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Fj(X)36183 54775 y Fi(k)24 b Fh(=1)39285 52309 y Fm(1)p 38355 52902 2512 54 v 38355 54119 a Fl(k)45 b Fm(2)39726 53735 y Fi(k)24 b(n)40999 53208 y Fl(;)800 57343 y Fm(log)19 b(\(2)3636 56861 y Fi(n)4604 57343 y Fg(\241)341 b Fm(1\))501 b(has)g(a)h(binary)f(BBP-form)-36 b(ula,)518 b(subsequen)-36 b(tly)501 b(yielding)h(form)-36 b(ulae)501 b(for)h(log)19 b(\(2)46727 56861 y Fi(n)47695 57343 y Fm(+)340 b(1\))502 b(and)800 58948 y(the)433 b(natural)g(logarithm)i(of)f(an)-36 b(y)434 b(in)-36 b(teger)433 b(of)i(the)e(form)18046 61641 y(\(2)19202 61159 y Fi(a)19702 61282 y Fd(1)20514 61641 y Fg(\241)295 b Fm(1\))222 b(\(2)24376 61159 y Fi(a)24876 61282 y Fd(2)25687 61641 y Fg(\241)296 b Fm(1\))221 b Fg(\242)g(\242)g(\242)h Fm(\(2)31320 61159 y Fi(a)31820 61315 y Fc(h)32708 61641 y Fg(\241)296 b Fm(1\))p 18046 62235 17147 54 v 18244 63451 a(\(2)19400 63068 y Fi(b)19803 63191 y Fd(1)20614 63451 y Fg(\241)g Fm(1\))221 b(\(2)24476 63068 y Fi(b)24879 63191 y Fd(2)25691 63451 y Fg(\241)295 b Fm(1\))221 b Fg(\242)g(\242)g(\242)i Fm(\()o(2)31323 63044 y Fi(b)31726 63179 y Fc(j)32510 63451 y Fg(\241)296 b Fm(1\))35326 62540 y Fl(:)15451 b Fm(\(1\))800 66219 y(The)450 b(pap)36 b(er)449 b([)p 0 1 0 0 TeXcolorcmyk(2)p (#cite.BBP) [[140 121 145 133] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])h(ga)-36 b(v)g(e)451 b(a)f(list)g(of)h(some)f(primes)g (whic)-36 b(h)449 b(ha)-36 b(v)g(e)450 b(this)g(form.)628 b(Bailey)451 b([)p 0 1 0 0 TeXcolorcmyk(1)p (#cite.Bailey) [[446 121 451 133] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])f(extended)f(this)h(list)800 67824 y(b)-36 b(y)434 b(using)f(the)g(expression)19607 70188 y(Re)21364 68315 y Fj(\265)22342 70188 y Fm(log)24243 68315 y Fj(\265)25221 70188 y Fm(1)296 b Fg(\247)27628 69290 y Fm(\(1)f(+)g Fl(i)p Fm(\))31336 68808 y Fi(k)p 27628 69883 4277 54 v 29128 71100 a Fm(2)29778 70716 y Fi(n)32037 68315 y Fj(\266\266)51138 70188 y Fm(\(2\))p Black 26475 74617 a(2)p Black eop %%Page: 3 3 3 2 bop Black 0 TeXcolorgray Black Black 800 1424 a Fm(suggested)503 b(b)-36 b(y)503 b(R.)h(Harley)g(and)f(J.)h(Borw)-36 b(ein.)787 b(This)504 b(expression)f(has)g(a)h(binary)f(BBP-form)-36 b(ula)504 b(since,)800 3029 y(for)434 b(example,)10613 6767 y(log)12515 4894 y Fj(\265)13492 6767 y Fm(1)296 b Fg(\241)15899 5868 y Fm(1)g(+)f Fl(i)p 15899 6461 2697 54 v 16609 7678 a Fm(2)17259 7294 y Fi(n)18728 4894 y Fj(\266)20813 6767 y Fm(=)1106 b Fg(\241)24675 5106 y Fk(1)24186 5505 y Fj(X)24328 8304 y Fi(j)51 b Fh(=1)26326 4894 y Fj(\265)27437 5868 y Fm(1)295 b(+)g Fl(i)p 27437 6461 V 28147 7678 a Fm(2)28797 7294 y Fi(n)30266 4894 y Fj(\266)31243 5191 y Fi(j)32085 5868 y Fm(1)p 32085 6461 651 54 v 32104 7678 a Fl(j)20813 11416 y Fm(=)1106 b Fg(\241)24675 9755 y Fk(1)24186 10154 y Fj(X)24328 12953 y Fi(j)51 b Fh(=0)27050 9755 y(8)26326 10154 y Fj(X)26428 12983 y Fi(k)24 b Fh(=1)28466 9543 y Fj(\265)29577 10517 y Fm(1)295 b(+)g Fl(i)p 29577 11110 2697 54 v 30287 12327 a Fm(2)30937 11943 y Fi(n)32406 9543 y Fj(\266)33384 9840 y Fh(8)p Fi(j)51 b Fh(+)p Fi(k)37408 10517 y Fm(1)p 35941 11110 3587 54 v 35941 12327 a(8)p Fl(j)370 b Fm(+)295 b Fl(k)20813 16065 y Fm(=)1106 b Fg(\241)24675 14405 y Fk(1)24186 14803 y Fj(X)24328 17602 y Fi(j)51 b Fh(=0)28190 15166 y Fm(1)p 26459 15759 4113 54 v 26459 16976 a(2)27109 16592 y Fh(\(8)p Fi(n)p Fk(\241)p Fh(4\))p Fi(j)31650 14405 y Fh(8)30926 14803 y Fj(X)31027 17632 y Fi(k)24 b Fh(=1)33066 14192 y Fj(\265)34177 15166 y Fm(1)295 b(+)g Fl(i)p 34177 15759 2697 54 v 34886 16976 a Fm(2)35536 16592 y Fi(n)37006 14192 y Fj(\266)37983 14489 y Fi(k)40374 15166 y Fm(1)p 38906 15759 3587 54 v 38906 16976 a(8)p Fl(j)371 b Fm(+)295 b Fl(k)42626 16065 y(:)2751 21149 y Fm(A)344 b(crucial)h(to)36 b(ol)346 b(for)f(factoring)g(n)-36 b(um)g(b)36 b(ers)343 b(of)i(the)f(form)h Fl(b)30220 20667 y Fi(n)30959 21149 y Fg(\241)113 b Fm(1)345 b(is)g(the)f (classical)i(theory)e(of)h(cyclotomic)800 22754 y(p)36 b(olynomials:)21718 24518 y Fl(b)22271 23969 y Fi(n)23192 24518 y Fg(\241)295 b Fm(1)370 b(=)26920 23256 y Fj(Y)27111 26134 y Fi(d)p Fk(j)p Fi(n)28839 24518 y Fm(\251)29778 24717 y Fi(d)30318 24518 y Fm(\()p Fl(b)p Fm(\))19255 b(\(3\))800 28358 y(where)434 b(\251)5497 28557 y Fi(d)6036 28358 y Fm(\()p Fl(x)p Fm(\),)g(the)f Fl(d)11498 27876 y Fi(th)12869 28358 y Fm(cyclotomic)i(p)36 b(olynomial,)435 b(is)f(de\257ned)e(as)21028 32555 y(\251)21967 32754 y Fi(d)22506 32555 y Fm(\()p Fl(x)p Fm(\))369 b(=)26007 30791 y Fi(\301)p Fh(\()p Fi(d)p Fh(\))26046 31293 y Fj(Y)26077 34093 y Fi(j)51 b Fh(=1)27782 32555 y Fm(\()p Fl(x)295 b Fg(\241)g Fl(\263)31218 32754 y Fi(j)31705 32555 y Fm(\))p Fl(:)800 36799 y Fm(The)598 b(terms)g Fl(\263)7991 36998 y Fi(j)9076 36799 y Fm(are)g(the)f(primitiv)-36 b(e)598 b Fl(d)20216 36317 y Fi(th)21752 36799 y Fm(ro)36 b(ots)598 b(of)h(unit)-36 b(y)597 b(and)h Fl(\301)p Fm(\()p Fg(\242)p Fm(\))f(is)h(the)g(Euler)f(totien)-36 b(t)598 b(function.)800 38404 y(Alternativ)-36 b(ely)-108 b(,)419 b(a)414 b(w)-36 b(ell-kno)g(wn)414 b(iden)-36 b(tit)g(y)414 b(for)g(these)g(p)36 b(olynomials)415 b(deriv)-36 b(ed)414 b(using)f(M\304)-650 b(obius)414 b(in)-36 b(v)g(ersion)414 b(is)19754 41614 y(\251)20693 41813 y Fi(d)21233 