%%Page: 1 1 1 0 bop Black 0 TeXcolorgray Black Black Black Black 10800 7607 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 03.1.3) 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 1998 11452 a Fu(A)648 b(NOTE)g(ON)h (ARITHMETIC)e(PR)-42 b(OGRESSIONS)648 b(ON)h(ELLIPTIC)d(CUR)-166 b(VES)21774 15599 y Ft(Garik)-72 b(ai)434 b(Campb)36 b(ell)p Black Black 14677 17531 a(Departmen)-36 b(t)433 b(of)h(Mathematics)g(and)f(Statistics)21110 19081 y(Sw)-36 b(arthmore)433 b(College)20270 20631 y(Sw)-36 b(arthmore,)433 b(P)-108 b(A)433 b(19081)25464 22180 y(USA)p Black 4785 25457 a Fs(Abstra)-25 b(ct.)p Black 554 w Fr(Andrew)316 b(Bremner)f(\()p Fq(Exp)-57 b(eriment.)349 b(Math.)473 b Fp(8)316 b Fr(\(1999\),)330 b(409{413\))319 b(has)d(describ)31 b(ed)315 b(a)h(tec)-31 b(h-)4785 26785 y(nique)548 b(for)f(pro)31 b(ducing)547 b(in\257nite)i(families)g(of)e(elliptic)j(curv)-31 b(es)546 b(con)-31 b(taining)550 b(length)f(7)e(and)g(length)i(8)4785 28114 y(arithmetic)387 b(progressions.)538 b(This)384 b(note)h(describ)31 b(es)384 b(another)h(w)-31 b(a)g(y)386 b(to)f(pro)31 b(duce)384 b(in\257nite)h(families)i(of)e(el-)4785 29442 y(liptic)307 b(curv)-31 b(es)305 b(con)-31 b(taining)308 b(length)f(7)f(and)f(length)i(8)f(arithmetic)h(progressions.)471 b(W)-92 b(e)305 b(illustrate)i(ho)-31 b(w)307 b(the)4785 30770 y(tec)-31 b(hnique)307 b(articulated)h(here)d(giv)-31 b(es)307 b(an)f(easy)f(w)-31 b(a)g(y)308 b(to)e(pro)31 b(duce)305 b(an)h(elliptic)i(curv)-31 b(e)306 b(con)-31 b(taining)309 b(a)d(length)4785 32099 y(12)282 b(progression)f(and)g (an)g(in\257nite)i(family)g(of)e(elliptic)i(curv)-31 b(es)281 b(con)-31 b(taining)284 b(a)d(length)h(9)f(progression,)300 b(with)4785 33427 y(the)370 b(ca)-31 b(v)g(eat)371 b(that)f(these)f (curv)-31 b(es)369 b(are)g(not)h(in)f(W)-92 b(eierstrass)368 b(form.)21243 36931 y Ft(1.)651 b Fo(Intr)-30 b(oduction.)2128 39255 y Ft(There)408 b(are)f(t)-36 b(w)g(o)408 b(\(a\261ne\))f(mo)36 b(dels)407 b(of)h(elliptic)g(curv)-36 b(e)408 b(that)e(are)i(v)-36 b(ery)408 b(common.)570 b(They)408 b(are)f Fn(y)47991 38773 y Fm(2)48886 39255 y Ft(=)368 b Fn(f)142 b Ft(\()p Fn(x)p Ft(\))800 40805 y(where)563 b Fn(f)142 b Ft(\()p Fn(x)p Ft(\))562 b(is)h(either)f(a)h(cubic)f(or)h(a)g(quartic.)966 b(W)-108 b(e)562 b(will)i(sa)-36 b(y)564 b(that)e Fl(p)-66 b(oints)582 b(on)i(a)g(p)-66 b(articular)583 b(mo)-66 b(del)800 42355 y(of)534 b(an)h(el)66 b(liptic)533 b(curve)h(ar)-66 b(e)535 b(in)e(arithmetic)g(pr)-66 b(o)g(gr)g(ession)508 b Ft(if)i(their)f Fn(x)p Ft(-co)36 b(ordinates)509 b(form)h(an)f (arithmetic)800 43905 y(progression.)543 b(F)-108 b(or)326 b(example,)349 b(Buhler,)e(Gross)326 b(and)g(Zagier)h([)p 0 1 0 0 TeXcolorcmyk(3)p (#cite.BU) [[345 322 351 334] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])f(found)g(that)g(the)g(p)36 b(oin)-36 b(ts)326 b(\()p Fk(\241)p Ft(3)p Fn(;)221 b Ft(0\))p Fn(;)328 b Ft(\()p Fk(\241)p Ft(2)p Fn(;)221 b Ft(3\))p Fn(;)800 45454 y Ft(\()p Fk(\241)p Ft(1)p Fn(;)g Ft(3\))p Fn(;)491 b Ft(\(0)p Fn(;)221 b Ft(2\))p Fn(;)490 b Ft(\(1)p Fn(;)221 b Ft(0\))p Fn(;)490 b Ft(\(2)p Fn(;)221 b Ft(0\))p Fn(;)490 b Ft(\(3)p Fn(;)221 b Ft(3\))p Fn(;)491 b Ft(and)d(\(4)p Fn(;)221 b Ft(6\))490 b(form)g(an)f(arithmetic)g(progression)g(of)h (length)f(8)g(on)800 47004 y(the)433 b(curv)-36 b(e)434 b Fn(y)7191 46522 y Fm(2)8011 47004 y Ft(+)295 b Fn(y)417 b Ft(=)369 b(\()p Fn(x)295 b Fk(\241)g Ft(1\)\()p Fn(x)g Fk(\241)h Ft(2\)\()p Fn(x)f Ft(+)g(3\).)578 b(Moreo)-36 b(v)g(er,)435 b(Bremner)d([)p 0 1 0 0 TeXcolorcmyk(2)p (#cite.BR) [[400 294 406 306] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])i(pro)-36 b(v)g(es:)p Black 800 49456 a Fu(Theorem)499 b(1.1.)p Black 554 w Fl(Each)466 b(p)-66 b(oint)463 b(on)i(the)g(el)66 b(liptic)464 b(curve)18899 51771 y Fn(C)g Ft(:)369 b Fn(y)21706 51222 y Fm(2)22601 51771 y Ft(=)g Fn(x)24721 51222 y Fm(3)25542 51771 y Fk(\241)295 b Fn(x)27609 51222 y Fm(2)28430 51771 y Fk(\241)h Ft(36)p Fn(x)f Ft(+)g(36)800 54086 y Fl(c)-66 b(orr)g(esp)g(onds)377 b(to)h(an)f(el)66 b(liptic)377 b(curve)h(in)e(Weierstr)-66 b(ass)377 b(form)g(c)-66 b(ontaining)375 b(at)j(le)-66 b(ast)377 b(8)h(p)-66 b(oints)377 b(in)g(arithmetic)800 55636 y(pr)-66 b(o)g(gr)g(ession.)2128 58088 y