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Fk(2)p Fj(n)p 21758 65581 1427 54 v 21903 66798 a(m)24987 65887 y Fh(\))1670 b Fj(m)368 b Fh(\267)h Fk(2)31676 64976 y Fe(2)p 31677 65137 406 40 v 31677 65686 a(3)32271 65887 y Fj(n)33180 64976 y Fe(1)p 33180 65137 V 33180 65686 a(3)35075 65887 y Fk(\(2\))p Fj(:)800 69514 y Fk(By)434 b(\(1\))g(and)f(\(2\))22088 71296 y Fj(n)22997 70385 y Fe(1)p 22997 70546 V 22997 71095 a(3)23960 71296 y Fh(\267)369 b Fj(m)g Fh(\267)g Fk(2)29054 70385 y Fe(2)p 29054 70546 V 29054 71095 a(3)29648 71296 y Fj(n)30557 70385 y Fe(1)p 30557 70546 V 30557 71095 a(3)31151 71296 y Fj(:)p Black 26475 74617 a Fk(2)p Black eop %%Page: 3 3 3 2 bop Black 0 TeXcolorgray Black Black 2751 1424 a Fk(\(2a\))434 b(Substitute)e Fj(v)416 b Fk(=)369 b Fj(m)295 b Fh(\241)g Fj(u)434 b Fk(in)f Fj(n=m)370 b Fk(=)e Fj(u)24723 942 y Fi(2)25545 1424 y Fh(\241)295 b Fj(uv)343 b Fk(+)295 b Fj(v)30569 942 y Fi(2)31527 1424 y Fk(to)434 b(obtain)g(the)f (equation)20509 3656 y Fj(n)p 20328 4249 1138 54 v 20328 5466 a(m)21968 4555 y Fk(=)368 b(3\()p Fj(u)25244 4006 y Fi(2)26066 4555 y Fh(\241)295 b Fj(mu)p Fk(\))g(+)g Fj(m)32518 4006 y Fi(2)33043 4555 y Fj(:)800 7847 y Fk(Hence)433 b(3)370 b Fh(j)f Fj(m)7561 7365 y Fi(2)8381 7847 y Fh(\241)296 b Fj(n=m)p Fk(.)579 b(Since,)433 b(b)-36 b(y)434 b(\(1a\))f Fj(n)370 b Fh(\267)f Fj(m)25209 7365 y Fi(3)25734 7847 y Fk(,)434 b Fj(l)399 b Fk(=)368 b(\()p Fj(m)30339 7365 y Fi(2)31160 7847 y Fh(\241)295 b Fj(n=m)p Fk(\))p Fj(=)p Fk(3)370 b Fh(\270)f Fk(0.)2751 9452 y(\(3a\))448 b(No)-36 b(w)449 b(consider)f(the)f(v)-72 b(alue)449 b(of)g Fj(l)29 b Fk(:)608 b Fj(l)423 b Fk(=)393 b Fh(\241)p Fj(u)26066 8970 y Fi(2)26898 9452 y Fk(+)304 b Fj(mu)p Fk(.)622 b(This)449 b(means)f Fj(u)g Fk(is)g(an)g(in)-36 b(teger)448 b(ro)36 b(ot)449 b(of)g(the)800 11057 y(quadratic)516 b(equation)g Fj(x)12831 10575 y Fi(2)13708 11057 y Fh(\241)351 b Fj(mx)g Fk(+)g Fj(l)538 b Fk(=)508 b(0)516 b(with)g(in)-36 b(teger)515 b(co)36 b(e\261cien)-36 b(ts,)537 b(so)516 b(the)f(discriminan)-36 b(t,)535 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b(d)443 b(63.)p Black 26475 74617 a(3)p Black eop %%Page: 4 4 4 3 