41614 y Fm(\()p Fl(x)p Fm(\))369 b(=)24734 40352 y Fj(Y)24953 43231 y Fi(k)24 b Fk(j)p Fi(d)26431 41614 y Fm(\(1)295 b Fg(\241)h Fl(x)29950 41066 y Fi(d=k)31473 41614 y Fm(\))31979 41066 y Fi(\271)p Fh(\()p Fi(k)24 b Fh(\))51138 41614 y Fm(\(4\))800 45354 y(where)434 b Fl(\271)p Fm(\()p Fg(\242)p Fm(\))e(is)i(the)f(M\304)-650 b(obius)433 b(function.)2751 46959 y(In)526 b(conjunction)g(with)g(expression)g(\()p 0 1 0 0 TeXcolorcmyk(1)p (#equation.1) [[266 294 272 306] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\),)549 b(Bailey)528 b Fe(et)549 b(al.)856 b Fm(state)526 b(that)f(log)241 b(\251)40536 47158 y Fi(m)41424 46959 y Fm(\(2\))526 b(admits)g(a)g(binary)800 48564 y(BBP-form)-36 b(ula)434 b(for)g(all)g(p)36 b(ositiv)-36 b(e)435 b(in)-36 b(tegers)433 b Fl(m)p Fm(.)578 b(One)433 b(ma)-36 b(y)434 b(easily)h(extend)f(this)f(to)23604 51363 y(log)674 b(\251)26878 51562 y Fi(m)27766 51363 y Fm(\(2)28922 50814 y Fi(k)29491 51363 y Fm(\))21141 b(\(5\))800 54161 y(for)465 b(all)g(in)-36 b(tegers)464 b Fl(k)45 b Fm(.)670 b(Ho)-36 b(w)g(ev)g(er,)472 b(the)464 b(cyclotomic)i(p)36 b(olynomials)465 b(ma)-36 b(y)465 b(b)36 b(e)464 b(used)f(to)h(obtain)g(man)-36 b(y)464 b(other)800 55767 y(v)-72 b(alues.)579 b(Using)434 b(the)f(M\304)-650 b(obius)433 b(form)-36 b(ula)434 b(\()p 0 1 0 0 TeXcolorcmyk(4)p (#equation.4) [[261 215 266 227] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))g(with)f Fl(x)369 b Fm(=)g(\()p Fg(\247)p Fm(1)296 b(+)e Fl(i)p Fm(\))p Fl(=)p Fm(2)34027 55285 y Fi(n)35087 55767 y Fm(yields)7340 59728 y(Re)9097 57855 y Fj(\265)10075 59728 y Fm(log)674 b(\251)13349 59927 y Fi(m)14458 57855 y Fj(\265)15569 58829 y Fg(\247)p Fm(1)295 b(+)g Fl(i)p 15569 59422 3730 54 v 16795 60639 a Fm(2)17445 60255 y Fi(n)19431 57855 y Fj(\266\266)21755 59728 y Fm(=)23136 58466 y Fj(X)23306 61344 y Fi(d)p Fk(j)p Fi(m)25276 59728 y Fl(\271)p Fm(\()p Fl(d)p Fm(\)Re)29503 57456 y Fj(\303)30554 59728 y Fm(log)32456 57456 y Fj(\303)33507 59728 y Fm(1)h Fg(\241)35781 57855 y Fj(\265)36892 58829 y Fg(\247)p Fm(1)g(+)e Fl(i)p 36892 59422 V 38118 60639 a Fm(2)38768 60255 y Fi(n)40754 57855 y Fj(\266)41732 58152 y Fi(m=d)43574 57456 y Fj(!!)45899 59728 y Fl(:)4878 b Fm(\(6\))800 63999 y(As)336 b(in)f(the)g(consideration)g(of)h(the)f(expression)h(\() p 0 1 0 0 TeXcolorcmyk(2)p (#equation.2) [[290 141 296 153] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\),)355 b(the)335 b(righ)-36 b(t)335 b(side)h(is)f(a)h(binary)f (BBP-form)-36 b(ula.)546 b(Though)800 65604 y(it)478 b(is)h(simply)g(a)f(linear)h(com)-36 b(bination)478 b(of)h(expressions) f(of)h(the)f(form)g(\()p 0 1 0 0 TeXcolorcmyk(2)p (#equation.2) [[393 127 399 139] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\),)490 b(the)478 b(adv)-72 b(an)-36 b(tage)478 b(here)g(is)h(that)800 67209 y(implicitly)435 b(some)f(cancellation)g (ma)-36 b(y)435 b(tak)-36 b(e)434 b(place.)578 b(F)-108 b(or)434 b(example,)g(w)-36 b(e)434 b(ha)-36 b(v)g(e)13170 70698 y(Re)14927 68825 y Fj(\265)15904 70698 y Fm(log)241 b(\251)18745 70897 y Fh(6)19492 68825 y Fj(\265)20603 69799 y Fm(1)296 b(+)e Fl(i)p 20603 70392 2697 54 v 21301 71609 a Fm(16)23432 68825 y Fj(\266\266)25757 70698 y Fm(=)27270 69799 y(1)p 27270 70392 651 54 v 27270 71609 a(2)28275 70698 y([log)240 b(14449)297 b Fg(\241)e Fm(14)221 b(log)242 b(2])222 b Fl(;)p Black 26475 74617 a Fm(3)p Black eop %%Page: 4 4 4 3 bop Black 0 TeXcolorgray Black Black 800 1424 a Fm(hence)474 b(14449)h(joins)g(the)f(list.)700 b(Similarly)-108 b(,)486 b(one)474 b(ma)-36 b(y)474 b(use)g(the)g(M\304)-650 b(obius)474 b(form)-36 b(ula)474 b(\()p 0 1 0 0 TeXcolorcmyk(4)p (#equation.4) [[450 704 456 716] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))g(with)h Fl(x)438 b Fm(=)f(\()p Fg(\247)p Fm(1)323 b(+)800 1930 y Fg(p)p 1907 1930 651 54 v 1099 x Fm(3)p Fl(i)p Fm(\))p Fl(=)p Fm(2)4807 2547 y Fi(n)5867 3029 y Fm(to)434 b(obtain)5140 7523 y(Re)6897 5252 y Fj(\303)7948 7523 y Fm(log)674 b(\251)11222 7722 y Fi(m)12332 5252 y Fj(\303)13516 6625 y Fg(\247)p Fm(1)296 b(+)16802 5526 y Fg(p)p 17909 5526 V 1099 x Fm(3)p Fl(i)p 13516 7218 5487 54 v 15621 8435 a Fm(2)16271 8051 y Fi(n)19136 5252 y Fj(!)o(!)21608 7523 y Fm(=)22988 6262 y Fj(X)23159 9140 y Fi(d)p Fk(j)p Fi(m)25128 7523 y Fl(\271)p Fm(\()p Fl(d)p Fm(\)Re)29355 4853 y Fj(0)29355 7245 y(@)30517 7523 y Fm(log)32419 4853 y Fj(0)32419 7245 y(@)33581 7523 y Fm(1)g Fg(\241)35855 5252 y Fj(\303)37040 6625 y Fg(\247)p Fm(1)f(+)40325 5526 y Fg(p)p 41432 5526 651 54 v 1099 x Fm(3)p Fl(i)p 37040 7218 5487 54 v 39144 8435 a Fm(2)39794 8051 y Fi(n)42659 5252 y Fj(!)43710 5549 y Fi(m=d)45553 4853 y Fj(1)45553 7245 y(A)46715 4853 y(1)46715 7245 y(A)48099 7523 y Fl(;)2678 b Fm(\(7\))800 11944 y(again)434 b(pro)36 b(ducing)433 b(binary)h(BBP-form)-36 b(ulae.)579 b(An)433 b(example)h(here)f(is)12902 15966 y(Re)14659 13694 y Fj(\303)15711 15966 y Fm(log)674 b(\251)18985 16165 y Fh(5)19732 13694 y Fj(\303)20917 15067 y Fm(1)295 b(+)23169 13968 y Fg(p)p 24276 13968 651 54 v 1099 x Fm(3)p Fl(i)p 20917 15660 4454 54 v 22818 16877 a Fm(4)25503 13694 y Fj(!)o(!)27975 15966 y Fm(=)29488 15067 y(1)p 29488 15660 651 54 v 29488 16877 a(2)30493 15966 y([log)241 b(331)296 b Fg(\241)f Fm(8)221 b(log)241 b(2])222 b Fl(;)800 