Ft(Before)470 b(pro)-36 b(ving)468 b(this)h(theorem,)477 b(Bremner)468 b(considers)g(the)g(follo)-36 b(wing)471 b(strategy)-108 b(.)683 b(First)468 b(he)h(remarks)800 59637 y(that)382 b(an)-36 b(y)382 b(monic)g(degree)g(8)g(p)36 b(olynomial,)395 b Fn(P)181 b Ft(\()p Fn(x)p Ft(\),)392 b(can)382 b(b)36 b(e)382 b(written)g(as)g Fn(Q)p Ft(\()p Fn(x)p Ft(\))38220 59155 y Fm(2)38936 59637 y Fk(\241)190 b Fn(R)11 b Ft(\()p Fn(x)p Ft(\))382 b(where)g(the)f(degree)800 61187 y(of)398 b Fn(R)11 b Ft(\()p Fn(x)p Ft(\))397 b(is)g(less)h(than) f(or)g(equal)h(to)f(3.)567 b(If)398 b Fn(R)11 b Ft(\()p Fn(x)p Ft(\))397 b(has)g(degree)g(precisely)h(3)f(and)g(no)g(rep)36 b(eated)397 b(zeros,)405 b(then)800 62737 y Fn(y)1482 62255 y Fm(2)2526 62737 y Ft(=)517 b Fn(R)11 b Ft(\()p Fn(x)p Ft(\))521 b(is)h(an)f(elliptic)h(curv)-36 b(e)521 b(and)f(for)i(eac)-36 b(h)521 b(zero,)544 b Fn(\256)8 b Ft(,)544 b(of)522 b Fn(P)181 b Ft(\()p Fn(x)p Ft(\),)543 b(this)521 b(elliptic)h(curv)-36 b(e)521 b(con)-36 b(tains)521 b(a)800 64287 y(pair)454 b(of)g(p)36 b(oin)-36 b(ts)453 b(with)h Fn(x)p Ft(-co)36 b(ordinate)454 b Fn(\256)8 b Ft(.)639 b(So)453 b(one)h(p)36 b(ossible)454 b(strategy)g(for)g(pro) 36 b(ducing)453 b(an)h(elliptic)g(curv)-36 b(e)800 65836 y(with)407 b(an)g(arithmetic)f(progression)h(of)h(length)f(8)g(migh)-36 b(t)406 b(b)36 b(e)407 b(to)g(let)g Fn(P)181 b Ft(\()p Fn(x)p Ft(\))368 b(=)h Fn(x)p Ft(\()p Fn(x)241 b Ft(+)f(1\)\()p Fn(x)g Ft(+)h(2\))221 b Fk(\242)g(\242)g(\242)h Ft(\()p Fn(x)241 b Ft(+)f(7\))800 67386 y(and)467 b(compute)f(the)h(corresp)36 b(onding)466 b Fn(R)11 b Ft(\()p Fn(x)p Ft(\))467 b(so)h(that)e Fn(P)181 b Ft(\()p Fn(x)p Ft(\))426 b(=)g Fn(Q)p Ft(\()p Fn(x)p Ft(\))34447 66904 y Fm(2)35291 67386 y Fk(\241)318 b Fn(R)11 b Ft(\()p Fn(x)p Ft(\).)679 b(Unfortunately)-108 b(,)475 b(in)467 b(this)800 68936 y(case,)461 b Fn(R)11 b Ft(\()p Fn(x)p Ft(\))454 b(is)h(linear)g(and)f(so)h(this)f(strategy)h (fails)h(for)f Fl(any)f Ft(degree)h(8)g(p)36 b(olynomial)456 b(whose)f(zeros)g(form)800 70486 y(an)525 b(arithmetic)f(progression.) 852 b(The)525 b(goal)h(of)f(this)g(note)f(is)h(to)g(illustrate)g(ho)-36 b(w)525 b(to)g(turn)e(this)h(strategy)800 72035 y(in)-36 b(to)434 b(a)f(successful)h(one.)p Black 26544 73364 a Fj(1)p Black eop %%Page: 2 2 2 1 bop Black 0 TeXcolorgray Black 800 886 a Fj(2)8453 b(A)444 b(NOTE)f(ON)h(ARITHMETIC)d(PR)-28 b(OGRESSIONS)442 b(ON)i(ELLIPTIC)d(CUR)-114 b(VES)p Black 13131 3542 a Ft(2.)651 b Fo(Arithmetic)503 b(Pr)-30 b(ogressions)503 b(of)f(Length)f(8)2128 5867 y Ft(The)372 b(statemen)-36 b(t)371 b(that)h(a)g(degree)g(8)g(p)36 b(olynomial)373 b(can)f(b)36 b(e)371 b(written)h(as)g Fn(Q)p Ft(\()p Fn(x)p Ft(\))39001 5385 y Fm(2)39696 5867 y Fk(\241)169 b Fn(P)181 b Ft(\()p Fn(x)p Ft(\))372 b(is)g(a)g(sp)36 b(ecial)373 b(case)800 7417 y(of)434 b(the)f(follo)-36 b(wing:)p Black 800 9861 a Fu(Prop)42 b(osition)577 b(2.1.)p Black 589 w Fl(If)525 b Fn(P)181 b Ft(\()p Fn(x)p Ft(\))527 b Fl(is)g(a)g(monic)e(p)-66 b(olynomial)526 b(of)h(de)-66 b(gr)g(e)g(e)524 b Ft(2)p Fn(n)k Fl(de\257ne)-66 b(d)525 b(over)i(a)g(\257eld)f Fn(k)45 b Fl(,)542 b(then)800 11411 y(ther)-66 b(e)464 b(ar)-66 b(e)464 b(unique)h(p)-66 b(olynomials)464 b Fn(Q)p Ft(\()p Fn(x)p Ft(\))h Fl(and)g Fn(R)11 b Ft(\()p Fn(x)p Ft(\))465 b Fl(de\257ne)-66 b(d)463 b(over)i Fn(k)510 b Fl(such)465 b(that)p Black 2779 13337 a Ft(\(1\))p Black 553 w Fn(P)181 b Ft(\()p Fn(x)p Ft(\))369 b(=)g Fn(Q)p Ft(\()p Fn(x)p Ft(\))12295 12855 y Fm(2)13115 13337 y Fk(\241)296 b Fn(R)11 b Ft(\()p Fn(x)p Ft(\))464 b Fl(and)p Black 2779 14887 a Ft(\(2\))p Black 553 w Fl(the)h(de)-66 b(gr)g(e)g(e)463 b(of)h Fn(R)11 b Ft(\()p Fn(x)p Ft(\))465 b Fl(is)g(strictly)f(less)h(than)g Fn(n)p Fl(.)2128 17331 y Ft(Since)326 b Fn(R)11 b Ft(\()p Fn(x)p Ft(\))325 b(is)h(a)g(square)g(at)g(ev)-36 b(ery)327 b(zero)f(of)g Fn(P)181 b Ft(\()p Fn(x)p Ft(\),)348 b(if)326 b Fn(R)11 b Ft(\()p Fn(x)p Ft(\))326 b(is)g(a)g(cubic)f(or)h(a)g (quartic)g(with)g(no)g(rep)36 b(eated)800 18881 y(zeros,)541 b(then)519 b(w)-36 b(e)519 b(can)g(pro)36 b(duce)519 b(elliptic)h(curv)-36 b(es)519 b Fn(y)26180 18399 y Fm(2)27220 18881 y Ft(=)c Fn(R)11 b Ft(\()p Fn(x)p Ft(\))518 b(with)i(great)f(con) -36 b(trol)520 b(o)-36 b(v)g(er)519 