p Fo(x)p Fq(\))f(that)g(w)-36 b(e)379 b(used)e(in)i(the)f(pro)36 b(of)379 b(of)g(the)f(Theorem)p 0 1 0 0 TeXcolorcmyk 378 w(23)p (#Thm.23) [[432 250 444 262] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)391 b(w)-36 b(e)379 b(get)f(the)g(classical)800 53470 y(Legendre)433 b(theorem)g(that)g(calculates)i(the)e(maximal)i(p) 36 b(o)-36 b(w)g(er)433 b(of)i(a)e(prime)h Fo(p)f Fq(in)g Fo(n)p Fq(!)p Fo(:)2751 55075 y Fq(On)576 b(the)g(other)g(hand)f(if)i (w)-36 b(e)577 b(use)f Fo(L)p Fq(\()p Fo(x)p Fq(\))612 b(=)f(ld\()p Fo(x)p Fq(\))577 b(w)-36 b(e)576 b(will)i(b)36 b(e)576 b(able,)613 b(follo)-36 b(wing)578 b(Barb)36 b(eau)577 b([)p 0 1 0 0 TeXcolorcmyk(1)p (#cite.Bar) [[519 221 525 233] [1 1 1 [3 3]] [0 0 1]] pdfm Black(],)613 b(to)800 56680 y(estimate)5977 55684 y Fi(P)7379 56034 y Fm(n)7379 57067 y(i)p Fl(=1)9179 56680 y Fq(ld\()p Fo(i)p Fq(\))p Fo(:)400 b Fq(Let)h Fo(m)368 b Fq(=)h Fk(b)p Fq(log)19935 56993 y Fl(2)20683 56680 y Fo(n)p Fk(c)p Fo(:)401 b Fq(Then)g(w)-36 b(e)401 b(can)g(c)-36 b(hange)401 b(in\257nit)-36 b(y)400 b(in)h(our)f(sums)h(to)g Fo(m:)g Fq(Using)800 58285 y(standard)433 b(estimates)21928 58888 y Fi(X)22000 61694 y Fm(p)p Fn(\267)p Fm(n)23847 59106 y Fn(0)24292 59252 y Fq(1)p 24290 59845 653 54 v 24290 61061 a Fo(p)25445 60150 y Fq(=)369 b Fo(O)36 b Fq(\(ln)221 b Fo(m)p Fq(\))p Fo(;)11067 63494 y Fi(X)11138 66283 y Fm(p>n)12986 63712 y Fn(0)15337 63857 y Fo(n)p 13429 64450 4592 54 v 13429 65667 a(p)p Fq(\()p Fo(p)295 b Fk(\241)g Fq(1\))18522 64756 y Fo(<)19903 63494 y Fi(X)19954 66323 y Fm(k)24 b(>n)24151 63857 y Fo(n)p 22176 64450 4728 54 v 22176 65667 a(k)45 b Fq(\()p Fo(k)340 b Fk(\241)295 b Fq(1\))27405 64756 y(=)28786 63494 y Fi(X)28837 66323 y Fm(k)24 b(>n)30926 64756 y Fo(n)31923 62883 y Fi(\265)33070 63857 y Fq(1)p 33034 64450 722 54 v 33034 65667 a Fo(k)34183 64756 y Fk(\241)36817 63857 y Fq(1)p 35645 64450 2995 54 v 35645 65667 a Fo(k)340 b Fk(\241)295 b Fq(1)38772 62883 y Fi(\266)40119 64756 y Fk(\267)369 b Fq(1)p Fo(;)10203 68022 y Fi(X)10274 70828 y Fm(p)p Fn(\267)p Fm(n)12122 68240 y Fn(0)15517 68385 y Fo(n)p 12565 68978 6682 54 v 12565 70195 a(p)13218 69811 y Fm(m)p Fl(+1)15308 70195 y Fq(\()p Fo(p)295 b Fk(\241)g Fq(1\))19748 69284 y Fo(<)21129 68022 y Fi(X)21200 70828 y Fm(p)p Fn(\267)p Fm(n)23047 68240 y Fn(0)26443 68385 y Fq(2)p Fo(n)p 23491 68978 7332 54 v 23491 70195 a Fq(2)24141 69811 y Fm(m)p Fl(+1)26231 70195 y Fo(p)p Fq(\()p Fo(p)g Fk(\241)g Fq(1\))31324 69284 y Fo(<)32705 68022 y Fi(X)32756 70851 y Fm(k)24 b Fn(\267)p Fm(n)37016 68385 y Fq(2)p Fo(n)p 34977 68978 5505 54 v 34977 70195 a(nk)45 b Fq(\()p Fo(k)341 b Fk(\241)295 b Fq(1\))40983 