19988 y Fm(so)498 b(331)h(comes)f(on)-36 b(to)498 b(the)f(list.)771 b(Again,)515 b(suc)-36 b(h)497 b(results)h(are)f(linear)i(com)-36 b(binations)497 b(of)i(earlier)f(form)-36 b(ulae)800 21593 y(since,)434 b(for)g(example,)11167 25615 y(Re)12924 23343 y Fj(\303)13975 25615 y Fm(log)15877 23343 y Fj(\303)16928 25615 y Fm(1)296 b Fg(\241)19335 24716 y Fm(1)g(+)21587 23617 y Fg(p)p 22694 23617 V 1099 x Fm(3)q Fl(i)p 19335 25309 4454 54 v 21237 26526 a Fm(2)23921 25615 y Fl(x)24660 23343 y Fj(!!)27133 25615 y Fm(=)368 b(log)19 b(\(1)296 b Fg(\241)f Fl(x)33712 25066 y Fh(3)34238 25615 y Fm(\))g Fg(\241)g Fm(log)19 b(\(1)295 b Fg(\241)h Fl(x)p Fm(\))p Fl(:)800 29637 y Fm(Mo)36 b(dest)634 b(calculations)h(with)f(Maple)g (pro)36 b(duced)632 b(the)i(follo)-36 b(wing)636 b(augmen)-36 b(tation)634 b(of)g(Bailey's)i(list)e(of)800 31242 y(primes)574 b(whose)i(logarithm)f(admits)g(a)g(binary)g(BBP-form)-36 b(ula)575 b(\(underlined)e(n)-36 b(um)g(b)36 b(ers)573 b(are)i(giv)-36 b(en)575 b(b)-36 b(y)800 32847 y(Bailey[)p 0 1 0 0 TeXcolorcmyk(1)p (#cite.Bailey) [[114 421 120 433] [1 1 1 [3 3]] [0 0 1]] pdfm Black 2 w(]\):)800 36052 y(2)p 800 36265 651 54 v(,)868 b(3)p 2679 36265 V(,)g(5)p 4558 36265 V(,)f(7)p 6436 36265 V 1 w(,)g(11)p 8315 36265 1301 54 v 1 w(,)g(13)p 10844 36265 V 1 w(,)g(17)p 13373 36265 V 1 w(,)g(19)p 15902 36265 V 1 w(,)g(29)p 18431 36265 V 1 w(,)g(31)p 20960 36265 V 1 w(,)g(37)p 23489 36265 V 1 w(,)g(41)p 26018 36265 V 1 w(,)h(43)p 28548 36265 V(,)g(61)p 31077 36265 V(,)g(73)p 33606 36265 V(,)g(109)p 36135 36265 1951 54 v 1 w(,)f(113)p 39314 36265 V 1 w(,)h(127)p 42494 36265 V 1 w(,)800 37657 y(151)p 800 37870 V 1 w(,)f(241)p 3979 37870 V 1 w(,)h(257)p 7159 37870 V 1 w(,)f(331)435 b(,)867 b(337)p 13951 37870 V 1 w(,)h(397)p 17131 37870 V 1 w(,)f(683)p 20310 37870 V 1 w(,)g(1321)p 23489 37870 2602 54 v 2 w(,)g(1429,)i(1613)p 31149 37870 V 1 w(,)f(2113)p 34979 37870 V 1 w(,)f(2731)p 38808 37870 V 2 w(,)g(5419)p 42638 37870 V 1 w(,)800 39262 y(8191)p 800 39475 V 1 w(,)h(14449,)h(26317,)g(38737,)g(43691)p 18070 39475 3252 54 v 2 w(,)e(61681)p 22550 39475 V 2 w(,)g(65537,)i(87211)p 31510 39475 V 2 w(,)e(131071)p 35990 39475 3902 54 v 2 w(,)h(174763)p 41121 39475 V 2 w(,)f(246241,)800 40867 y(262657)p 800 41080 V 2 w(,)781 b(268501,)j(279073,)f(312709,)h (524287)p 20978 41080 V 2 w(,)d(525313,)j(599479,)f(2796203)p 36111 41080 4553 54 v 2 w(,)f(4327489,)i(7416361,)800 42473 y(15790321)p 800 42685 5203 54 v 3 w(,)653 b(18837001)p 7017 42685 V 3 w(,)g(22366891)p 13234 42685 V 3 w(,)g(29247661,)k (47392381,)f(107367629,)g(536903681,)h(1326700741,)800 44078 y(4278255361)p 800 44290 6504 54 v 3 w(,)868 b(4562284561)p 8532 44290 V 3 w(,)g(40388473189,)j(77158673929,)g(118750098349,)g (415878438361,)800 45683 y(1133836730401,)h(2932031007403)p 10483 45895 8455 54 v 4 w(,)867 b(3630105520141,)872 b(4363953127297)p 29848 45895 V 5 w(,)867 b(4432676798593)p 39531 45895 V 4 w(,)800 47288 y(4981857697937,)612 b(108140989558681,)g (140737471578113,)g(1041815865690181,)h(96076791871613611,)800 48893 y(18446744069414584321,)874 b(5302306226370307681801,)800 50498 y(2048568835297380486760231,)i(17059410504738323992180849,)800 52103 y(84159375948762099254554456081,)h (134304196845099262572814573351,)800 53708 y (19177458387940268116349766612211,)h (304832756195865229284807891468769,)800 55313 y (1339272539833668386958920468400193,)h (3652124453410972878264128353955921,)800 56918 y (1772303994379887829769795077302561451,)816 b (6113142872404227834840443898241613032969,)800 58523 y(1461503031127477825099979369543473122548042956801,)800 60129 y(867988564747274927163124868127898657976489313137639569)2751 62534 y(Of)621 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Fm(8681)296 b Fg(\243)g Fm(49477)p Fl(;)657 b Fm(10169)296 b Fg(\243)g Fm(43249589)p Fl(;)3311 16725 y Fm(13367)h Fg(\243)e Fm(164511353)p Fl(;)659 b Fm(32377)296 b Fg(\243)g Fm(1212847)p Fl(;)657 b Fm(92737)297 b Fg(\243)f Fm(649657)p Fl(;)657 b Fm(179951)297 b Fg(\243)e Fm(3203431780337)p Fl(;)800 18823 y Fm(181549)i Fg(\243)f Fm(12112549)2751 22033 y(Note)391 b(that)f(all)i(the)e(primes)g(up)g(to)h(101)h(are)f (on)f(the)g(list,)400 b(either)391 b(alone)g(or)g(m)-36 b(ultiplied)390 b(b)-36 b(y)391 b(one)f(other)800 23638 y(prime.)593 b(Indeed,)439 b(ev)-36 b(ery)439 b(o)36 b(dd)438 b(prime)g Fl(p)g Fm(is)h(on)f(the)g(list,)i(either)e(alone)h (or)g(in)f(some)h(m)-36 b(ultiple)438 b(pro)36 b(duct)437 b(of)800 25243 y(primes)461 b(since)h(2)8891 24761 y Fi(p)p Fk(\241)p Fh(1)10937 25243 y Fg(\241)315 b Fm(1)462 b(is)g(on)f(the)g(list)i(and)e Fl(p)p Fg(j)p Fm(\(2)25774 24761 y Fi(p)p Fk(\241)p Fh(1)27819 25243 y Fg(\241)315 b Fm(1\).)663 b(Carl)462 b(P)-36 b(omerance)462 b(\(see)g([)p 0 1 0 0 TeXcolorcmyk(2)p (#cite.BBP) [[467 490 473 502] [1 1 1 [3 3]] [0 0 1]] pdfm Black(]\))g(sho)-36 b(w)g(ed)461 b(that)800 26848 y(23)504 b(could)e(not)h(b)36 b(e)502 b(written)h(in)g(the)f(form)h(\()p 0 1 0 0 TeXcolorcmyk(1)p (#equation.1) [[275 475 281 487] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\);)538 b(ho)-36 b(w)g(ev)g(er,)521 b(it)503 b(is)g(still)h (unkno)-36 b(wn)502 b(whether)g(log)240 b(23)504 b(has)f(a)800 28453 y(binary)390 b(BBP-form)-36 b(ula.)564 