b(man)-36 b(y)520 b(of)g(the)800 20431 y Fn(x)p Ft(-co)36 b(ordinates.)p Black 800 22876 a Fl(R)-66 b(emark)617 b Ft(2.2)p Fl(.)p Black 560 w Ft(W)-108 b(e)444 b(note)h(that)f(Mestre)h([)p 0 1 0 0 TeXcolorcmyk(9)p (#cite.ME1) [[261 511 267 523] [1 1 1 [3 3]] [0 0 1]] pdfm Black(])g(w)-36 b(as)446 b(\257rst)e(to)g(observ)-36 b(e)445 b(that)g(this)f(relativ)-36 b(ely)447 b(simple)e(prop)36 b(o-)800 24425 y(sition)610 b(could)g(b)36 b(e)610 b(used)f(to)h(pro)36 b(duce)609 b(elliptic)h(curv)-36 b(es)610 b(of)h(large)g(rank.)1107 b(Since)610 b(Mestre's)f(\257rst)h(pap)36 b(er)800 25975 y(exploiting)491 b(this)f(idea,)505 b(man)-36 b(y)490 b(others)g(\([)p 0 1 0 0 TeXcolorcmyk(4)p (#cite.CA1) [[262 483 268 495] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(],)505 b([)p 0 1 0 0 TeXcolorcmyk(6)p (#cite.FE) [[282 483 288 495] [1 1 1 [3 3]] [0 0 1]] pdfm Black(],)g([)p 0 1 0 0 TeXcolorcmyk(7)p (#cite.KI1) [[302 483 308 495] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(],)g([)p 0 1 0 0 TeXcolorcmyk(8)p (#cite.KU) [[322 483 328 495] [1 1 1 [3 3]] [0 0 1]] pdfm Black(],)g([)p 0 1 0 0 TeXcolorcmyk(11)p (#cite.NA1) [[342 483 354 495] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(]\))491 b(ha)-36 b(v)g(e)490 b(used)f(the)h(prop)36 b(osition)490 b(in)g(clev)-36 b(er)800 27525 y(w)g(a)g(ys)404 b(to)g(pro)36 b(duce)402 b(elliptic)i(curv)-36 b(es)403 b(and)g(in\257nite)g(families)i(of)f(elliptic)g(curv)-36 b(es)403 b(with)g(the)g(largest)h(kno)-36 b(wn)800 29075 y(rank)434 b(\(often)f(with)h(some)g(condition)f(on)h(the)f(torsion)h (subgroup\).)2128 31519 y(No)-36 b(w)435 b(consider)e(the)g(p)36 b(olynomial)16191 35005 y Fn(p)16844 35204 y Fi(t)17239 35005 y Ft(\()p Fn(x)p Ft(\))368 b(=)h(\()p Fn(x)295 b Fk(\241)h Fn(t)p Ft(\))24584 34456 y Fm(2)25944 33344 y(5)25330 33743 y Fh(Y)25362 36542 y Fi(j)51 b Fm(=0)27028 35005 y Ft(\()p Fn(x)295 b Fk(\241)g Fn(j)75 b Ft(\))802 b Fk(2)369 b Fg(Q)p Ft(\()p Fn(t)p Ft(\)[)p Fn(x)p Ft(])p Fn(:)800 38633 y Ft(In)433 b(this)h(case,)g(w)-36 b(e)434 b(can)g(write)20516 40392 y Fn(p)21169 40591 y Fi(t)21564 40392 y Ft(\()p Fn(x)p Ft(\))368 b(=)h Fn(q)25641 40591 y Fi(t)26036 40392 y Ft(\()p Fn(x)p Ft(\))27787 39844 y Fm(2)28608 40392 y Fk(\241)295 b Fn(f)30577 40591 y Fi(t)30972 40392 y Ft(\()p Fn(x)p Ft(\))p Fn(;)800 42423 y Ft(where)434 b Fn(f)5199 42622 y Fi(t)5594 42423 y Ft(\()p Fn(x)p Ft(\))f(is)h(a)g(p)36 b(olynomial)435 b(of)f(degree)g(3)g(in)f Fg(Q)p Ft(\()p Fn(t)p Ft(\)[)p Fn(x)p Ft(])h(suc)-36 b(h)433 b(that)p Black 2779 44349 a(\(1\))p Black 553 w(the)g(discriminan)-36 b(t)433 b(of)i Fn(f)16843 44548 y Fi(t)17238 44349 y Ft(\()p Fn(x)p Ft(\))e(is)h(an)f(irreducible)h(p)36 b(olynomial)435 b(in)e Fg(Q)p Ft([)p Fn(t)p Ft(])p Black 2779 45899 a(\(2\))p Black 553 w(the)g(co)36 b(e\261cien)-36 b(t)434 b(of)g Fn(x)15524 45417 y Fm(3)16484 45899 y Ft(is)g Fn(c)p Ft(\(2)p Fn(t)294 b Fk(\241)i Ft(5\),)434 b(where)f Fn(c)369 b Fk(2)g Fg(Q)p Ft(.)800 47825 y(Therefore,)434 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)434 b(that)p Black 800 50270 a Fu(Theorem)593 b(2.3.)p Black 595 w Fl(The)540 b(curve)f Fn(E)17420 50469 y Fi(t)18354 50270 y Fl(de\257ne)-66 b(d)538 b(by)i Fn(y)25263 49788 y Fm(2)26296 50270 y Ft(=)506 b Fn(f)28455 50469 y Fi(t)28851 50270 y Ft(\()p Fn(x)p Ft(\))539 b Fl(is)g(an)h(el)66 b(liptic)538 b(curve)i(de\257ne) -66 b(d)538 b(over)h Fg(Q)p Ft(\()p Fn(t)p Ft(\))p Fl(,)800 51820 y(c)-66 b(ontaining)531 b(at)j(le)-66 b(ast)533 b(six)h(p)-66 b(oints)533 b(in)g(arithmetic)f(pr)-66 b(o)g(gr)g(ession)533 b(and)g(for)h(e)-66 b(ach)533 b Fn(t)40817 52019 y Fm(0)41840 51820 y Fk(2)496 b Fg(Q)p Fl(,)551 b Fn(t)45675 52019 y Fm(0)46697 51820 y Fk(6)p Ft(=)497 b(5)p Fn(=)p Ft(2)p Fl(,)551 b(the)800 53369 y(sp)-66 b(e)g(cialization)543 b(of)i Fn(E)11388 53568 y Fi(t)12327 53369 y Fl(at)g Fn(t)517 b Ft(=)g Fn(t)16954 53568 y Fm(0)18024 53369 y Fl(gives)544 b(an)h(el)66 b(liptic)544 b(curve)g(de\257ne)-66 b(d)544 b(over)g Fg(Q)h Fl(c)-66 b(ontaining)542 b(at)j(le)-66 b(ast)544 b(six)800 54919 y(p)-66 b(oints)464 b(in)g(arithmetic)f(pr)-66 b(o)g(gr)g(ession.)2128 57364 y Ft(W)-108 b(e)607 b(next)g(observ)-36 b(e)607 b(that)f Fn(f)16078 57563 y Fi(t)16473 57364 y Ft(\(6\))h(is)g(a)g(conic)g(in)g