69284 y Fk(\267)369 b Fq(2)p Fo(;)p Black 26150 74617 a Fq(21)p Black eop %%Page: 22 22 22 21 bop Black 0 TeXcolorgray Black Black 800 1424 a Fq(w)-36 b(e)434 b(get)11563 2457 y Fm(n)10889 2856 y Fi(X)11087 5655 y Fm(i)p Fl(=1)13029 4118 y Fq(ld\()p Fo(i)p Fq(\))368 b(=)17318 2856 y Fi(X)17389 5662 y Fm(p)p Fn(\267)p Fm(n)19236 3073 y Fn(0)19681 3219 y Fq(1)p 19680 3812 653 54 v 19680 5029 a Fo(p)21230 2457 y Fm(m)20687 2856 y Fi(X)20829 5655 y Fm(j)51 b Fl(=1)22827 2245 y Fi(\271)23916 3219 y Fo(n)p 23735 3812 1140 54 v 23735 5029 a(p)24388 4645 y Fm(j)25007 2245 y Fi(\272)26151 4118 y Fq(=)27532 2856 y Fi(X)27603 5662 y Fm(p)p Fn(\267)p Fm(n)29450 3073 y Fn(0)29895 3219 y Fq(1)p 29894 3812 653 54 v 29894 5029 a Fo(p)30901 1846 y Fi(\303)32496 2457 y Fm(m)31952 2856 y Fi(X)32095 5655 y Fm(j)g Fl(=1)34407 3219 y Fo(n)p 34225 3812 1140 54 v 34225 5029 a(p)34878 4645 y Fm(j)35793 4118 y Fq(+)295 b Fo(O)36 b Fq(\()p Fo(m)p Fq(\))40280 1846 y Fi(!)41700 4118 y Fq(=)11485 7461 y Fi(X)11556 10267 y Fm(p)p Fn(\267)p Fm(n)13404 7679 y Fn(0)14831 7825 y Fo(n)p 13847 8418 2743 54 v 13847 9634 a(p)14500 9251 y Fm(m)p Fl(+1)16944 6850 y Fi(\265)18055 7825 y Fo(p)18708 7343 y Fm(m)19891 7825 y Fk(\241)295 b Fq(1)p 18055 8418 3815 54 v 18499 9634 a Fo(p)g Fk(\241)g Fq(1)22002 6850 y Fi(\266)23275 8723 y Fq(+)g Fo(O)36 b Fq(\(ln)221 b Fo(m)p Fq(\))p Fo(O)36 b Fq(\()p Fo(m)p Fq(\))368 b(=)33996 7461 y Fi(X)34718 10251 y Fm(p)35914 7679 y Fn(0)38265 7825 y Fo(n)p 36358 8418 4592 54 v 36358 9634 a(p)p Fq(\()p Fo(p)294 b Fk(\241)i Fq(1\))41082 8723 y Fk(\241)11958 12939 y(\241)12991 11677 y Fi(X)13062 14466 y Fm(p>n)14910 11895 y Fn(0)17260 12040 y Fo(n)p 15353 12633 V 15353 13850 a(p)p Fq(\()p Fo(p)f Fk(\241)g Fq(1\))20372 12939 y Fk(\241)21760 11677 y Fi(X)21701 14483 y Fm(p)p Fn(\267)p Fm(m)23738 11895 y Fn(0)27134 12040 y Fo(n)p 24182 12633 6682 54 v 24182 13850 a(p)24835 13466 y Fm(m)p Fl(+1)26924 13850 y Fq(\()p Fo(p)g Fk(\241)h Fq(1\))31291 12939 y(+)f Fo(O)36 b Fq(\(ln)221 b Fo(m)p Fq(\))p Fo(O)36 b Fq(\()p Fo(m)p Fq(\))368 b(=)19353 15892 y Fi(X)20076 18682 y Fm(p)21272 16110 y Fn(0)23623 16256 y Fo(n)p 21716 16849 4592 54 v 21716 18066 a(p)p Fq(\()p Fo(p)294 b Fk(\241)i Fq(1\))26735 17154 y(+)f Fo(O)36 b Fq(\()p Fo(m)221 b Fq(ln)g Fo(m)p Fq(\))p Fo(:)p Black 800 22529 a Fr(Theorem)499 b(24)p Black 651 w Fj([)p 0 1 0 0 TeXcolorcmyk(1)p (#cite.Bar) [[159 514 165 526] [1 1 1 [3 3]] [0 0 1]] pdfm Black(])464 b(L)-66 b(et)18639 24727 y Fo(C)464 b Fq(=)21415 23465 y Fi(X)22137 26254 y Fm(p)23333 23682 y Fn(0)25747 23828 y Fq(1)p 23777 24421 V 23777 25638 a Fo(p)p Fq(\()p Fo(p)294 b Fk(\241)i Fq(1\))28870 24727 y(=)368 b(0)p Fo(:)p Fq(749)221 b Fo(:)g(:)g(:)800 28312 y Fj(Then)14602 30597 y(ld)p Fq(\()p Fo(n)p Fq(!