b(Related)390 b(questions)g(are)g(extensiv)-36 b(ely)391 b(dealt)f(with)g(b)-36 b(y)389 b(Borw)-36 b(ein,)399 b(Borw)-36 b(ein)391 b(and)800 30058 y(Galw)-36 b(a)g(y[)p 0 1 0 0 TeXcolorcmyk(4)p (#cite.BBG) [[121 446 126 458] [1 1 1 [3 3]] [0 0 1]] pdfm Black 2 w(].)2751 31663 y(An)433 b(imp)36 b(ortan)-36 b(t)433 b(sub)36 b(class)434 b(of)g(binary)g(BBP-form)-36 b(ulae)434 b(concerns)f(the)g(expression)20002 34758 y(Re)21759 32885 y Fj(\265)22737 34758 y Fm(log)674 b(\251)26011 34957 y Fi(m)27120 32885 y Fj(\265)28231 33860 y 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b Fh(+1\))40865 64309 y Fg(\241)296 b Fm(1)42844 63233 y Fj(\242)43748 64309 y Fg(\241)f Fm(2)p Fl(k)267 b Fm(log)240 b(2)17707 67418 y(=)19959 66519 y(1)p 19959 67112 V 19959 68329 a(2)20963 67418 y(log)g(\(\(1)296 b(+)e Fl(i)p Fm(\))27078 66869 y Fi(p)27902 67418 y Fg(\241)h Fm(1\))222 b(\(\(1)295 b Fg(\241)h Fl(i)p Fm(\))34844 66869 y Fi(p)35667 67418 y Fg(\241)g Fm(1\))f Fg(\241)g Fm(2)p Fl(k)267 b Fm(log)240 b(2)17707 70698 y(=)19959 69799 y(1)p 19959 70392 V 19959 71609 a(2)20963 70698 y(log)22864 68825 y Fj(\265)23975 69799 y Fm(\(\(1)295 b(+)g Fl(i)p Fm(\))28189 69317 y Fi(p)29013 69799 y Fg(\241)g Fm(1\))222 b(\()o(\(1)296 b Fg(\241)f Fl(i)p Fm(\))35954 69317 y Fi(p)36778 69799 y Fg(\241)g Fm(1\))p 23975 70392 15288 54 v 30774 71609 a(2)31424 71225 y Fh(4)p Fi(k)39395 68825 y Fj(\266)40594 70698 y Fl(:)p Black 26475 74617 a Fm(5)p Black eop %%Page: 6 6 6 5 bop Black 0 TeXcolorgray Black Black 800 1424 a Fm(A)447 b(similar)h(calculation)g(ma)-36 b(y)448 b(b)36 b(e)447 b(done)f(for)i(primes)f(of)g(the)g(form)g Fl(p)392 b Fm(=)g(4)p Fl(k)350 b Fg(\241)304 b Fm(1.)619 b(W)-108 b(e)447 b(then)f(ha)-36 b(v)g(e)448 b(that)e(if)800 3029 y Fl(q)499 b Fm(is)452 b(a)g(Gaussian-Mersenne)e(prime,)456 b(then)451 b(log)241 b Fl(q)499 b Fm(admits)451 b(a)h(binary)g (BBP-form)-36 b(ula.)633 b(This)452 b(connection)800 4634 y(implies)560 b(a)g(larger)g(question:)830 b(F)-108 b(or)559 b(whic)-36 b(h)560 b(p)36 b(ositiv)-36 b(e)560 b(in)-36 b(tegers)559 b Fl(m)h Fm(is)f(the)g(n)-36 b(umerator)559 b(of)h(the)f(rational)800 6239 y(expression)18103 8534 y(exp)20312 6661 y Fj(\265)21289 8534 y Fm(2Re)23696 6661 y Fj(\265)24674 8534 y Fm(log)241 b(\251)27515 8733 y Fi(m)28624 6661 y Fj(\265)29735 7636 y Fm(1)295 b(+)g Fl(i)p 29735 8229 2697 54 v 30758 9446 a Fm(2)32564 6661 y Fj(\266)o(\266\266)51138 8534 y Fm(\(9\))800 11331 y(prime?)888 b(Besides)537 b(the)g(Gaussian-Mersenne)e(primes,)563 b(man)-36 b(y)537 b(comp)36 b(osite)537 b Fl(m)g Fm(satisfy)h(this)f 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Fm(651)p Fl(;)g Fm(679)p Fl(;)g Fm(741)p Fl(;)g Fm(779)p Fl(;)g Fm(819)p Fl(;)g Fm(871)p Fl(;)g Fm(885)p Fl(;)g Fm(997)p 25360 26532 V 14 w Fl(;)g Fm(1062)p Fl(;)g Fm(1114)p Fl(;)g Fm(1126)p Fl(;)g Fm(1150)p Fl(;)g Fm(1226)p Fl(;)g Fm(1275)p Fl(;)g Fm(1317)p Fl(;)5104 28256 y Fm(1329)p Fl(;)g Fm(1367)p 8286 28469 2602 54 v 3 w Fl(;)g Fm(1382)p Fl(;)g Fm(1434)p Fl(;)g Fm(1477)p Fl(;)g Fm(1710)p Fl(;)g Fm(1926)p Fl(;)g Fm(1970)p Fl(;)g Fm(2331)p Fl(;)g Fm(2422)p Fl(;)g Fm(2446)p Fl(;)g Fm(2995)p Fl(:)2751 32618 y Fm(Some)562 b(of)g(the)g(primes)f (pro)36 b(duced)561 b(b)-36 b(y)562 b(expression)g(\()p 0 1 0 0 TeXcolorcmyk(9)p (#equation.9) [[336 423 341 435] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))g(ha)-36 b(v)g(e)562 b(b)36 b(een)561 b(put)g(on)h(the)f (previous)h(list)g(of)800 34223 y(primes.)578 b(When)433 b Fl(m)369 b Fm(=)g(2995,)435 b(this)e(pro)36 b(duces)433 b(a)h(prime)f(with)h(o)-36 b(v)g(er)434 b(700)g(digits.)800 38585 y Ff(3)2152 b(Aurifeuillian)714 b(F)-179 b(actorizations)800 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[1 1 1 [3 3]] [0 0 1]] pdfm Black(])434 b(and)f(Bren)-36 b(t[)p 0 1 0 0 TeXcolorcmyk(5)p (#cite.Brent) [[479 184 485 196] [1 1 1 [3 3]] [0 0 1]] pdfm Black(].)2751 60833 y(Earlier)434 b(w)-36 b(e)434 b(sa)-36 b(w)434 b(that)f(log)19 b(\(2)17101 60351 y Fi(n)18023 60833 y Fg(\247)295 b Fm(1\))434 b(has)f(a)h(binary)g(BBP-form)-36 b(ula.)579 b(Bailey[)p 0 1 0 0 TeXcolorcmyk(1)p (#cite.Bailey) [[439 170 445 182] [1 1 1 [3 3]] [0 0 1]] pdfm Black 2 w(]\))433 b(notes)h(that)8229 64022 y(Re)9986 62149 y Fj(\265)10963 64022 y Fm(log)12865 62149 y Fj(\265)13843 64022 y Fm(1)295 b Fg(\247)16249 63123 y Fm(1)h(+)f Fl(i)p 16249 63716 2697 54 v 16959 64933 a Fm(2)17609 64549 y Fi(n)19078 62149 y Fj(\266\266)21403 64022 y Fm(=)22783 62149 y Fj(\265)23894 63123 y Fm(1)p 23894 63716 651 54 v 23894 64933 a(2)24972 64022 y Fg(\241)h Fl(n)27077 62149 y Fj(\266)28276 64022 y Fm(log)241 b(2)295 b(+)32563 63123 y(1)p 32563 63716 V 32563 64933 a(2)33568 64022 y(log)35469 62946 y Fj(\241)36078 64022 y 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Fl(:)p Black 26475 74617 a Fm(6)p Black eop %%Page: 7 7 7 6 bop Black 0 TeXcolorgray Black Black 800 1424 a Fm(This)590 b(demonstrates)f(wh)-36 b(y)590 b(some)g(calculations)g(used)f(in)h (the)f(last)h(section)g(to)f(generate)h(the)f(list)h(of)800 3029 y(primes)393 b(w)-36 b(ere)394 b(redundan)-36 b(t.)563 b(Indeed,)401 b(in)393 b(searc)-36 b(hing)394 b(for)g(v)-72 b(arious)394 b(families)h(of)f(factors,)403 b(similar)394 b(iden)-36 b(tities)800 4634 y(arise.)552 b(W)-108 b(e)354 