Fg(Q)p Ft([)p Fn(t)p Ft(])g(whic)-36 b(h)607 b(is)g(a)g(rational)g(square)g(when)g Fn(t)663 b Ft(=)g(6.)800 58913 y(Therefore,)434 b(w)-36 b(e)434 b(can)g(parameterize)g(all)g(rational)g(solutions)g(to)g Fn(y)33435 58431 y Fm(2)34330 58913 y Ft(=)368 b Fn(f)36351 59112 y Fi(t)36747 58913 y Ft(\(6\))433 b(b)-36 b(y)434 b(letting)18363 62060 y Fn(t)1107 b Ft(=)22191 61161 y(6)p Fn(m)23979 60679 y Fm(2)24800 61161 y Fk(\241)296 b Ft(126)p Fn(m)f Fk(\241)h Ft(285360)p 22191 61755 12552 54 v 25198 62971 a Fn(m)26336 62588 y Fm(2)27156 62971 y Fk(\241)g Ft(72256)34876 62060 y Fn(:)p Black 14889 w Ft(\(2.1\))p Black 800 64941 a(Since)433 b(no)h(rational)g(v)-72 b(alue)434 b(of)g Fn(m)g Ft(giv)-36 b(es)434 b Fn(t)369 b Ft(=)g(5)p Fn(=)p Ft(2,)434 b(w)-36 b(e)434 b(ha)-36 b(v)g(e:)p Black 800 67386 a Fu(Corrolary)678 b(2.4.)p Black 625 w Fl(L)-66 b(et)589 b Fn(g)13751 67585 y Fi(m)14639 67386 y Ft(\()p Fn(x)p Ft(\))h Fl(b)-66 b(e)591 b(the)f(p)-66 b(olynomial)590 b Fn(f)28294 67585 y Fi(t)28690 67386 y Ft(\()p Fn(x)p Ft(\))g Fl(with)h Fn(t)g Fl(given)e(by)i(\()p 0 1 0 0 TeXcolorcmyk(2.1)p (#equation.2.1) [[440 111 456 123] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\).)975 b(The)591 b(curve)g Fn(E)51913 67585 y Fi(m)800 68936 y Fl(de\257ne)-66 b(d)432 b(by)h Fn(y)7496 68454 y Fm(2)8391 68936 y Ft(=)369 b Fn(g)10395 69135 y Fi(m)11282 68936 y Ft(\()p Fn(x)p Ft(\))433 b Fl(is)h(an)f(el)66 b(liptic)433 b(curve)g(de\257ne)-66 b(d)432 b(over)h Fg(Q)p Ft(\()p Fn(m)p Ft(\))g Fl(c)-66 b(ontaining)431 b(at)i(le)-66 b(ast)433 b(seven)g(p)-66 b(oints)800 70486 y(in)435 b(arithmetic)f(pr)-66 b(o)g(gr)g(ession)435 b(and)h(for)f(e)-66 b(ach)436 b Fn(m)23882 70685 y Fm(0)24776 70486 y Fk(2)369 b Fg(Q)p Fl(,)441 b(the)436 b(sp)-66 b(e)g(cialization)434 b(of)i Fn(E)40404 70685 y Fi(m)41727 70486 y Fl(at)g Fn(m)369 b Ft(=)g Fn(m)47285 70685 y Fm(0)48246 70486 y Fl(gives)435 b(an)800 72035 y(el)66 b(liptic)464 b(curve)h(de\257ne)-66 b(d)463 b(over)i Fg(Q)g Fl(c)-66 b(ontaining)462 b(at)j(le)-66 b(ast)464 b(seven)g(p)-66 b(oints)464 b(in)g(arithmetic)f(pr)-66 b(o)g(gr)g(ession.)p Black Black eop %%Page: 3 3 3 2 bop Black 0 TeXcolorgray Black 9765 886 a Fj(A)444 b(NOTE)f(ON)h(ARITHMETIC)d(PR)-28 b(OGRESSIONS)442 b(ON)i(ELLIPTIC)d (CUR)-114 b(VES)8452 b(3)p Black 2128 3542 a Ft(If)421 b(w)-36 b(e)419 b(con)-36 b(tin)g(ue)419 b(in)g(this)h(v)-36 b(ein)420 b(and)f(explore)h(the)f(conditions)g(imp)36 b(osed)420 b(b)-36 b(y)419 b Fn(y)40274 3060 y Fm(2)41169 3542 y Ft(=)368 b Fn(g)43172 3741 y Fi(m)44060 3542 y Ft(\(7\),)422 b(w)-36 b(e)420 b(\257nd)f(the)800 5092 y(follo)-36 b(wing.)p Black 800 7558 a Fu(Theorem)499 b(2.5.)p Black 554 w Fl(L)-66 b(et)464 b Fn(D)501 b Fl(b)-66 b(e)464 b(the)h(el)66 b(liptic)464 b(curve)g(de\257ne)-66 b(d)464 b(by)10353 9898 y Fn(D)405 b Ft(:)369 b Fn(y)13250 9350 y Fm(2)14145 9898 y Ft(=)g Fk(\241)p Ft(264815)p Fn(m)21597 9350 y Fm(4)22419 9898 y Fk(\241)296 b Ft(19343520)p Fn(m)30086 9350 y Fm(3)30909 9898 y Ft(+)f(62846856064)p Fn(m)40504 9350 y Fm(2)18668 11835 y Fk(\241)p Ft(2906312951808)p Fn(m)300 b Fk(\241)295 b Ft(495507443511296)p Fn(:)800 14175 y Fl(L)-66 b(et)10338 16515 y Fn(g)10961 16714 y Fm(3)12593 16515 y Ft(=)1107 b Fk(\241)p Ft(18816)p Fn(m)20133 15967 y Fm(4)20956 16515 y Ft(+)294 b(677376)p Fn(m)27300 15967 y Fm(3)28123 16515 y Ft(+)h(1922543616)p Fn(m)37068 15967 y Fm(2)17502 18452 y Fk(\241)p Ft(48944480256)p Fn(m)j Fk(\241)e Ft(40678301368320)p Fn(;)10338 20390 y(g)10961 20589 y Fm(2)12593 20390 y Ft(=)1107 b(236896)p Fn(m)19750 19841 y Fm(4)20573 20390 y Fk(\241)295 b Ft(9821952)p Fn(m)27589 19841 y Fm(3)28412 20390 y Fk(\241)g Ft(22598349824)p Fn(m)38028 19841 y Fm(2)17502 22327 y Ft(+508953231360)p Fn(m)j Ft(+)d(520252184657920)p Fn(;)10338 24264 y(g)10961 24463 y Fm(1)12593 24264 y Ft(=)1107 b Fk(\241)p Ft(958800)p Fn(m)20783 23716 y Fm(4)21606 24264 y Ft(+)295 b(40985280)p Fn(m)29251 23716 y Fm(3)30074 24264 y Ft(+)g(89932669440)p Fn(m)39669 23716 y Fm(2)17502 26201 y Fk(\241)p Ft(1957723729920)p Fn(m)k Fk(\241)c Ft(2113363439616000)p Fn(;)692 b Fl(and)10338 28138 y Fn(g)10961 28337 y Fm(0)12593 28138 y Ft(=)1107 b(1292769)p Fn(m)20400 27590 y Fm(4)21223 28138 y Fk(\241)295 b Ft(57304800)p