\))370 b(=)20171 28937 y Fm(n)19497 29335 y Fi(X)19695 32135 y Fm(i)p Fl(=1)21637 30597 y Fj(ld)q Fq(\()p Fo(i)p Fq(\))e(=)h Fo(C)95 b(n)295 b Fq(+)g Fo(O)257 b Fq(\(\(ln)221 b Fo(n)p Fq(\)\(ln)h(ln)f Fo(n)p Fq(\)\))20653 33612 y Fm(n)19979 34010 y Fi(X)20080 36839 y Fm(k)24 b Fl(=1)22119 35272 y Fo(k)22840 34724 y Fn(0)23520 35272 y Fq(=)25033 34374 y Fo(C)p 25033 34967 1026 54 v 25221 36184 a Fq(2)26192 35272 y Fo(n)26968 34724 y Fl(2)27789 35272 y Fq(+)295 b Fo(O)36 b Fq(\()p Fo(n)31408 34724 y Fl(1+)p Fm(\261)33116 35272 y Fq(\))800 38725 y Fj(for)465 b(any)f Fo(\261)419 b(>)369 b Fq(0)p Fo(:)800 41437 y Fj(Pr)-66 b(o)g(of.)597 b Fq(The)433 b(\257rst)g(form)-36 b(ula)434 b(is)g(already)g(pro)-36 b(v)g(ed.)578 b(As)434 b(to)g(the)f(second)g(w)-36 b(e)434 b(ha)-36 b(v)g(e)16720 43648 y Fm(n)16046 44047 y Fi(X)16147 46876 y Fm(k)24 b Fl(=1)18186 45309 y Fo(k)18907 44760 y Fn(0)19587 45309 y Fq(=)21641 43648 y Fm(n)20967 44047 y Fi(X)21069 46876 y Fm(k)g Fl(=1)23107 45309 y Fo(k)341 b Fk(\242)295 b Fq(ld\()p Fo(k)45 b Fq(\))368 b(=)30028 43648 y Fm(n)29354 44047 y Fi(X)29455 46876 y Fm(k)24 b Fl(=1)32168 43648 y Fm(n)31494 44047 y Fi(X)31671 46876 y Fm(i)p Fl(=)p Fm(k)33634 45309 y Fq(ld\()p Fo(i)p Fq(\))368 b(=)13308 49151 y Fm(n)12634 49550 y Fi(X)12736 52379 y Fm(k)24 b Fl(=1)14553 50812 y Fq(\(ld\()p Fo(n)p Fq(!\))295 b Fk(\241)g Fq(ld\(\()p Fo(k)340 b Fk(\241)296 b Fq(1\)!\)\))369 b(=)f Fo(n)p Fq(ld\()p Fo(n)p Fq(!\))296 b Fk(\241)34340 49151 y Fm(n)p Fn(\241)p Fl(1)34267 49550 y Fi(X)34369 52379 y Fm(k)24 b Fl(=1)36407 50812 y Fq(ld\()p Fo(k)45 b Fq(!)q(\))368 b(=)16239 55706 y Fo(n)p Fq(\()p Fo(C)95 b(n)296 b Fq(+)f Fo(O)36 b Fq(\()p Fo(n)23238 55157 y Fm(\261)23743 55706 y Fq(\)\))295 b Fk(\241)26451 54045 y Fm(n)p Fn(\241)p Fl(1)26378 54444 y Fi(X)26479 57273 y Fm(k)24 b Fl(=1)28297 55706 y Fq(\()p Fo(C)95 b(k)340 b Fq(+)295 b Fo(O)36 b Fq(\()p Fo(n)34464 55157 y Fm(\261)34969 55706 y Fq(\)\))368 b(=)13280 60131 y Fo(C)95 b(n)15082 59582 y Fl(2)15903 60131 y Fk(\241)295 b Fo(C)18390 59232 y(n)p Fq(\()p Fo(n)h Fk(\241)f Fq(1\))p 18390 59825 4839 54 v 20484 61042 a(2)23656 60131 y(+)g Fo(O)36 b Fq(\()p Fo(n)27275 59582 y Fl(1+)p Fm(\261)28983 60131 y Fq(\))368 b(=)31371 59232 y Fo(C)p 31371 59825 1026 54 v 31559 61042 a Fq(2)32530 60131 y Fo(n)33306 59582 y Fl(2)34127 60131 y Fq(+)295 b Fo(O)36 b Fq(\()p Fo(n)37746 59582 y Fl(1+)p Fm(\261)39453 60131 y Fq(\))p Fo(:)p 800 62455 630 630 v 2751 64724 a Fq(W)-108 b(e)434 b(lea)-36 b(v)g(e)434 b(to)g(the)f(reader)g(the)g(pleasure)h(to)f(pla)-36 