bop Black 0 TeXcolorgray Black Black Black 800 1424 a Fg(Theorem)499 b(3.2.)p Black 554 w Ff(L)-66 b(et)464 b Fj(m;)221 b(n)466 b Ff(b)-66 b(e)464 b(such)i(that)e(ther)-66 b(e)464 b(exist)g Fj(u;)221 b(v)48 b(;)221 b(x;)g(y)515 b Ff(with)19933 4143 y Fj(m)1106 b Fh(\264)h Fj(u)25057 3595 y Fi(3)25879 4143 y Fk(+)294 b Fj(v)27862 3595 y Fi(3)29716 4143 y Fk(mo)36 b(d)443 b(7)20294 6081 y Fj(n)1107 b Fh(\264)g Fj(x)25056 5532 y Fi(3)25877 6081 y Fk(+)295 b Fj(y)27866 5532 y Fi(3)29720 6081 y Fk(mo)36 b(d)443 b(9)p Fj(:)800 8800 y Ff(Then)464 b(ther)-66 b(e)464 b(exist)g(inte)-66 b(gers)463 b Fj(A;)221 b(B)533 b Ff(such)466 b(that)17540 11519 y Fk(28)p Fj(m)296 b Fh(\241)f Fk(27)p Fj(n)370 b Fh(\264)f Fj(A)26425 10971 y Fi(3)27246 11519 y Fk(+)295 b Fj(B)29608 10971 y Fi(3)31462 11519 y Fk(mo)36 b(d)443 b(63)p Fj(:)800 14239 y Ff(F)-100 b(urthermor)-66 b(e,)464 b(every)g(sum)i(of)e(two)h(cub)-66 b(es)465 b(mo)-66 b(dulo)465 b(63)g(arises)h(in)e(this)g(manner.)p Black 800 16767 a(Pr)-66 b(o)g(of.)p Black 649 w Fk(Let)416 b Fj(A)369 b Fk(=)g(28)p Fj(u)261 b Fh(\241)f Fk(27)p Fj(x)417 b Fk(and)f Fj(B)436 b Fk(=)369 b(28)p Fj(v)308 b Fh(\241)261 b Fk(27)p Fj(y)465 b Fk(and)416 b(expand)g Fj(A)35193 16285 y Fi(3)35979 16767 y Fk(+)259 b Fj(B)38305 16285 y Fi(3)39248 16767 y Fk(mo)36 b(dulo)416 b(63)h(to)g(deriv)-36 b(e)417 b(the)800 18372 y(giv)-36 b(en)434 b(equation.)579 b(A)434 b(computation)f(v)-36 b(eri\257es)434 b(the)f(last)h(claim)g (of)g(the)g(theorem)f(statemen)-36 b(t.)p 51860 18372 45 878 v 51905 17538 781 45 v 51905 18372 V 52684 18372 45 878 v 2751 20912 a(If)434 b Fj(N)508 b Fh(\270)369 b Fk(2)434 b(let)8532 24269 y Fj(\261)50 b Fk(\()p Fj(N)139 b Fk(\))368 b(=)13232 23371 y(#)p Fh(f)p Fj(n)i Fh(2)e(f)p Fk(1)p Fj(;)221 b(:)g(:)g(:)k(;)c(N)139 b Fh(g)369 b Fk(:)g Fj(n)g Fh(\264)h Fj(x)27839 22889 y Fi(3)28660 23371 y Fk(+)295 b Fj(y)30649 22889 y Fi(3)31543 23371 y Fk(mo)36 b(d)369 b Fj(N)572 b Fk(has)433 b(a)h(solution)q Fh(g)p 13232 23964 31342 54 v 28313 25180 a Fj(N)44707 24269 y(:)p Black 800 27328 a Fg(Lemma)715 b(3.1.)p Black 650 w Ff(L)-66 b(et)637 b Fj(n)689 b Fh(2)g Fb(Z)637 b Ff(b)-66 b(e)637 b(given)f(and)h Fj(p)h