Fn(m)28889 27590 y Fm(3)29713 28138 y Fk(\241)g Ft(118795148928)p Fn(m)39979 27590 y Fm(2)17502 30076 y Ft(+2647001548800)p Fn(m)j Ft(+)d(2758336954896384)p Fn(:)800 32416 y Fl(Then)16634 34213 y Fn(E)17675 33665 y Ff(0)17597 34541 y Fi(m)18853 34213 y Ft(:)369 b Fn(y)20265 33665 y Fm(2)21160 34213 y Ft(=)g Fn(g)23164 34412 y Fm(3)24154 34213 y Fn(x)24893 33665 y Fm(3)25714 34213 y Ft(+)295 b Fn(g)27644 34412 y Fm(2)28635 34213 y Fn(x)29374 33665 y Fm(2)30195 34213 y Ft(+)g Fn(g)32125 34412 y Fm(1)33115 34213 y Fn(x)h Ft(+)e Fn(g)36079 34412 y Fm(0)36605 34213 y Fn(;)800 36282 y Fl(is)441 b(an)h(el)66 b(liptic)440 b(curve)i(de\257ne)-66 b(d)440 b(over)h Fg(Q)p Ft(\()p Fn(D)36 b Ft(\))442 b Fl(c)-66 b(ontaining)438 b(the)j(8)h(p)-66 b(oints)441 b(in)g(arithmetic)e(pr)-66 b(o)g(gr)g(ession)440 b(with)800 37832 y Fn(x)p Fl(-c)-66 b(o)g(or)g(dinates)463 b(0,)i(1,)f(2,)h Fn(:)221 b(:)g(:)467 b Fl(,)d(7.)p Black 800 40298 a(Pr)-66 b(o)g(of.)p Black 552 w Fn(E)5816 39816 y Ff(0)5738 40626 y Fi(m)6979 40298 y Ft(is)354 b(isomorphic)g(to)g Fn(E)17183 40497 y Fi(m)18424 40298 y Ft(via)h(the)e(c)-36 b(hange)353 b(of)h(v)-72 b(ariables)355 b Fn(y)417 b Fk(7!)369 b Fn(y)48 b(=)p Ft(\()p Fn(m)39291 39816 y Fm(2)39948 40298 y Fk(\241)132 b Ft(72256\).)554 b(Substituting)800 41848 y Fn(x)369 b Ft(=)g(7)434 b(in)-36 b(to)433 b Fn(E)8051 41366 y Ff(0)7973 42176 y Fi(m)8861 41848 y Ft(,)g(w)-36 b(e)434 b(get)g(the)f(curv)-36 b(e)434 b Fn(D)36 b Ft(.)30843 b Fe(\244)2128 44365 y Ft(Moreo)-36 b(v)g(er,)472 b(if)464 b(w)-36 b(e)464 b(let)f Fn(D)36 b Ft(\()p Fg(Q)p Ft(\))464 b(b)36 b(e)463 b(the)g(group)g(of)h (rational)g(p)36 b(oin)-36 b(ts)463 b(on)g Fn(D)36 b Ft(,)472 b(then)462 b(w)-36 b(e)464 b(ha)-36 b(v)g(e)463 b(that)g Fn(D)36 b Ft(\()p Fg(Q)p Ft(\))800 45915 y(is)434 b(in\257nite.)578 b(More)433 b(sp)36 b(eci\257cally)-108 b(,)435 b(w)-36 b(e)434 b(ha)-36 b(v)g(e:)p Black 800 48381 a Fu(Prop)42 b(osition)500 b(2.6.)p Black 554 w Fn(D)h Fl(has)465 b(r)-66 b(ank)465 b(2)g(and)g(torsion)f(sub)-66 b(gr)g(oup)465 b Fg(Z)p Fn(=)p Ft(2)p Fg(Z)p Fl(.)p Black 800 50847 a(Pr)-66 b(o)g(of.)p Black 552 w Ft(A)330 b(short)e(computer) h(searc)-36 b(h)329 b(rev)-36 b(eals)330 b(that)f Fn(O)405 b Ft(=)369 b(\()p Fk(\241)p Ft(88)p Fn(;)221 b Ft(15628032\))333 b(is)d(a)f(p)36 b(oin)-36 b(t)329 b(in)h Fn(D)36 b Ft(\()p Fg(Q)p Ft(\).)543 b(T)-108 b(aking)800 52397 y Fn(O)470 b Ft(tak)-36 b(en)433 b(to)h(b)36 b(e)433 b(the)h(iden)-36 b(tit)g(y)-108 b(,)433 b Fn(D)36 b Ft(\()p Fg(Q)p Ft(\))433 b(is)h(generated)f(b)-36 b(y)11955 54737 y Fn(P)12793 54936 y Fm(0)14426 54737 y Ft(=)1107 b(\(10984)p Fn(=)p Ft(79)p Fn(;)221 b Fk(\241)p Ft(80015523840)p Fn(=)p Ft(6241\))442 b(and)11955 56674 y Fn(P)12793 56873 y Fm(1)14426 56674 y Ft(=)1107 b(\()p Fk(\241)p Ft(1363640)p Fn(=)p Ft(2531)p Fn(;)221 b Ft(31969540657152)p Fn(=)p Ft(6405961\))p Fn(;)800 59014 y Ft(and)433 b(con)-36 b(tains)434 b(the)f(p)36 b(oin)-36 b(t)433 b(of)h(order)f(t)-36 b(w)g(o:)15723 61354 y Fn(P)16561 61553 y Fm(2)18195 61354 y Ft(=)1106 b(\(10984)p Fn(=)p Ft(79)p Fn(;)221 b Ft(80015523840)p Fn(=)p Ft(6241\))p Fn(:)51767 63694 y Fe(\244)2128 66211 y Ft(\(The)434 b(calculations)g(ab)36 b(o)-36 b(v)g(e)435 b(w)-36 b(ere)433 b(p)36 b(erformed)434 b(with)f(the)g(help)g(of)i Fd(mwrank)c Ft([)p 0 1 0 0 TeXcolorcmyk(5)p (#cite.CR) [[429 121 435 133] [1 1 1 [3 3]] [0 0 1]] pdfm Black(])j(and)f(GP)g([)p 0 1 0 0 TeXcolorcmyk(1)p (#cite.GP) [[489 121 495 133] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(].\))2128 67761 y(An)h(immediate)f(consequence)h(of)g(the)f (prop)36 b(osition)434 b(ab)36 b(o)-36 b(v)g(e)434 b(is)g(the)f(follo) -36 b(wing:)p Black 800 70227 a Fu(Corrolary)406 b(2.7.)p Black 477 w Fl(Each)373 b(p)-66 b(oint)371 b(on)h(the)h(el)66 b(liptic)371 b(curve)i Fn(D)408 b Fl(c)-66 b(orr)g(esp)g(onds)372 b(to)h(an)f(el)66 b(liptic)372 b(curve)g(in)g(Weier-)800 71777 y(str)-66 b(ass)465 b(form)g(c)-66 b(ontaining)462 b(at)i(le)-66 b(ast)465 b(8)g(p)-66 b(oints)464 b(in)g(arithmetic)f(pr) -66 b(o)g(gr)g(ession.)p Black Black eop %%Page: 4 4 4 3 bop Black 0 TeXcolorgray Black 800 886 a Fj(4)8453 b(A)444 b(NOTE)f(ON)h(ARITHMETIC)d(PR)-28 b(OGRESSIONS)442 b(ON)i(ELLIPTIC)d(CUR)-114 b(VES)p Black Black 800 