b(y)434 b(with)g Fo(\263)31254 64923 y Fm(D)32107 64724 y Fq(\()p Fo(s)p Fq(\))369 b(=)35481 63728 y Fi(P)37310 64201 y Fm(n)37881 63889 y Fe(0)p 37238 64419 1015 54 v 37238 65183 a Fm(n)37809 64930 y Fh(s)38385 64724 y Fo(:)p Black 26150 74617 a Fq(22)p Black eop %%Page: 23 23 23 22 bop Black 0 TeXcolorgray Black Black 800 1424 a Fp(13)2152 b(Logical)717 b(dep)60 b(endence)715 b(of)h(the)h (conjectures)800 4345 y Fq(Here)302 b(w)-36 b(e)302 b(w)-36 b(ould)301 b(lik)-36 b(e)303 b(to)f(exhibit)f(some)h(of)h(the)e (logical)i(dep)36 b(endence)300 b(b)36 b(et)-36 b(w)g(een)302 b(the)f(di\256eren)-36 b(t)300 b(conjectures)800 5950 y(w)-36 b(e)430 b(ha)-36 b(v)g(e)430 b(men)-36 b(tioned)429 b(ab)36 b(o)-36 b(v)g(e.)578 b(As)429 b(w)-36 b(e)431 b(see)e(the)h(Conjectures)p 0 1 0 0 TeXcolorcmyk 430 w(8)p (#Cnj.8) [[356 663 361 675] [1 1 1 [3 3]] [0 0 1]] pdfm Black 430 w(and)p 0 1 0 0 TeXcolorcmyk 429 w(9)p (#Cnj.9) [[388 663 394 675] [1 1 1 [3 3]] [0 0 1]] pdfm Black 430 w(seem)g(to)g(b)36 b(e)429 b(the)h(k)-36 b(ey)430 b(problems.)p Black 800 8886 a Fr(Theorem)499 b(25)p Black 651 w Fj(The)587 b(fol)66 b(lowing)586 b(pictur)-66 b(e)586 b(describ)-66 b(es)585 b(the)i(lo)-66 b(gic)g(al)585 b(dep)-66 b(endenc)g(e)585 b(b)-66 b(etwe)g(en)585 b(the)h(di\256er)-66 b(ent)800 10491 y(c)g(onje)g(ctur)g(es.)22060 12096 y Fq(\()p 0 1 0 0 TeXcolorcmyk(2)p (#Cnj.2) [[275 608 281 620] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))369 b Fk(\))g Fq(\()p 0 1 0 0 TeXcolorcmyk(3)p (#Cnj.3) [[309 608 314 620] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))g Fk(\))h Fq(\()p 0 1 0 0 TeXcolorcmyk(4)p (#Cnj.4) [[342 608 348 620] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))p Fo(;)23274 14380 y Fq(\()p 0 1 0 0 TeXcolorcmyk(12)p (#Cnj.12) [[286 588 298 600] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))f Fk(\))h Fq(\()p 0 1 0 0 TeXcolorcmyk(13)p (#Cnj.13) [[325 588 337 600] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))p Fo(;)21071 16664 y Fq(\()p 0 1 0 0 TeXcolorcmyk(5)p (#Cnj.5) [[266 567 272 579] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))f Fk(\()h Fq(\()p 0 1 0 0 TeXcolorcmyk(6)p (#Cnj.6) [[300 567 306 579] [1 1 1 [3 3]] [0 0 1]] pdfm Black Fo(;)221 b Fj(Goldb)-66 b(ach)q Fq(\))p Fo(;)23274 18949 y Fq(\()p 0 1 0 0 TeXcolorcmyk(15)p (#Cnj.15) [[286 546 298 558] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))369 b Fk(\()h Fq(\()p 0 1 0 0 TeXcolorcmyk(16)p (#Cnj.16) [[325 546 337 558] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))p Fo(;)25375 21194 y Fq(\()p 0 1 0 0 TeXcolorcmyk(11)p (#Cnj.11) [[305 526 317 538] [1 1 1 [3 3]] [0 0 1]] pdfm Black