Ff(b)-66 b(e)637 b(a)h(prime)e(with)i Fj(p)689 b Fh(6)p Fk(=)f(3)p Ff(.)1116 b(Then)637 b(if)g(the)g(e)-66 b(quation)800 28933 y Fj(n)497 b Fh(\264)f Fj(x)4341 28451 y Fi(3)5213 28933 y Fk(+)346 b Fj(y)7253 28451 y Fi(3)8148 28933 y Fk(mo)36 b(d)369 b Fj(p)533 b Ff(has)i(a)f(solution)f(so)i(also)f(do)-66 b(es)534 b(the)f(e)-66 b(quation)533 b Fj(n)496 b Fh(\264)h Fj(x)39169 28451 y Fi(3)40041 28933 y Fk(+)346 b Fj(y)42081 28451 y Fi(3)42975 28933 y Fk(mo)36 b(d)369 b Fj(p)46490 28451 y Fc(\256)47683 28933 y Ff(for)534 b(every)800 30538 y Fj(\256)378 b Fh(\270)369 b Fk(1)p Ff(.)p Black 800 33066 a(Pr)-66 b(o)g(of.)p Black 649 w Fk(Let)466 b Fj(n)426 b Fh(\264)f Fj(x)10634 32584 y Fi(3)11478 33066 y Fk(+)317 b Fj(y)13489 32584 y Fi(3)14384 33066 y Fk(mo)36 b(d)369 b Fj(p)p Fk(.)677 b(Assume)467 b(that)f Fj(p)425 b Fb(-)g Fj(x)p Fk(.)678 b(\(If)467 b Fj(p)425 b Fh(j)g Fj(x)467 b Fk(and)f Fj(p)425 b Fh(j)g Fj(y)515 b Fk(then)465 b Fj(p)425 b Fh(j)h Fj(n)p Fk(,)475 b(so)467 b(w)-36 b(e)467 b(can)800 34672 y(use)475 b Fj(x)441 b Fk(=)g(1)476 b(and)f Fj(y)489 b Fk(=)440 b Fh(\241)p Fk(1.\))705 b(Assume,)486 b(using)476 b(induction,)486 b(that)475 b Fj(n)441 b Fh(\264)g Fj(x)36413 34190 y Fi(3)37263 34672 y Fk(+)323 b Fj(y)39280 34190 y Fi(3)40175 34672 y Fk(mo)36 b(d)369 b Fj(p)43690 34190 y Fc(\256)44825 34672 y Fk(has)476 b(a)g(solution)800 36277 y(for)434 b(some)g Fj(\256)378 b Fh(\270)369 b Fk(1)434 b(with)f Fj(p)369 b Fb(-)g Fj(x)p Fk(.)579 b(Then)21823 37882 y Fj(x)22562 37333 y Fi(3)23384 37882 y Fk(+)294 b Fj(y)25372 37333 y Fi(3)26193 37882 y Fh(\241)i Fj(n)369 b Fk(=)g Fj(l)29 b(p)31117 37333 y Fc(\256)800 40115 y Fk(for)434 b(some)g Fj(l)399 b Fh(2)368 b Fb(Z)433 b Fk(and)g(so,)i(if)f Fj(m)f Fk(is)h(an)g(in)-36 b(teger)433 b(to)h(b)36 b(e)433 b(c)-36 b(hosen)433 b(later,)10240 42834 y(\()p Fj(x)295 b Fk(+)g Fj(mp)14878 42285 y Fc(\256)15537 42834 y Fk(\))16043 42285 y Fi(3)16937 42834 y Fk(=)369 b Fj(y)19000 42285 y Fi(3)19821 42834 y Fh(\241)295 b Fj(n)1108 b Fh(\264)f Fj(x)25912 42285 y Fi(3)26733 42834 y Fk(+)295 b Fj(y)28722 42285 y Fi(3)29543 42834 y Fh(\241)g Fj(n)g Fk(+)g(3)p Fj(mx)35776 42285 y Fi(2)36302 42834 y