3542 a Fl(R)-66 b(emark)515 b Ft(2.8)p Fl(.)p Black 478 w Ft(This)335 b(condition)f(is)h(v)-36 b(ery)335 b(similar)g(to)f(the)g (condition)g(found)g(in)g(Bremner's)g(construction{)800 5092 y(namely)-108 b(,)534 b(that)514 b(p)36 b(oin)-36 b(ts)513 b(on)h(the)f(curv)-36 b(e)514 b Fn(C)608 b Ft(giv)-36 b(e)515 b(rise)f(to)g(elliptic)g(curv)-36 b(es)514 b(with)g(8)g(p)36 b(oin)-36 b(ts)513 b(in)h(arithmetic)800 6642 y(progression.)554 b(The)359 b(di\256erences)g(are)h(that)e Fn(C)455 b Ft(has)359 b(rank)h(1)f(and)g(torsion)h(subgroup)e Fg(Z)p Fn(=)p Ft(2)p Fg(Z)144 b Fk(\243)g Fg(Z)p Fn(=)p Ft(2)p Fg(Z)p Ft(,)373 b(while)800 8192 y Fn(D)470 b Ft(has)433 b(rank)h(2)g(and)f (torsion)h(subgroup)e Fg(Z)p Fn(=)p Ft(2)p Fg(Z)p Ft(.)18800 11608 y(3.)651 b Fo(Longer)501 b(Pr)-30 b(ogressions)2128 13932 y Ft(This)527 b(construction)f(can)h(also)h(b)36 b(e)526 b(used)g(to)h(pro)36 b(duce)526 b(progressions)h(of)h(length)e (greater)h(than)f(8)h(on)800 15482 y(elliptic)434 b(curv)-36 b(es)434 b(of)g(the)f(form)h Fn(y)16519 15000 y Fm(2)17414 15482 y Ft(=)368 b Fn(f)142 b Ft(\()p Fn(x)p Ft(\))434 b(where)f Fn(f)142 b Ft(\()p Fn(x)p Ft(\))433 b(is)h(a)g(quartic.)579 b(More)434 b(sp)36 b(eci\257cally)-108 b(,)434 b(w)-36 b(e)434 b(ha)-36 b(v)g(e:)p Black 800 17963 a Fu(Theorem)567 b(3.1.)p Black 584 w Fl(Ther)-66 b(e)518 b(exists)g(an)g(el)66 b(liptic)518 b(curve)h(in)e(the)i(form)f Fn(y)35050 17481 y Fm(2)36044 17963 y Ft(=)469 b Fn(w)36 b Ft(\()p Fn(x)p Ft(\))p Fl(,)531 b(with)518 b Fn(w)36 b Ft(\()p Fn(x)p Ft(\))518 b Fl(a)h(quartic,)800 19513 y(c)-66 b(ontaining)462 b(12)j(p)-66 b(oints)464 b(in)g(arithmetic)f(pr)-66 b(o)g(gr)g(ession.) p Black 800 21994 a(Pr)g(o)g(of.)p Black 552 w Ft(Let)21453 25000 y Fn(g)22076 25199 y Fm(0)22601 25000 y Ft(\()p Fn(x)p Ft(\))369 b(=)26480 23340 y Fm(11)26102 23738 y Fh(Y)26133 26538 y Fi(j)51 b Fm(=0)27799 25000 y Ft(\()p Fn(x)295 b Fk(\241)g Fn(j)75 b Ft(\))p Fn(:)800 28551 y Ft(Then)433 b Fn(g)4819 28750 y Fm(0)5345 28551 y Ft(\()p Fn(x)p Ft(\))369 b(=)f Fn(u)9585 28750 y Fm(0)10111 28551 y Ft(\()p Fn(x)p Ft(\))11862 28069 y Fm(2)12683 28551 y Fk(\241)295 b Ft(\(81)p Fn(=)p Ft(4\))i Fk(\242)e Fn(v)19213 28750 y Fm(0)19738 28551 y Ft(\()p Fn(x)p Ft(\),)434 b(with)10202 30929 y Fn(u)10942 31128 y Fm(0)11468 30929 y Ft(\()p Fn(x)p Ft(\))1107 b(=)f Fn(x)17183 30381 y Fm(6)18004 30929 y Fk(\241)296 b Ft(33)p Fn(x)21372 30381 y Fm(5)22193 30929 y Ft(+)f(418)p Fn(x)26189 30381 y Fm(4)27011 30929 y Fk(\241)h Ft(2541)p Fn(x)31679 30381 y Fm(3)32501 30929 y Ft(+)f(\(14993)p Fn(=)p Ft(2\))p Fn(x)40109 30381 y Fm(2)18612 32866 y Fk(\241)p Ft(\(18513)p Fn(=)p Ft(2\))p Fn(x)i Ft(+)e(\(4851)p Fn(=)p Ft(2\))p Fn(;)657 b Ft(and)10313 34816 y Fn(v)10942 35015 y Fm(0)11468 34816 y Ft(\()p Fn(x)p Ft(\))1107 b(=)f(429)p Fn(x)19133 34267 y Fm(4)19955 34816 y Fk(\241)296 b Ft(9438)p Fn(x)24623 34267 y Fm(3)25445 34816 y Ft(+)f(74295)p Fn(x)30741 34267 y Fm(2)31564 34816 y Fk(\241)g Ft(246246)p Fn(x)i Ft(+)e(290521)p Fn(:)800 37181 y Ft(Since)396 b(the)g(discriminan)-36 b(t)396 b(of)i Fn(v)15885 37380 y Fm(0)16410 37181 y Ft(\()p Fn(x)p Ft(\))f(is)f(nonzero,)405 b(the)396 b(curv)-36 b(e)396 b Fn(E)447 b Ft(:)369 b Fn(y)33458 36699 y Fm(2)34353 37181 y Ft(=)f Fn(v)36362 37380 y Fm(0)36888 37181 y Ft(\()p Fn(x)p Ft(\))396 b(is)h(an)g(elliptic)g(curv)-36 b(e.)566 b(This)800 38731 y(elliptic)434 b(curv)-36 b(e)434 b(then)e(con)-36 b(tains)434 b(a)g(length)f(12)h(arithmetic)g (progression.)15358 b Fe(\244)2128 41280 y Ft(\(Note)566 b(that)f(b)-36 b(y)566 b(using)f Fd(mwrank)p Ft(,)596 b(w)-36 b(e)566 b(computed)f(the)g(rank)h(of)g(this)g(curv)-36 b(e)565 b(to)h(b)36 b(e)565 b(4)h(with)g(torsion)800 42830 y(subgroup)432 b Fg(Z)p Fn(=)p Ft(2)p Fg(Z)p Ft(.