Fo(;)221 b Fj(T)q Fo(r)36 b(ipl)29 b(es)p Fq(\))3542 b(\()p 0 1 0 0 TeXcolorcmyk(8)p (#Cnj.8) [[401 526 407 538] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))28557 22799 y Fk(+)7149 b(+)15484 24404 y Fq(\()p 0 1 0 0 TeXcolorcmyk(10)p (#Cnj.10) [[216 497 228 509] [1 1 1 [3 3]] [0 0 1]] pdfm Black Fo(;)221 b Fj(Twins)q Fq(\))1107 b Fk(\))3864 b Fq(\()p 0 1 0 0 TeXcolorcmyk(9)p (#Cnj.9) [[330 497 336 509] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))g Fk(\))1107 b Fq(\()p 0 1 0 0 TeXcolorcmyk(1)p (#Cnj.1) [[401 497 407 509] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))38308 22810 y Fo(:)800 26672 y Fj(A)-66 b(dditional)66 b(ly)556 b(if)h(Conje)-66 b(ctur)g(e)p 0 1 0 0 TeXcolorcmyk 555 w(1)p (#Cnj.1) [[218 477 224 489] [1 1 1 [3 3]] [0 0 1]] pdfm Black 557 w(is)557 b(valid)h(then)e(either)g(Conje)-66 b(ctur)g(e)p 0 1 0 0 TeXcolorcmyk 556 w(8)p (#Cnj.8) [[392 477 398 489] [1 1 1 [3 3]] [0 0 1]] pdfm Black 557 w(or)558 b(Conje)-66 b(ctur)g(e)p 0 1 0 0 TeXcolorcmyk 555 w(9)p (#Cnj.9) [[478 477 484 489] [1 1 1 [3 3]] [0 0 1]] pdfm Black 557 w(is)557 b(valid)h(\(or)800 28277 y(b)-66 b(oth\).)800 31213 y(Pr)g(o)g(of.)594 b Fq(The)426 b(only)g(non)-36 b(trivial)426 b(dep)36 b(endence)424 b(is)i(the)f(last)h(one.)576 b(Supp)36 b(ose)425 b(that)g(Conjecture)p 0 1 0 0 TeXcolorcmyk 425 w(9)p (#Cnj.9) [[491 436 497 448] [1 1 1 [3 3]] [0 0 1]] pdfm Black 426 w(is)h(wrong,)800 32818 y(but)526 b(Conjecture)p 0 1 0 0 TeXcolorcmyk 527 w(1)p (#Cnj.1) [[162 422 168 434] [1 1 1 [3 3]] [0 0 1]] pdfm Black 526 w(is)h(true.)857 b(W)-108 b(e)527 b(need)f(to)h(sho)-36 b(w)527 b(that)f(Conjecture)p 0 1 0 0 TeXcolorcmyk 527 w(8)p (#Cnj.8) [[398 422 403 434] [1 1 1 [3 3]] [0 0 1]] pdfm Black 527 w(is)h(v)-72 b(alid.)858 b(Let)526 b(\241)h(b)36 b(e)526 b(the)g(tree)800 34423 y(ha)-36 b(ving)484 b(v)-36 b(ertices)484 b(1)f(\(the)g(ro)36 b(ot\),)497 b(the)482 b(primes)h Fo(p)h Fq(with)f Fo(i)p Fq(\()p Fo(p)p Fq(\))453 b Fo(>)g Fq(0)484 b(and)f(all)h(comp)36 b(osite)484 b Fo(n)g Fq(with)f Fo(n)48929 33941 y Fl(\()p Fm(k)24 b Fl(\))50684 34423 y Fq(=)454 b(0)800 36028 y(for)485 b(some)g Fo(k)500 b Fk(\270)456 b Fq(1)p Fo(:)485 b Fq(F)-108 b(urther,)496 b(let)484 b(\241)g(ha)-36 b(v)g(e)485 b(edges)f(from)h Fo(n)g Fq(to)g Fo(n)32051 35546 y Fn(0)32362 36028 y Fo(:)f Fq(By)h(Conjecture)p 0 1 0 0 TeXcolorcmyk 485 w(1)p (#Cnj.1) [[450 393 456 405] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)498 b(\241)484 b(is)g(in\257nite.)731 b(By)800 37633 y(Theorem)p 0 1 0 0 TeXcolorcmyk 554 w(8)p (#Thm.8) [[130 378 136 390] [1 1 1 [3 3]] [0 0 1]] pdfm Black 554 w(and)553 b(Corollary)p 0 1 0 0 TeXcolorcmyk 