Fj(p)36955 42285 y Fc(\256)37984 42834 y Fk(mo)36 b(d)369 b Fj(p)41499 42285 y Fc(\256)p Fi(+1)23033 44771 y Fh(\264)1107 b Fj(p)25826 44223 y Fc(\256)26485 44771 y Fk(\()p Fj(l)325 b Fk(+)295 b(3)p Fj(mx)31537 44223 y Fi(2)32063 44771 y Fk(\))368 b(mo)36 b(d)369 b Fj(p)36452 44223 y Fc(\256)p Fi(+1)38314 44771 y Fj(:)2751 47490 y Fk(But)433 b Fj(p)369 b Fh(6)p Fk(=)f(3)434 b(and)f Fj(p)369 b Fb(-)g Fj(x)434 b Fk(so)g(w)-36 b(e)434 b(can)f(c)-36 b(ho)36 b(ose)434 b Fj(m)f Fk(with)h Fj(l)325 b Fk(+)295 b(3)p Fj(mx)33384 47008 y Fi(2)34279 47490 y Fh(\264)369 b Fk(0)g(mo)36 b(d)369 b Fj(p)433 b Fk(giving)18084 50210 y Fj(n)370 b Fh(\264)f Fk(\()p Fj(x)295 b Fk(+)g Fj(mp)25270 49661 y Fc(\256)25929 50210 y Fk(\))26435 49661 y Fi(3)27256 50210 y Fk(+)g Fj(y)29245 49661 y Fi(3)30139 50210 y Fk(mo)36 b(d)369 b Fj(p)33654 49661 y Fc(\256)p Fi(+1)800 52929 y Fk(and)433 b Fj(p)369 b Fb(-)g Fj(x)295 b Fk(+)g Fj(mp)9221 52447 y Fc(\256)10314 52929 y Fk(since)433 b Fj(p)369 b Fb(-)g Fj(x)p Fk(.)579 b(This)433 b(completes)h(the)f (inductiv)-36 b(e)433 b(step.)p 51860 52929 45 878 v 51905 52095 781 45 v 51905 52929 V 52684 52929 45 878 v Black 800 55458 a Fg(Lemma)548 b(3.2.)p Black 576 w Ff(L)-66 b(et)504 b Fj(n)442 b Fh(2)g Fb(Z)504 b Ff(b)-66 b(e)503 b(given.)714 b(Then)504 b(if)f(the)h(e)-66 b(quation)503 b Fj(n)443 b Fh(\264)f Fj(x)36512 54975 y Fi(3)37362 55458 y Fk(+)324 b Fj(y)39380 54975 y Fi(3)40275 55458 y Fk(mo)36 b(d)369 b(3)43787 54975 y Fi(2)44817 55458 y Ff(has)505 b(a)g(solution)800 57063 y(so)466 b(also)f(do)-66 b(es)465 b(the)g(e)-66 b(quation)463 b Fj(n)370 b Fh(\264)f Fj(x)18599 56581 y Fi(3)19420 57063 y Fk(+)295 b Fj(y)21409 56581 y Fi(3)22304 57063 y Fk(mo)36 b(d)368 b(3)25815 56581 y Fc(\256)26940 57063 y Ff(for)465 b(every)f Fj(\256)378 b Fh(\270)369 b Fk(2)p Ff(.)p Black 800 59591 a(Pr)-66 b(o)g(of.)p Black 649 w Fk(Let)437 b Fj(n)376 b Fh(\264)g Fj(x)10506 59109 y Fi(3)11330 59591 y Fk(+)297 b Fj(y)13321 59109 y Fi(3)14216 59591 y Fk(mo)36 b(d)369 b(3)17728 59109 y Fi(2)18254 59591 y Fk(.)590 b(Assume)437 b(that)g(3)376 b Fb(-)g Fj(x)p Fk(.)590 b(\(If)438 b(3)376 b Fh(j)f Fj(x)438 b Fk(and)f(3)376 b Fh(j)f Fj(y)485 b Fk(then)437 b(3)44107 59109 y Fi(2)45009 59591 y Fh(j)375 b Fj(n)p Fk(,)439 b(so)f(w)-36 b(e)438 b(can)800 61196 y(use)503 b Fj(x)489 b Fk(=)e(1)504 b(and)g Fj(y)536 b Fk(=)487 b Fh(\241)p Fk(1.