\))2128 44380 y(The)335 b(construction)e(ab)36 b(o)-36 b(v)g(e)335 b(pro)36 b(duces)333 b(a)h(single)h(curv)-36 b(e)334 b(and)g(it)g(is)g(unclear)g(ho)-36 b(w)334 b(to)h(pro)36 b(duce)333 b(an)h(in\257nite)800 45929 y(family)539 b(of)g(curv)-36 b(es)537 b(con)-36 b(taining)538 b(a)g(length)f(12)h(progression)g (using)f(this)h(idea.)891 b(The)537 b(problem)g(is)h(that,)800 47479 y(in)482 b(general,)495 b(if)483 b(the)e Fn(P)181 b Ft(\()p Fn(x)p Ft(\))482 b(of)h(prop)36 b(osition)p 0 1 0 0 TeXcolorcmyk 482 w(2.1)p (#theorem.2.1) [[274 290 289 302] [1 1 1 [3 3]] [0 0 1]] pdfm Black 483 w(is)483 b(tak)-36 b(en)482 b(to)g(ha)-36 b(v)g(e)482 b(degree)g(12,)496 b(then)481 b(the)g Fn(R)11 b Ft(\()p Fn(x)p Ft(\))482 b(is)g(only)800 49029 y(guaran)-36 b(teed)511 b(to)g(ha)-36 b(v)g(e)511 b(degree)g(less)h(than)e(or)h(equal)h(to)f (5,)531 b(not)511 b(4.)811 b(Therefore,)532 b(the)510 b(curv)-36 b(e)511 b Fn(y)47509 48547 y Fm(2)48536 49029 y Ft(=)500 b Fn(R)11 b Ft(\()p Fn(x)p Ft(\))800 50579 y(need)433 b(not)g(b)36 b(e)434 b(an)f(elliptic)h(curv)-36 b(e.)579 b(W)-108 b(e)433 b(can,)h(ho)-36 b(w)g(ev)g(er,)434 b(pro)-36 b(v)g(e)434 b(the)f(follo)-36 b(wing.)p Black 800 53060 a Fu(Theorem)546 b(3.2.)p Black 575 w Fl(Ther)-66 b(e)501 b(ar)-66 b(e)502 b(in\257nitely)d(many)j(el)66 b(liptic)501 b(curves)h(of)g(the)g(form)f Fn(y)41184 52578 y Fm(2)42148 53060 y Ft(=)438 b Fn(w)36 b Ft(\()p Fn(x)p Ft(\))p Fl(,)510 b(with)502 b Fn(w)36 b Ft(\()p Fn(x)p Ft(\))800 54609 y Fl(a)465 b(quartic,)f(c)-66 b(ontaining)462 b(9)j(p)-66 b(oints)464 b(in)g(arithmetic)f(pr)-66 b(o)g(gr)g(ession.)p Black 800 57090 a(Pr)g(o)g(of.)p Black 552 w Ft(Let)19184 60097 y Fn(g)48 b Ft(\()p Fn(x)p Ft(\))368 b(=)g(\()p Fn(x)295 b Fk(\241)h Fn(a)p Ft(\))f Fk(\242)28984 58436 y Fm(8)28371 58835 y Fh(Y)28403 61634 y Fi(j)51 b Fm(=0)30068 60097 y Ft(\()p Fn(x)295 b Fk(\241)h Fn(j)75 b Ft(\))p Fn(;)800 63648 y Ft(and)364 b(write)i Fn(g)48 b Ft(\()p Fn(x)p Ft(\))364 b(as)h Fn(u)p Ft(\()p Fn(x)p Ft(\))13321 63166 y Fm(2)14002 63648 y Fk(\241)155 b Fn(v)48 b Ft(\()p Fn(x)p Ft(\).)555 b Fn(v)48 b Ft(\()p Fn(x)p Ft(\))364 b(is)h(a)g(degree)g(four)g(p)36 b(olynomial)367 b(in)d Fg(Q)p Ft(\()p Fn(a)p Ft(\)[)p Fn(x)p Ft(])i(with)f(discriminan) -36 b(t)800 65197 y(zero)434 b(only)g(for)g Fn(a)369 b Fk(2)g(f)p Ft(0)p Fn(;)221 b Ft(4)p Fn(;)g Ft(8)p Fk(g)p Ft(.)36270 b Fe(\244)2128 67746 y Ft(The)434 b(w)-36 b(ork)434 b(here)f(\(and)g(that)g(of)i(Bremner\))d(lea)-36 b(v)g(es)435 b(op)36 b(en)433 b(the)g(follo)-36 b(wing)436 b(questions:)p Black 800 70227 a Fu(Op)42 b(en)586 b(Question)i(3.3.)p Black 594 w Fl(Is)535 b(ther)-66 b(e)534 b(an)i(el)66 b(liptic)534 b(curve)h(of)h(the)f(form)f Fn(y)36646 69745 y Fm(2)37672 70227 y Ft(=)499 b Fn(f)142 b Ft(\()p Fn(x)p Ft(\))p Fn(;)221 b(f)142 b Ft(\()p Fn(x)p Ft(\))536 b Fl(a)f(cubic,)552 b(c)-66 b(on-)800 71777 y(taining)463 b(a)i(length)f(9)h(arithmetic)e(pr)-66 b(o)g(gr)g(ession?)597 b(A)-33 b(r)-66 b(e)463 b(ther)-66 b(e)464 b(in\257nitely)e(many?)p Black Black eop %%Page: 5 5 5 4 bop Black 0 TeXcolorgray Black 9765 886 a Fj(A)444 b(NOTE)f(ON)h(ARITHMETIC)d(PR)-28 b(OGRESSIONS)442 b(ON)i(ELLIPTIC)d (CUR)-114 b(VES)8452 b(5)p Black Black 800 3542 a Fu(Op)42 b(en)728 b(Question)h(3.4.)p Black 657 w Fl(Is)648 b(ther)-66 b(e)648 b(an)h(el)66 b(liptic)647 b(curve)i(of)f(the)h(form)f Fn(y)37899 3060 y Fm(2)39134 3542 y Ft(=)709 b Fn(f)142 b Ft(\()p Fn(x)p Ft(\))p Fn(;)221 b(f)142 b Ft(\()p Fn(x)p Ft(\))648 b Fl(a)h(quartic,)800 5092 y(c)-66 b(ontaining)430 b(a)k(length)e(13)i(arithmetic)d(pr)-66 b(o)g(gr)g(ession?)586 b(A)-33 b(r)-66 b(e)432 b(ther)-66 b(e)432 b(in\257nitely)f(many)i (curves)g(in)g(this)g(form)800 6642 y(c)-66 b(ontaining)462 b(a)j(length)f(10)h(pr)-66 b(o)g(gr)g(ession?)2128 8966 y Ft(And)433 b(\257nally)-108 b(,)p Black 800 11291 a Fu(Op)42 b(en)447 b(Question)i(3.5.)p Black 520 w Fl(What)424 b(is)h(the)e(longest)h(arithmetic)e(pr)-66 b(o)g(gr)g(ession)423 b(one)h(c)-66 b(an)423 b(\257nd)h(on)g(an)g(el)66 b(liptic)800 12841 y(curve)465 b(in)f(the)g(form)g Fn(y)11876 12359 y Fm(2)12771 12841 y Ft(=)369 b Fn(f)142 b Ft(\()p Fn(x)p Ft(\))p Fl(,)464 b(wher)-66 b(e)464 b Fn(f)142 b Ft(\()p Fn(x)p Ft(\))465 b Fl(is)g(a)g(cubic?)597 b(a)465 b(quartic?)19741 15475 y Ft(4.)651 b Fo(A)-39 b(ckno)-30 b(wledgments)2128 17800 y Ft(This)478 b(w)-36 b(ork)478 b(w)-36 b(as)477 b(completed)g(with)g(the)f(supp)36 b(ort)476 b(of)i(the)f(Lindbac)-36 b(k)476 b(F)-108 b(oundation)476 b(Minorit)-36 b(y)477 b(Junior)800 19350 y(F)-108 b(acult)-36 b(y)433 b(Gran)-36 b(t.)22927 21984 y Fo(References)p Black 1354 24087 a Fr([1])p Black 554 w(C.)1158 b(Batut,)1356 b(K.)1157 b(Belabas,)1355 b(D.)1157 b(Bernardi,)1354 b(H.)1157 b(Cohen,)1355 