b(no)-36 b(w)353 b(dev)-36 b(elop)354 b(other)e(Aurifeuillian)j(iden) -36 b(tities,)369 b(in)-36 b(teresting)353 b(for)h(their)f(o)-36 b(wn)353 b(sak)-36 b(e,)371 b(and)352 b(mak)-36 b(e)800 6239 y(connections)578 b(to)g(expressions)g(used)g(in)g(the)f(last)i (section.)1011 b(Let)578 b(us)g(start)g(with)g(a)g(general)g(theorem) 800 7844 y(regarding)434 b(cyclotomic)h(p)36 b(olynomials.)p Black 800 10889 a Fn(Theorem)499 b(3.1)p Black 651 w Fe(L)-66 b(et)532 b Fl(m;)221 b(n)497 b Fg(2)e Fm(Z)-664 b(Z)17305 10331 y Fh(+)18625 10889 y Fe(satisfy)533 b Fm(gcd\()p Fl(m;)221 b(n)p Fm(\))496 b(=)f(1)534 b Fe(and)f(at)g(le)-66 b(ast)534 b(one)e(of)h Fl(m)g Fe(or)h Fl(n)g Fe(is)f(gr)-66 b(e)g(ater)800 12494 y(than)465 b(2.)597 b(Then)20493 15165 y Fm(\251)21432 15364 y Fi(mn)22891 15165 y Fm(\()p Fl(x)p Fm(\))368 b(=)26391 13401 y Fi(\301)p Fh(\()p Fi(m)p Fh(\))26604 13903 y Fj(Y)26636 16703 y Fi(j)51 b Fh(=1)28736 15165 y Fm(\251)29675 15364 y Fi(n)30302 15165 y Fm(\()p Fl(x\263)32115 15364 y Fi(j)32601 15165 y Fm(\))17381 b(\(10\))800 18935 y Fe(wher)-66 b(e)465 b Fl(\263)5038 19134 y Fi(j)5990 18935 y Fe(ar)-66 b(e)464 b(the)h(primitive)d Fl(m)17057 18453 y Fi(th)18459 18935 y Fe(r)-66 b(o)g(ots)465 b(of)f(unity.)800 21979 y Fn(Pro)42 b(of:)1258 b Fm(Since)459 b(the)f(degree)h(of)h(\251)18302 22178 y Fi(n)18929 21979 y Fm(\()p Fl(x)p Fm(\))e(is)i Fl(\301)p Fm(\()p Fl(n)p Fm(\),)465 b(the)459 b(degree)g(of)h(the)e (left)i(side)f(p)36 b(olynomial)461 b(of)e(\()p 0 1 0 0 TeXcolorcmyk(10)p 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b(remains)f(to)h(sho)-36 b(w)434 b(that)f(the)g(ro)36 b(ots)434 b(of)g(eac)-36 b(h)434 b(side)f(are)h(the)f(same.)2751 37820 y(The)495 b(ro)36 b(ots)496 b(of)g(\251)11325 38019 y Fi(mn)12783 37820 y Fm(\()p Fl(x)p Fm(\))f(are)h(simply)f Fl(e)22085 37338 y Fi(k)24 b(i)p Fh(2)p Fi(\274)32 b(=mn)26386 37820 y Fm(with)495 b(gcd\()p Fl(k)45 b(;)221 b(mn)p Fm(\))474 b(=)f(1.)764 b(W)-108 b(e)495 b(will)h(sho)-36 b(w)495 b(that)g(eac)-36 b(h)800 39425 y(of)448 b(these)e Fl(\301)p Fm(\()p Fl(mn)p Fm(\))g(distinct)g(ro)36 b(ots)447 b(is)h(also)f(a)g (ro)36 b(ot)447 b(of)h(the)e(righ)-36 b(t)447 b(side.)617 b(T)-108 b(o)447 b(expand)g(the)f(righ)-36 b(t)446 b(side,)451 b(\257rst)800 41030 y(note)440 b(that)g(eac)-36 b(h)440 b Fl(\263)10023 41229 y Fi(j)10950 41030 y Fm(has)g(the)f(form)i Fl(e)19203 40548 y Fi(l)11 b(i)p Fh(2)p Fi(\274)32 b(=m)22655 41030 y Fm(for)441 b(some)f Fl(l)470 b Fm(satisfying)442 b(gcd\()p Fl(l)29 b(;)221 b(m)p Fm(\))381 b(=)e(1.)599 b(This)440 b(com)-36 b(bines)440 b(to)800 42635 y(giv)-36 b(e)21661 44240 y Fl(x\263)22968 44439 y Fi(j)23824 44240 y Fm(=)369 b Fl(e)25808 43692 y Fh(\()p Fi(k)24 b Fh(+)p Fi(l)11 b(n)p Fh(\))p Fi(i)p Fh(2)p Fi(\274)32 b(=mn)800 46565 y Fm(so)434 b(it)g(is)g(su\261ces)f(to)g(sho)-36 b(w)434 b(that)f(for)h(eac)-36 b(h)434 b Fl(k)479 b Fm(there)433 b(exists)h(an)f Fl(l)463 b Fm(suc)-36 b(h)433 b(that)20785 49930 y(gcd)22957 48057 y Fj(\265)24068 49031 y Fl(k)340 b Fm(+)295 b Fl(l)29 b(n)p 24068 49624 3517 54 v 25257 50841 a(m)27717 49930 y(;)221 b(n)29075 48057 y Fj(\266)30423 49930 y Fm(=)368 b(1)p Fl(:)800 53627 y Fm(Since)k(gcd\()p Fl(k)45 b(;)221 b(mn)p Fm(\))369 b(=)g(1,)385 b(w)-36 b(e)372 b(ha)-36 b(v)g(e)372 b(gcd\()p Fl(k)45 b(;)221 b(n)p Fm(\))370 b(=)e(1.)559 b(This)372 b(implies)g(that)g(gcd\()p Fl(k)214 b Fm(+)169 b Fl(l)29 b(n;)221 b(n)p Fm(\))371 b(=)e(1)j(and,)384 b(using)800 55232 y(the)426 b(Euclidean)h(algorithm) g(with)g(gcd\()p Fl(m;)221 b(n)p Fm(\))369 b(=)g(1,)428 b(there)e(exists)i(an)e Fl(l)457 b Fm(suc)-36 b(h)426 b(that)g Fl(k)326 b Fm(+)281 b Fl(l)29 b(n)427 b Fm(is)g(a)g(m)-36 b(ultiple)800 56837 y(of)434 b Fl(m)p Fm(.)578 b(This)434 b(completes)g(the)f(pro)36 b(of.)p 20053 56726 665 665 v 2751 58442 a(With)434 b(the)f(iden)-36 b(tities)17442 60047 y(\251)18381 60303 y Fh(2)18851 60051 y Fc(k)19360 60303 y Fi(n)19986 60047 y Fm(\()p Fl(x)p Fm(\))369 b(=)f(\251)24425 60246 y Fi(n)25274 58573 y Fj(\263)26067 60047 y Fg(\241)p Fl(x)27839 59499 y Fh(2)28309 59186 y Fc(k)14 b Fb(\241)p Fd(1)29918 58573 y Fj(\264)30933 60047 y Fl(;)1523 b(n)434 b Fm(o)36 b(dd)800 62664 y(and)17040 64269 y(\251)17979 64468 y Fi(pn)19079 64269 y Fm(\()p Fl(x)p Fm(\))369 b(=)g(\251)23519 64468 y Fi(n)24145 64269 y Fm(\()p Fl(x)25390 63720 y Fi(p)25919 64269 y Fm(\))p Fl(;)1522 b(p)434 b Fm(prime)o Fl(;)656 b(p)368 b Fg(6)h(j)p Fl(n;)800 66593 y Fm(w)-36 b(e)434 b(construct)f(sev)-36 b(eral)434 b(examples.)p Black 26475 74617 a(7)p Black eop %%Page: 8 8 8 7 bop Black 0 TeXcolorgray Black Black Black 800 1424 a Fn(Example)499 b(3.1)p Black 651 w Fm(The)402 b(case)h Fl(m)368 b Fm(=)h(4)403 b(w)-36 b(as)403 b(foreshado)-36 b(w)g(ed)402 b(b)-36 b(y)402 b(Sc)-36 b(hinzel[)p 0 1 0 0 TeXcolorcmyk(7)p (#cite.Schinzel) [[396 704 402 716] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)409 b(form)-36 b(ula)403 b(\(12\)].)569 b(Letting)402 b Fl(n)g Fm(b)36 b(e)800 3029 y(o)g(dd)433 b(in)h(Theorem)p 0 1 0 0 TeXcolorcmyk 433 w(3.1)p (#thm.3.1) [[165 690 180 702] [1 1 1 [3 3]] [0 0 1]] pdfm Black 435 w(giv)-36 b(es)16262 5947 y(\251)17201 6146 y Fi(n)17828 5947 y Fm(\()p Fg(\241)p Fl(x)20106 5399 y Fh(2)20632 5947 y Fm(\))368 b(=)h(\251)23826 6146 