556 w(3)p (#Crl.3) [[217 378 223 390] [1 1 1 [3 3]] [0 0 1]] pdfm Black 554 w(the)g(degree)h(at)g(eac)-36 b(h)554 b(v)-36 b(ertex)554 b(di\256eren)-36 b(t)552 b(from)j(1)f(is)g(\257nite.)938 b(Also)555 b(the)800 39238 y(v)-36 b(ertex)483 b(1)g(has)f(\257nite)g (degree)g(since)p 0 1 0 0 TeXcolorcmyk 482 w(9)p (#Cnj.9) [[243 364 249 376] [1 1 1 [3 3]] [0 0 1]] pdfm Black 483 w(is)g(false.)726 b(By)483 b(K\304)-650 b(oning)483 b(in\257nit)-36 b(y)482 b(lemma)g(\241)g(con)-36 b(tains)483 b(an)f(in\257nite)800 40844 y(c)-36 b(hain,)434 b(ending)f(in)g(1)p Fo(;)h Fq(whic)-36 b(h)434 b(is)g(Conjecture)p 0 1 0 0 TeXcolorcmyk 433 w(8)p (#Cnj.8) [[282 349 288 361] [1 1 1 [3 3]] [0 0 1]] pdfm Black(.)p 24978 40844 630 630 v 800 45904 a Fp(14)2152 b(Concluding)715 b(remarks)800 48824 y Fq(This)543 b(article)g(is)g (our)f(expression)h(of)h(the)e(pleasure)g(b)36 b(eing)543 b(a)g(mathematician.)905 b(W)-108 b(e)543 b(ha)-36 b(v)g(e)543 b(written)f(it)800 50430 y(b)36 b(ecause)401 b(w)-36 b(e)401 b(found)f(the)g(sub)72 b(ject)401 b(to)g(b)36 b(e)400 b(v)-36 b(ery)402 b(attractiv)-36 b(e)401 b(and)f(w)-36 b(an)g(ted)401 b(to)f(share)h(our)f(jo)-36 b(y)402 b(with)f(others.)800 52035 y(T)-108 b(o)497 b(our)f(surprise)g(w)-36 b(e)497 b(did)g(not)f(\257nd)f(man)-36 b(y)497 b(references.)768 b(In)497 b(the)f(article)h(of)g(A.)h(Buium)e([)p 0 1 0 0 TeXcolorcmyk(2)p (#cite.Buium) [[484 249 489 261] [1 1 1 [3 3]] [0 0 1]] pdfm Black(])h(and)f(other)800 53640 y(articles)529 b(of)g(this)f(author)f (\(whic)-36 b(h)528 b(are)h(highly)f(recommended\))f(w)-36 b(e)529 b(at)f(least)h(ha)-36 b(v)g(e)528 b(found)g(that)g(there)800 55245 y(exists)461 b(authors)f(who)h(can)f(imagine)i(a)e(deriv)-72 b(ativ)-36 b(e)462 b(without)e(the)g(linearit)-36 b(y)461 b(prop)36 b(ert)-36 b(y)-108 b(.)658 b(But)460 b(the)g(article)800 56850 y(of)511 b(E.)f(J.)g(Barb)36 b(eau)510 b([)p 0 1 0 0 TeXcolorcmyk(1)p (#cite.Bar) [[174 205 180 217] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])g(w)-36 b(as)511 b(the)e(only)i(article)g(that)e(has)h (direct)g(connection)g(to)g(our)g(topic.)807 b(Most)510 b(of)800 58455 y(the)489 b(material)i(from)f(this)f(article)i(w)-36 b(e)490 b(ha)-36 b(v)g(e)490 b(rep)36 b(eated)489 b(here)g(\(not)h(alw) -36 b(a)g(ys)491 b(citing\).)747 b(W)-108 b(e)490 b(omitted)f(only)800 60060 y(the)529 b(description)g(of)i(the)e(n)-36 b(um)g(b)36 b(ers)528 b(with)i(deriv)-72 b(ativ)-36 b(es)531 b(that)e(are)h (divisible)h(b)-36 b(y)529 b(4)h(and)g(his)f(conjecture)800 61665 y(that)518 b(for)g(ev)-36 b(ery)519 b Fo(n)g Fq(there)e(exists)i (a)g(prime)f Fo(p)g Fq(suc)-36 b(h)517 b(that)h(all)h(deriv)-72 b(ativ)-36 b(es)519 