b(and)1157 b(M.)g(Olivier.)h(The)f(P)-31 b(ari)1158 b(system.)3075 25416 y Fc (ftp://megrez.math.u-bordeaux.fr/pub/pari/)p Fr(,)376 b(2000.)p Black 1354 26744 a([2])p Black 554 w(Andrew)470 b(Bremner.)f(On)g(arithmetic)j(progressions)d(on)h(elliptic)i(curv)-31 b(es.)469 b Fq(Exp)-57 b(eriment.)489 b(Math.)p Fr(,)494 b Fp(8)470 b Fr(\(1999\),)498 b(409)471 b({)3075 28072 y(413.)p Black 1354 29401 a([3])p Black 554 w(J.)432 b(P)-92 b(.)433 b(Buhler,)447 b(B.)433 b(H.)f(Gross,)447 b(and)432 b(D.)f(B.)h(Zagier.)i(On)e(the)g(conjecture)g(of)g(Birc)-31 b(h)432 b(and)g(Swinnerton-Dy)-31 b(er)433 b(for)f(an)3075 30729 y(elliptic)372 b(curv)-31 b(e)369 b(of)h(rank)f(3.)h Fq(Math.)395 b(Comp.)p Fr(,)370 b Fp(44)g Fr(\(1985\),)i(473)f({)e (481.)p Black 1354 32058 a([4])p Black 554 w(Garik)-61 b(ai)1278 b(Campb)31 b(ell.)1279 b Fq(Finding)1232 b(el)57 b(liptic)1231 b(curves)g(and)f(in\257nite)j(families)f(of)f(el)57 b(liptic)1231 b(curves)g(de-)3075 33386 y(\257ne)-57 b(d)1067 b(over)e Fb(Q)h Fq(of)f(lar)-57 b(ge)1066 b(r)-57 b(ank)p Fr(.)1097 b(PhD)f(thesis,)1279 b(Rutgers)1097 b(Univ)-31 b(ersit)g(y)-92 b(,)1280 b(June)1096 b(1999.)j(Av)-61 b(ailable)1099 b(at)3075 34714 y Fc (http://math.swarthmore.edu/kai/thesis.html)p Fr(.)p Black 1354 36043 a([5])p Black 554 w(John)370 b(Cremona.)h(Home)f (page.)g Fc(http://www.maths.nottingham.ac.uk/personal/jec/ftp/progs/)p Fr(.)p Black 1354 37371 a([6])p Black 554 w(Stefane)g(F)-31 b(\266)-523 b(ermigier.)371 b(Un)d(exemple)i(de)e(courb)31 b(e)369 b(elliptique)i(d)-31 b(\266)-523 b(e\257nie)369 b(sur)f Fb(Q)h Fr(de)f(rang)h Fa(\270)308 b Fr(19.)369 b Fq(C.)397 b(R.)f(A)-57 b(c)g(ad.)396 b(Sci.)i(Paris)3075 38699 y(S)-28 b(\266)-537 b(er.)396 b(I)p Fr(,)370 b Fp(315)g Fr(\(1992\),)i(719)f({)e(722.)p Black 1354 40028 a([7])p Black 554 w(Shoic)-31 b(hi)385 b(Kihara.)g(On)e(an)h (in\257nite)g(family)i(of)e(elliptic)h(curv)-31 b(es)383 b(with)i(rank)e Fa(\270)331 b Fr(14)384 b(o)-31 b(v)g(er)384 b Fb(Q)p Fr(.)g Fq(Pr)-57 b(o)g(c.)410 b(Jap)-57 b(an)409 b(A)-57 b(c)g(ad.)410 b(Ser.)3075 41356 y(A.)p Fr(,)370 b Fp(73)g Fr(\(1997\))i(32.)p Black 1354 42684 a([8])p Black 554 w(L.)409 b(Kulesz.)g Fq(A)-28 b(rithm)g(\266)-537 b(etique)431 b(des)h(c)-57 b(ourb)g(es)432 b(alg)-28 b(\266)-537 b(ebriques)432 b(de)g(genr)-57 b(e)433 b(au)f(moins)g(deux) p Fr(.)408 b(PhD)g(thesis,)419 b(Univ)-31 b(ersit)g(\266)-523 b(e)410 b(P)-31 b(aris)3075 44013 y(7,)370 b(1998.)p Black 1354 45341 a([9])p Black 554 w(Jean-F)-92 b(ran\273)-492 b(cois)399 b(Mestre.)e(Construction)j(d'une)f(courb)31 b(e)397 b(elliptique)k(de)d(rang)g Fa(\270)356 b Fr(12.)399 b Fq(C.)424 b(R.)f(A)-57 b(c)g(ad.)424 b(Sci.)g(Paris)e(S)-28 b(\266)-537 b(er.)3075 46669 y(I)p Fr(,)370 b Fp(295)g Fr(\(1982\),)j(643)d({)f(644.)p Black 800 47998 a([10])p Black 555 w(Jean-F)-92 b(ran\273)-492 b(cois)437 b(Mestre.)f(Courb)31 b(es)436 b(elliptiques)j(de)d(rang)h Fa(\270)420 b Fr(11)437 b(sur)f Fb(Q)p Fr(\()p Fb(t)p Fr(\).)i Fq(C.)458 b(R.)h(A)-57 b(c)g(ad.)459 b(Sci.)h(Paris)d(S)-28 b(\266)-537 b(er.)458 b(I)p Fr(,)437 b Fp(313)3075 49326 y Fr(\(1991\),)373 b(139)d({)g(142.)p Black 800 50654 a([11])p Black 555 w(Koh-Ic)-31 b(hi)473 b(Nagao.)i(Examples)e(of)f(elliptic)j(curv)-31 b(es)471 b(o)-31 b(v)g(er)472 b Fb(Q)g Fr(with)i(rank)e Fa(\270)478 b Fr(17.)c Fq(Pr)-57 b(o)g(c.)491 b(Jap)-57 b(an)491 b(A)-57 b(c)g(ad.)491 b(Ser.)h(A.)p Fr(,)498 b Fp(68)3075 51983 y Fr(\(1997\),)373 b(287)d({)g(289.)p 800 53621 52000 45 v 800 55628 a Ft(2000)435 b Fl(Mathematics)464 b(Subje)-66 b(ct)463 b(Classi\257c)-66 b(ation)p Ft(:)577 b(11G05,)434 b(11B25)h(.)800 57178 y Fl(Keywor)-66 b(ds:)597 b(el)66 b(liptic)464 b(curves,)h(arithmetic)e(pr)-66 b(o)g(gr)g(ession)p 800 58566 V 800 61373 a Ft(Receiv)-36 b(ed)435 b(F)-108 b(ebruary)433 b(5,)h(2003;)i(revised)f(v)-36 b(ersion)434 b(receiv)-36 b(ed)434 b(F)-108 b(ebruary)433 b(7,)i(2003.)581 b(Published)433 b(in)h Fl(Journal)800 62923 y(of)465 b(Inte)-66 b(ger)462 b(Se)-66 b(quenc)g(es)432 b Ft(F)-108 b(ebruary)432 b(25,)j(2003.)p 800 64310 V 800 66317 a(Return)e(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 [[133 120 338 132] [1 1 1 [3 3]] [0 0 1]] (http://www.math.uwaterloo.ca/JIS/) pdfm Black(.)p Black Black eop %%Trailer end end