y Fh(4)p Fi(n)24923 5947 y Fm(\()p Fl(x)p Fm(\))g(=)f(\251)29362 6146 y Fi(n)29989 5947 y Fm(\()p Fl(ix)p Fm(\)\251)33123 6146 y Fi(n)33749 5947 y Fm(\()p Fg(\241)p Fl(ix)p Fm(\))p Fl(:)800 8865 y Fm(Replacing)434 b Fl(x)g Fm(with)g Fl(ix)f Fm(giv)-36 b(es)19991 10470 y(\251)20930 10669 y Fi(n)21556 10470 y Fm(\()p Fl(x)22801 9922 y Fh(2)23327 10470 y Fm(\))369 b(=)f(\251)26521 10669 y Fi(n)27148 10470 y Fm(\()p Fl(x)p Fm(\)\251)29838 10669 y Fi(n)30464 10470 y Fm(\()p Fg(\241)p Fl(x)p Fm(\))p Fl(:)p Black 800 13496 a Fn(Example)499 b(3.2)p Black 651 w Fm(Letting)433 b Fl(m)368 b Fm(=)h(8,)434 b Fl(n)g Fm(o)36 b(dd,)434 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)6065 17104 y(\251)7004 17303 y Fh(8)p Fi(n)8101 17104 y Fm(\()p Fl(x)p Fm(\))368 b(=)11601 15231 y Fj(\267)12302 17104 y Fm(\251)13241 17303 y Fi(n)14089 15231 y Fj(\265)15067 17104 y Fl(x)15939 16206 y Fm(1)295 b(+)g Fl(i)p 15939 16799 2697 54 v 16408 17064 a Fg(p)p 17515 17064 651 54 v 1099 x Fm(2)18768 15231 y Fj(\266)19967 17104 y Fm(\251)20906 17303 y Fi(n)21754 15231 y Fj(\265)22732 17104 y Fl(x)23604 16206 y Fm(1)g Fg(\241)h Fl(i)p 23604 16799 2718 54 v 24084 17064 a Fg(p)p 25191 17064 651 54 v 1099 x Fm(2)26454 15231 y Fj(\266\270)221 b(\267)29056 17104 y Fm(\251)29995 17303 y Fi(n)30843 15231 y Fj(\265)31820 17104 y Fl(x)32692 16206 y Fg(\241)p Fm(1)296 b(+)f Fl(i)p 32692 16799 3730 54 v 33678 17064 a Fg(p)p 34785 17064 651 54 v 1099 x Fm(2)36554 15231 y Fj(\266)37754 17104 y Fm(\251)38693 17303 y Fi(n)39541 15231 y Fj(\265)40518 17104 y Fl(x)41390 16206 y Fg(\241)p Fm(1)h Fg(\241)f Fl(i)p 41390 16799 3751 54 v 42387 17064 a Fg(p)p 43494 17064 651 54 v 1099 x Fm(2)45274 15231 y Fj(\266\270)47174 17104 y Fl(:)2953 b Fm(\(11\))800 20995 y(Replacing)434 b Fl(x)g Fm(with)10951 19896 y Fg(p)p 12058 19896 V 1099 x Fm(2)p Fl(x)g Fm(yields)8775 23913 y(\251)9714 24112 y Fi(n)10341 23913 y Fm(\()p Fg(\241)p Fm(4)p Fl(x)13269 23364 y Fh(4)13795 23913 y Fm(\))369 b(=)g(\251)16990 24112 y Fh(4)p Fi(n)18087 23913 y Fm(\(2)p Fl(x)19982 23364 y Fh(2)20508 23913 y Fm(\))10989 25850 y(=)g([\251)13670 26049 y Fi(n)14297 25850 y Fm(\()p Fl(x)p Fm(\(1)295 b(+)g Fl(i)p Fm(\)\)\251)20695 26049 y Fi(n)21321 25850 y Fm(\()p Fl(x)p Fm(\(1)g Fg(\241)g Fl(i)p Fm(\)\)])221 b([)q(\251)28684 26049 y Fi(n)29310 25850 y Fm(\()p Fl(x)p Fm(\()p Fg(\241)p Fm(1)296 b(+)f Fl(i)p Fm(\)\)\251)36742 26049 y Fi(n)37368 25850 y Fm(\()p Fl(x)p Fm(\()p Fg(\241)p Fm(1)g Fg(\241)g Fl(i)p Fm(\)\)])221 b Fl(:)p Black 5664 w Fm(\(12\))p Black 800 28768 a(If)484 b Fl(x)f Fm(is)g(real,)496 b(the)483 b(t)-36 b(w)g(o)483 b(righ)-36 b(t)482 b(side)h(expressions)h (m)-36 b(ust)482 b(b)36 b(e)483 b(in)-36 b(teger)482 b(p)36 b(olynomials)485 b(since)e(they)g(are)g(eac)-36 b(h)800 30373 y(the)433 b(pro)36 b(duct)433 b(of)h(complex)g (conjugates.)579 b(Example)434 b(sub)36 b(cases)434 b(with)f Fl(x)369 b Fm(=)g(2)38164 29891 y Fi(k)39167 30373 y Fm(are)800 31978 y Fn(n=1)p Fm(:)13163 33583 y(2)13813 33034 y Fh(4)p Fi(k)24 b Fh(+2)16350 33583 y Fm(+)295 b(1)369 b(=)g(\(2)21213 33034 y Fh(2)p Fi(k)24 b Fh(+1)23750 33583 y Fm(+)295 b(2)25707 33034 y Fi(k)24 b Fh(+1)27773 33583 y Fm(+)295 b(1\)\(2)31392 33034 y Fh(2)p Fi(k)24 b Fh(+1)33929 33583 y Fg(\241)296 b Fm(2)35908 33034 y Fi(k)24 b Fh(+1)37974 33583 y Fm(+)295 b(1\))800 35901 y Fn(n=15)p Fm(:)8781 37506 y(2)9431 36958 y Fh(32)p Fi(k)24 b Fh(+16)12910 37506 y Fm(+)294 b(2)14866 36958 y Fh(28)p Fi(k)24 b Fh(+14)18344 37506 y Fg(\241)296 b Fm(2)20323 36958 y Fh(20)p Fi(k)24 b Fh(+10)23801 37506 y Fg(\241)295 b Fm(2)25779 36958 y Fh(16)p Fi(k)24 b Fh(+8)28787 37506 y Fg(\241)295 b Fm(2)30765 36958 y Fh(12)p Fi(k)24 b Fh(+6)33773 37506 y Fm(+)295 b(2)35730 36958 y Fh(4)p Fi(k)24 b Fh(+2)38267 37506 y Fm(+)294 b(1)370 b(=)e Fl(L)296 b Fg(\242)f Fl(R)800 39824 y Fm(where)3729 42742 y Fl(L;)221 b(R)1118 b Fm(=)1107 b(2)10073 42193 y Fh(16)p Fi(k)24 b Fh(+8)13080 42742 y Fg(\247)296 b Fm(2)15059 42193 y Fh(15)p Fi(k)24 b Fh(+8)18066 42742 y Fm(+)295 b(2)20023 42193 y Fh(14)p Fi(k)24 b Fh(+7)23031 42742 y Fg(\247)295 b Fm(2)25009 42193 y Fh(13)p Fi(k)24 b Fh(+7)28017 42742 y Fm(+)295 b(2)29974 42193 y Fh(12)p Fi(k)24 b Fh(+7)32981 42742 y Fg(\247)295 b Fm(2)34959 42193 y Fh(11)p Fi(k)24 b Fh(+7)37967 42742 y Fm(+)295 b(3)g Fg(\242)h Fm(2)41534 42193 y Fh(10)p Fi(k)24 b Fh(+5)44541 42742 y Fg(\247)296 b Fm(2)46520 42193 y Fh(9)p Fi(k)24 b Fh(+6)9423 44679 y Fm(+3)295 b Fg(\242)g Fm(2)12694 44131 y Fh(8)p Fi(k)24 b Fh(+4)15231 44679 y Fg(\247)296 b Fm(2)17210 44131 y Fh(7)p Fi(k)24 b Fh(+5)19747 44679 y Fm(+)295 b(3)g Fg(\242)g Fm(2)23313 44131 y Fh(6)p Fi(k)24 b Fh(+3)25850 44679 y Fg(\247)296 b Fm(2)27829 44131 y Fh(5)p Fi(k)24 b Fh(+4)30366 44679 y Fm(+)295 b(2)32323 44131 y Fh(4)p Fi(k)24 b Fh(+3)34860 44679 y Fg(\247)295 b Fm(2)36838 44131 y Fh(3)p Fi(k)24 b Fh(+2)39375 44679 y Fm(+)295 b(2)41332 44131 y Fh(2)p Fi(k)24 b Fh(+1)43869 44679 y Fg(\247)296 b Fm(2)45848 44131 y Fi(k)24 b Fh(+1)47914 44679 y Fm(+)295 b(1)2751 47597 y(Getting)589 b(bac)-36 b(k)590 b(to)g(the)f(redundancies)g(in)g(our)h(earlier)g(calculations,) 630 b(let)590 b Fl(x)635 b Fm(=)g(1)p Fl(=)p Fm(2)45909 47115 y Fi(k)47068 47597 y Fm(in)590 b(\()p 0 1 0 0 TeXcolorcmyk(12)p (#equation.12) [[515 289 527 301] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))g(to)800 49202 y(pro)36 b(duce)3719 52810 y(Re)5475 50937 y Fj(\265)6453 52810 y Fm(log)674 b(\251)9727 53009 y Fh(4)p Fi(n)11046 50937 y Fj(\265)13277 51911 y Fm(1)p 12156 52504 2892 54 v 12156 53721 a(2)12806 53337 y Fh(2)p Fi(k)24 b Fk(\241)p Fh(1)15181 50937 y Fj(\266\266)17506 52810 y Fm(=)368 b(2)19757 50937 y Fj(\267)20459 52810 y Fm(Re)22216 50937 y Fj(\265)23194 52810 y Fm(log)674 b(\251)26468 53009 y Fi(n)27316 50937 y Fj(\265)28426 51911 y Fm(1)296 b(+)f Fl(i)p 28426 52504 2697 54 v 29165 53721 a Fm(2)29815 53337 y Fi(k)31255 50937 y Fj(\266\266)33506 52810 y Fm(+)g(Re)36569 50937 y Fj(\265)37547 52810 y Fm(log)674 b(\251)40821 53009 y Fi(n)41669 50937 y Fj(\265)42780 51911 y Fg(\241)p Fm(1)296 b(+)e Fl(i)p 42780 52504 3730 54 v 44035 53721 a Fm(2)44685 53337 y Fi(k)46642 50937 y Fj(\266\266\270)49520 52810 y Fl(:)800 56491 y Fm(This)434 b(sho)-36 b(ws)434 b(ho)-36 b(w)433 b(terms)h(from)g(\()p 0 1 0 0 TeXcolorcmyk(5)p (#equation.5) [[229 209 235 221] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))f(and)g(\()p 0 1 0 0 TeXcolorcmyk(6)p (#equation.6) [[270 209 276 221] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))h(app)36 b(ear)433 b(in)h(some)f(factorizations.)p Black 800 59518 a Fn(Example)499 b(3.3)p Black 651 w Fm(Letting)433 b Fl(m)368 b Fm(=)h(12)434 b(and)f(2)p Fl(;)221 b Fm(3)371 b Fg(6)e(j)p Fl(n)p Fm(,)434 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)3858 63598 y(\251)4797 63797 y Fh(12)p Fi(n)6364 63598 y Fm(\()p Fl(x)p Fm(\))369 b(=)f(\251)10803 63797 y Fi(n)11651 61327 y Fj(\303)12703 63598 y Fl(x)13575 61601 y Fg(p)p 14682 61601 651 54 v 1098 x Fm(3)296 b(+)e Fl(i)p 13575 63292 3804 54 v 15151 64509 a Fm(2)17511 61327 y Fj(!)18784 63598 y Fm(\251)19723 63797 y Fi(n)20571 61327 y Fj(\303)21623 63598 y Fl(x)22495 61601 y Fg(p)p 23601 61601 651 54 v 23601 62699 a Fm(3)i Fg(\241)f Fl(i)p 22495 63292 3825 54 v 24082 64509 a Fm(2)26452 61327 y Fj(!)27725 63598 y Fm(\251)28664 63797 y Fi(n)29512 61327 y Fj(\303)30564 63598 y Fl(x)31436 62699 y Fg(\241)32469 61601 y(p)p 33576 61601 651 54 v 1098 x Fm(3)g(+)g Fl(i)p 31436 63292 4837 54 v 33528 64509 a Fm(2)36404 61327 y Fj(!)37677 63598 y Fm(\251)38616 63797 y Fi(n)39464 61327 y Fj(\303)40516 63598 y Fl(x)41388 62699 y Fg(\241)42421 61601 y(p)p 43528 61601 651 54 v 1098 x Fm(3)h Fg(\241)f Fl(i)p 41388 63292 4858 54 v 43492 64509 a Fm(2)46379 61327 y Fj(!)50488 63598 y Fm(\(13\))800 67887 y(Replacing)434 b Fl(x)g Fm(with)g(2)11601 66788 y Fg(p)p 12708 66788 651 54 v 1099 x Fm(3)p Fl(x)g Fm(giv)-36 b(es)1322 71096 y(\251)2261 71295 y Fh(6)p Fi(n)3358 71096 y Fm(\(12)p Fl(x)5903 70548 y Fh(2)6429 71096 y Fm(\))369 b(=)8685 69622 y Fj(h)9312 71096 y Fm(\251)10251 71295 y Fi(n)11099 69622 y Fj(\263)11892 71096 y Fl(x)p Fm(\(3)296 b(+)15390 69933 y Fg(p)p 16496 69933 V 16496 71096 a Fm(3)q Fl(i)p Fm(\))18097 69622 y Fj(\264)19111 71096 y Fm(\251)20050 71295 y Fi(n)20898 69622 y Fj(\263)21691 71096 y Fl(x)p Fm(\(3)g Fg(\241)25210 69933 y(p)p 26317 69933 V 1163 x Fm(3)q Fl(i)p Fm(\))27918 69622 y Fj(\264)o(i)222 b(h)30186 71096 y Fm(\251)31125 71295 y Fi(n)31973 69622 y Fj(\263)32767 71096 y Fl(x)p Fm(\()p Fg(\241)p Fm(3)295 b(+)37297 69933 y Fg(p)p 38404 69933 V 1163 x Fm(3)p Fl(i)p Fm(\))40004 69622 y Fj(\264)41019 71096 y Fm(\251)41958 71295 y Fi(n)42806 69622 y Fj(\263)43599 71096 y Fl(x)p Fm(\()p Fg(\241)p Fm(3)h Fg(\241)48151 69933 y(p)p 49258 69933 V 1163 x Fm(3)p Fl(i)p Fm(\))50858 69622 y Fj(\264i)p Black 26475 74617 a Fm(8)p Black eop %%Page: 9 9 9 8 bop Black 0 TeXcolorgray Black Black 800 1424 a Fm(Again,)469 b(the)461 b(t)-36 b(w)g(o)462 b(righ)-36 b(t)461 b(side)g(expressions)h (m)-36 b(ust)461 b(b)36 b(e)461 b(in)-36 b(teger)461 b(p)36 b(olynomials.)664 b(Example)462 b(sub)36 b(cases)461 b(with)800 3029 y Fl(x)369 b Fm(=)g(2)3939 2547 y Fi(k)24 b Fk(\241)p Fh(1)6144 3029 y Fm(are)800 4634 y Fn(n=1)p Fm(:)9683 6239 y(9)296 b Fg(\242)f Fm(2)11943 5691 y Fh(4)p Fi(k)13278 6239 y Fg(\241)g Fm(3)g Fg(\242)h Fm(2)16866 5691 y Fh(2)p Fi(k)18200 6239 y Fm(+)f(1)370 b(=)e(\(3)295 b Fg(\242)h Fm(2)24673 5691 y Fh(2)p Fi(k)26008 6239 y Fg(\241)f Fm(3)g Fg(\242)h Fm(2)29596 5691 y Fi(k)30460 6239 y Fm(+)f(1\)\(3)g Fg(\242)g Fm(2)35688 5691 y Fh(2)p Fi(k)37023 6239 y Fm(+)g(3)h Fg(\242)f Fm(2)40590 5691 y Fi(k)41454 6239 y Fm(+)g(1\))800 8505 y Fn(n=5)p Fm(:)5990 11302 y(2)6640 10754 y Fh(16)p Fi(k)8150 11302 y Fm(3)8800 10754 y Fh(8)9621 11302 y Fm(+)g(2)11578 10754 y Fh(14)p Fi(k)13088 11302 y Fm(3)13738 10754 y Fh(7)14560 11302 y Fg(\241)g Fm(2)16538 10754 y Fh(10)p Fi(k)18048 11302 y Fm(3)18698 10754 y Fh(5)19520 11302 y Fg(\241)g Fm(2)21498 10754 y Fh(8)p Fi(k)22538 11302 y Fm(3)23188 10754 y Fh(4)24009 11302 y Fg(\241)g Fm(2)25987 10754 y Fh(6)p Fi(k)27027 11302 y Fm(3)27677 10754 y Fh(3)28498 11302 y Fm(+)g(2)30455 10754 y Fh(2)p Fi(k)31495 11302 y Fm(3)g(+)g(1)8204 13240 y(\(2)9360 12691 y Fh(8)p Fi(k)10399 13240 y Fm(3)11049 12691 y Fh(4)11871 13240 y Fm(+)g(2)13828 12691 y Fh(7)p Fi(k)14867 13240 y Fm(3)15517 12691 y Fh(4)16339 13240 y Fm(+)f(2)18295 12691 y Fh(6)p Fi(k)24 b Fh(+1)20537 13240 y Fm(3)21187 12691 y Fh(3)22009 13240 y