b Fo(n)38927 61183 y Fl(\()p Fm(k)24 b Fl(\))40746 61665 y Fq(are)518 b(divisible)h(b)-36 b(y)519 b Fo(p)e Fq(for)800 63270 y(su\261cien)-36 b(tly)456 b(large)h Fo(k)45 b(:)457 b Fq(In)f(fact)h(according)f(to)h(Theorems)p 0 1 0 0 TeXcolorcmyk 456 w(4)p (#Thm.4) [[340 148 346 160] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)p 0 1 0 0 TeXcolorcmyk 463 w(5)p (#Thm.5) [[354 148 360 160] [1 1 1 [3 3]] [0 0 1]] pdfm Black 456 w(it)f(is)h(equiv)-72 b(alen)-36 b(t)457 b(to)f(b)36 b(e)456 b(divisible)h(b)-36 b(y)456 b Fo(p)52271 62788 y Fm(p)800 64875 y Fq(for)434 b(su\261cien)-36 b(tly)434 b(large)g Fo(k)45 b(:)434 b Fq(Th)-36 b(us)433 b(this)g(conjecture)h (is)g(a)g(bit)f(stronger)g(then)g(Conjecture)p 0 1 0 0 TeXcolorcmyk 433 w(2)p (#Cnj.2) [[478 133 484 145] [1 1 1 [3 3]] [0 0 1]] pdfm Black(.)2751 66480 y(The)327 b(de\257nition)e(of)j(the)e(arithmetic)g (deriv)-72 b(ativ)-36 b(e)328 b(itself)f(and)f(its)h(elemen)-36 b(tary)327 b(prop)36 b(erties)326 b(w)-36 b(as)327 b(already)800 68086 y(in)549 b(the)g(Putnam)f(Prize)h(comp)36 b(etition)550 b(\(it)f(w)-36 b(as)550 b(Problem)f(5)g(of)h(the)f(morning)g(session)g (in)g(Marc)-36 b(h)549 b(25,)800 69691 y(1950,)492 b([)p 0 1 0 0 TeXcolorcmyk(4)p (#cite.Gleason) [[114 90 119 102] [1 1 1 [3 3]] [0 0 1]] pdfm Black(])480 b(\))e(and)h(probably)f(w)-36 b(as)480 b(kno)-36 b(wn)479 b(in)g(folklore)i(ev)-36 b(en)479 b(earlier.)714 b(What)479 b(w)-36 b(e)480 b(ha)-36 b(v)g(e)479 b(done)f(is)h(mainly) 800 71296 y(to)536 b(generalize)g(this)f(de\257nition)g(in)g (di\256eren)-36 b(t)535 b(directions,)561 b(to)535 b(solv)-36 b(e)537 b(some)f(di\256eren)-36 b(tial)535 b(equations,)562 b(to)p Black 26150 74617 a(23)p Black eop %%Page: 24 24 24 23 bop Black 0 TeXcolorgray Black Black 800 1424 a Fq(calculate)420 b(the)e(generating)h(function)f(and)g(to)h(in)-36 b(vite)419 b(the)f(reader)g(to)h(con)-36 b(tin)g(ue)418 b(w)-36 b(ork)419 b(in)g(this)f(area.)574 b(W)-108 b(e)800 3029 y(are)347 b(grateful)g(to)g(our)f(colleagues)i(for)f(useful)f (discussion,)365 b(esp)36 b(ecially)348 b(to)e(G.)h(Almkvist,)365 b(A.)347 b(Chap)36 b(o)-36 b(v)-72 b(alo)-36 b(v,)800 4634 y(S.)477 b(Dun)-36 b(bar,)487 b(G.)477 b(Galp)36 b(erin,)487 b(S.)477 b(Shimorin)f(and)h(the)f(referee,)489 b(who)477 b(help)36 b(ed)476 b(to)h(impro)-36 b(v)g(e)477 b(the)g(text.)708 b(W)-108 b(e)800 6239 y(are)411 b(esp)36 b(ecially)413 b(grateful)e(to)g(J.)g(Bac)-36 b(k)g(elin,)417 b(who)411 b(help)36 b(ed)410 b(us)g(to)h(reduce)f(the)h(n)-36 b(um)g(b)36 b(er)409 b(of)i(conjectures)g(b)-36 b(y)800 7844 y(suggesting)434 b(ideas)g(that)f(translated)g(them)g(in)-36 b(to)434 b(theorems.)800 12282 y Fp(References)p Black 800 15202 a Fq([1])p Black 1302 w(E.)418 b(J.)h(Barb)36 b(eau,)422 b(Remark)d(on)g(an)f(arithmetic)g(deriv)-72 b(ativ)-36 b(e,)423 b Fj(Canad.)451 b(Math.)f(Bul)66 b(l.)419 b Fr(4)g Fq(\(1961\),)k(117{)3474 16807 y(122.)p Black 800 19519 a([2])p Black 1302 w(A.)434 b(Buium,)f(Arithmetic)g (analogues)i(of)f(deriv)-72 b(ations,)434 b Fj(J.)465 b(A)-33 b(lgebr)-66 b(a)896 b Fr(198)435 b Fq(\(1997\),)f(290{299.)p Black 800 22231 a([3])p Black 1302 w(J.)g(H.)g(Con)-36 b(w)g(a)g(y)434 b(and)f(R.)h(K.)g(Guy,)g Fj(The)464 b(Bo)-66 b(ok)466 b(of)e(Numb)-66 b(ers,)432 b Fq(Springer,)h(1996.)p Black 800 24943 a([4])p Black 1302 w(A.)365 b(M.)g(Gleason,)380 b(R.)365 b(E.)g(Green)-36 b(w)g(o)36 b(o)g(d,)379 b(and)365 b(L.)g(M.)g(Kelly,)380 b Fj(The)402 b(Wil)66 b(liam)401 b(L)-66 b(owel)66 b(l)403 b(Putnam)e(Mathe-)3474 26548 y(matic)-66 b(al)468 b(Comp)-66 b(etition:)604 b(Pr)-66 b(oblems)470 b(and)f(Solutions)h(1938{1964)p Fq(,)442 b(Mathematical)d(Asso)36 b(ciation)440 b(of)3474 28154 y(America,)434 b(1980.)p Black 800 30866 a([5])p Black 1302 w(J.)461 b(Renze,)467 b(S.)461 b(W)-108 b(agon,)468 b(and)460 b(B.)h(Wic)-36 b(k,)468 b(The)461 b(Gaussian)f(zo)36 b(o,)469 b Fj(Exp)-66 b(eriment.)487 b(Math.)460 b Fr(10:2)h Fq(\(2001\),)3474 32471 y(161{173.)p 800 35282 52000 45 v 800 37533 a(2000)428 b Fj(Mathematics)458 b(Subje)-66 b(ct)457 b(Classi\257c)-66 b(ation)p Fq(:)573 b(Primary)428 b(11A25;)j(Secondary)c(11A41,)i(11N05,)h(11N56,)800 39138 y(11Y55.)800 40743 y Fj(Keywor)-66 b(ds:)569 b(A)-33 b(rithmetic)405 b(derivative,)416 b(Goldb)-66 b(ach's)408 b(Conje)-66 b(ctur)g(e,)416 b(the)408 b(Twin)f(Prime)g(Conje)-66 b(ctur)g(e,)416 b(prime)800 42348 y(numb)-66 b(ers,)464 b(L)-66 b(eibnitz)462 b(rule,)j(inte)-66 b(ger)462 b(se)-66 b(quenc)g(e,)463 b(gener)-66 b(ating)462 b(function.)p 800 43979 V 800 46304 a Fq(\(Concerned)433 b(with)h(sequence)p 0 1 0 0 TeXcolorcmyk 433 w(A003415)p 16090 46516 4878 54 v [[217 300 261 312] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A003415) pdfm Black 2 w(.\))p 800 48009 52000 45 v 800 51060 a(Receiv)-36 b(ed)603 b(April)g(4,)645 b(2003;)688 b(revised)603 b(v)-36 b(ersion)603 b(receiv)-36 b(ed)603 b(July)f(27,)646 b(2003.)1086 b(Published)602 b(in)g Fj(Journal)620 b(of)800 52665 y(Inte)-66 b(ger)463 b(Se)-66 b(quenc)g(es)p Fq(,)431 b(Septem)-36 b(b)36 b(er)433 b(17,)h(2003.)p 800 54296 V 800 56546 a(Return)f(to)p 0 1 0 0 TeXcolorcmyk 433 w(Journal)h(of)g(In)-36 b(teger)434 b(Sequences)f(home)g(page)p [[133 208 338 220] [1 1 1 [3 3]] [0 0 1]] (http://www.math.uwaterloo.ca/JIS/) pdfm Black(.)p Black 26150 74617 a(24)p Black eop %%Trailer end end