Fm(+)294 b(2)23965 12691 y Fh(5)p Fi(k)25005 13240 y Fm(3)25655 12691 y Fh(3)26476 13240 y Fm(+)h(2)28433 12691 y Fh(4)p Fi(k)29473 13240 y Fm(3)30123 12691 y Fh(2)30944 13240 y Fm(+)g(2)32901 12691 y Fh(3)p Fi(k)33941 13240 y Fm(3)34591 12691 y Fh(2)35412 13240 y Fm(+)g(2)37369 12691 y Fh(2)p Fi(k)24 b Fh(+1)39611 13240 y Fm(3)295 b(+)g(2)42513 12691 y Fi(k)43082 13240 y Fm(3)h(+)f(1\))8204 15177 y Fg(\243)p Fm(\(2)10393 14628 y Fh(8)p Fi(k)11433 15177 y Fm(3)12083 14628 y Fh(4)12904 15177 y Fg(\241)g Fm(2)14882 14628 y Fh(7)p Fi(k)15922 15177 y Fm(3)16572 14628 y Fh(4)17393 15177 y Fm(+)g(2)19350 14628 y Fh(6)p Fi(k)24 b Fh(+1)21592 15177 y Fm(3)22242 14628 y Fh(3)23063 15177 y Fg(\241)296 b Fm(2)25042 14628 y Fh(5)p Fi(k)26081 15177 y Fm(3)26731 14628 y Fh(3)27553 15177 y Fm(+)e(2)29509 14628 y Fh(4)p Fi(k)30549 15177 y Fm(3)31199 14628 y Fh(2)32020 15177 y Fg(\241)i Fm(2)33999 14628 y Fh(3)p Fi(k)35038 15177 y Fm(3)35688 14628 y Fh(2)36510 15177 y Fm(+)f(2)38467 14628 y Fh(2)p Fi(k)24 b Fh(+1)40709 15177 y Fm(3)295 b Fg(\241)g Fm(2)43632 14628 y Fi(k)44202 15177 y Fm(3)g(+)g(1\))2751 17974 y(T)-108 b(o)401 b(sho)-36 b(w)401 b(ho)-36 b(w)400 b(equation)h(\()p 0 1 0 0 TeXcolorcmyk(13)p (#equation.13) [[219 555 230 567] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))g(pro)36 b(duces)400 b(redundancies,)406 b(replace)401 b Fl(x)g Fm(with)f Fl(x)p Fm(\()41755 16875 y Fg(p)p 42862 16875 651 54 v 1099 x Fm(3)228 b(+)g Fl(i)p Fm(\))p Fl(=)p Fm(2)400 b(to)h(obtain)948 21933 y(\251)1887 22132 y Fh(6)p Fi(n)3205 19661 y Fj(\303)4390 21034 y Fm(1)295 b(+)6642 19935 y Fg(p)p 7749 19935 V 1099 x Fm(3)p Fl(i)p 4390 21627 4454 54 v 6291 22844 a Fm(2)8976 21933 y Fl(x)9715 21384 y Fh(2)10241 19661 y Fj(!)11662 21933 y Fm(=)368 b(\251)13981 22132 y Fh(12)p Fi(n)15770 19661 y Fj(\303)16955 19935 y Fg(p)p 18062 19935 651 54 v 1099 x Fm(3)295 b(+)g Fl(i)p 16955 21627 3804 54 v 18531 22844 a Fm(2)20890 21933 y Fl(x)21629 19661 y Fj(!)23050 21933 y Fm(=)369 b(\251)25370 22132 y Fi(n)26218 19661 y Fj(\303)27402 21034 y Fm(1)296 b(+)29655 19935 y Fg(p)p 30761 19935 651 54 v 30761 21034 a Fm(3)q Fl(i)p 27402 21627 4454 54 v 29304 22844 a Fm(2)31988 21933 y Fl(x)32727 19661 y Fj(!)34000 21933 y Fm(\251)34939 22132 y Fi(n)35788 19661 y Fj(\303)36972 21034 y Fg(\241)p Fm(1)g Fg(\241)40279 19935 y(p)p 41386 19935 651 54 v 1099 x Fm(3)p Fl(i)p 36972 21627 5509 54 v 39401 22844 a Fm(2)42613 21933 y Fl(x)43352 19661 y Fj(!)44625 21933 y Fm(\251)45564 22132 y Fi(n)46191 21933 y Fm(\()p Fl(x)p Fm(\)\251)48881 22132 y Fi(n)49507 21933 y Fm(\()p Fg(\241)p Fl(x)p Fm(\))p Fl(:)800 25993 y Fm(With)434 b Fl(x)369 b Fm(=)f(1)p Fl(=)p Fm(2)8598 25511 y Fi(k)24 b Fk(\241)p Fh(1)10804 25993 y Fm(this)433 b(yields)i(the)e(relationship)1318 29952 y(Re)3508 27681 y Fj(\303)4559 29952 y Fm(log)241 b(\251)7400 30151 y Fh(6)p Fi(n)8718 27681 y Fj(\303)9903 29054 y Fm(1)295 b(+)12155 27955 y Fg(p)p 13262 27955 651 54 v 1099 x Fm(3)p Fl(i)p 9903 29647 4454 54 v 10683 30863 a Fm(2)11333 30480 y Fh(2)p Fi(k)24 b Fk(\241)p Fh(1)14489 27681 y Fj(!!)17699 29952 y Fm(=)1107 b(Re)22008 27681 y Fj(\303)23060 29952 y Fm(log)240 b(\251)25900 30151 y Fi(n)26748 27681 y Fj(\303)27932 29054 y Fm(1)296 b(+)30185 27955 y Fg(p)p 31292 27955 651 54 v 1099 x Fm(3)p Fl(i)p 27932 29647 4454 54 v 29550 30863 a Fm(2)30200 30480 y Fi(k)32519 27681 y Fj(!)o(!)34917 29952 y Fm(+)f(Re)38414 27681 y Fj(\303)39466 29952 y Fm(log)240 b(\251)42306 30151 y Fi(n)43154 27681 y Fj(\303)44339 29054 y Fg(\241)p Fm(1)295 b(+)47624 27955 y Fg(p)p 48731 27955 651 54 v 1099 x Fm(3)p Fl(i)p 44339 29647 5487 54 v 46472 30863 a Fm(2)47122 30480 y Fi(k)49958 27681 y Fj(!!)19818 33982 y Fm(+)221 b(log)240 b(\251)23891 34181 y Fi(n)24739 32109 y Fj(\265)26735 33083 y Fm(1)p 25850 33676 2422 54 v 25850 34893 a(2)26500 34509 y Fi(k)24 b Fk(\241)p Fh(1)28404 32109 y Fj(\266)29677 33982 y Fm(+)295 b(log)240 b(\251)33824 34181 y Fi(n)34672 32109 y Fj(\265)36152 33083 y Fg(\241)p Fm(1)p 35783 33676 V 35783 34893 a(2)36433 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b(Brillhart,)f(D.H.)h(Lehmer,)e(J.L.)h (Selfridge,)h(B.)f(T)-108 b(uc)-36 b(k)g(erman,)513 b(and)496 b(S.S.)h(W)-108 b(agsta\256,Jr.)766 b Fe(F)-100 b(actor-)2823 3029 y(izations)465 b(of)f Fl(b)9719 2547 y Fi(n)10641 3029 y Fg(\247)295 b Fm(1.)577 b(American)434 b(Mathematical)g(So)36 b(ciet)-36 b(y)-108 b(,)434 b(Pro)-36 b(vidence,)434 b(1983.)p Black 800 5741 a([7])p Black 651 w(A.)365 b(Sc)-36 b(hinzel.)464 b(On)364 b(primitiv)-36 b(e)365 b(prime)f(factors)h(of)h Fl(a)27409 5259 y Fi(n)28189 5741 y Fg(\241)154 b Fl(b)29929 5259 y Fi(n)30556 5741 y Fm(.)464 b Fe(Pr)-66 b(o)g(c.)400 b(Cambridge)h(Phil.)g(So)-66 b(c.)363 b Fn(58)i Fm(\(1962\),)2823 7346 y(555{562.)p Black 800 10058 a([8])p Black 651 w(P)-108 b(.)720 b(Stev)-36 b(enhagen.)457 b(On)360 b(Aurifeuillian)h (factorizations.)459 b Fe(Pr)-66 b(o)g(c.)396 b(Kon.)g(A)-33 b(kad.)397 b(Wetensch.)358 b Fn(90)j Fm(\(1987\),)2823 11663 y(451{468.)p 800 14475 52000 45 v 800 16726 a(2000)439 b Fe(Mathematics)467 b(Subje)-66 b(ct)466 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(http://www.math.uwaterloo.ca/JIS/) pdfm Black(.)p Black 26150 74617 a(10)p Black eop %%Trailer end end