\))789 b(Assume)503 b(that)h Fj(n)488 b Fh(\264)h Fj(x)26909 60714 y Fi(3)27777 61196 y Fk(+)343 b Fj(y)29814 60714 y Fi(3)30708 61196 y Fk(mo)36 b(d)369 b(3)34220 60714 y Fc(\256)35384 61196 y Fk(has)504 b(a)f(solution)h (for)g(some)g Fj(\256)497 b Fh(\270)489 b Fk(2)800 62801 y(with)434 b(3)369 b Fb(-)g Fj(x)p Fk(.)578 b(Then)21825 64406 y Fj(x)22564 63858 y Fi(3)23385 64406 y Fk(+)295 b Fj(y)25374 63858 y Fi(3)26194 64406 y Fh(\241)h Fj(n)369 b Fk(=)g Fj(l)29 b Fk(3)31115 63858 y Fc(\256)800 66639 y Fk(for)434 b(some)g Fj(l)399 b Fh(2)368 b Fb(Z)433 b Fk(and)g(so,)i(if)f Fj(m)f Fk(is)h(an)g(in)-36 b(teger)433 b(to)h(b)36 b(e)433 b(c)-36 b(hosen)433 b(later,)9967 69359 y(\()p Fj(x)296 b Fk(+)e Fj(m)p Fk(3)14602 68810 y Fc(\256)p Fa(\241)p Fi(1)16464 69359 y Fk(\))16970 68810 y Fi(3)17865 69359 y Fk(=)369 b Fj(y)19928 68810 y Fi(3)20748 69359 y Fh(\241)296 b Fj(n)1107 b Fh(\264)g Fj(x)26839 68810 y Fi(3)27660 69359 y Fk(+)295 b Fj(y)29649 68810 y Fi(3)30470 69359 y Fh(\241)g Fj(n)h Fk(+)f Fj(mx)36054 68810 y Fi(2)36579 69359 y Fk(3)37229 68810 y Fc(\256)38258 69359 y Fk(mo)36 b(d)369 b(3)41770 68810 y Fc(\256)p Fi(+1)23960 71296 y Fh(\264)1107 b Fk(3)26750 70747 y Fc(\256)27410 71296 y Fk(\()p Fj(l)325 b Fk(+)295 b Fj(mx)31812 70747 y Fi(2)32337 71296 y Fk(\))369 b(mo)36 b(d)369 b(3)36724 70747 y Fc(\256)p Fi(+1)38586 71296 y Fj(:)p Black 26475 74617 a Fk(4)p Black eop %%Page: 5 5 5 4 bop Black 0 TeXcolorgray Black Black 2751 1424 a Fk(Cho)36 b(ose)434 b Fj(m)g Fk(with)f Fj(l)325 b Fk(+)295 b Fj(mx)15704 942 y Fi(2)16599 1424 y Fh(\264)369 b Fk(0)g(mo)36 b(d)369 b(3)434 b(giving)17305 4358 y Fj(n)369 b Fh(\264)g Fk(\()p Fj(x)296 b Fk(+)e Fj(m)p Fk(3)24487 3809 y Fc(\256)p Fa(\241)p Fi(1)26349 4358 y Fk(\))26855 3809 y Fi(3)27676 4358 y Fk(+)h Fj(y)29665 3809 y Fi(3)30560 4358 y Fk(mo)36 b(d)369 b(3)34072 3809 y Fc(\256)p Fi(+1)35934 4358 y Fj(:)p 51860 7291 45 878 v 51905 6457 781 45 v 51905 7291 V 52684 7291 45 878 v Black 800 10003 a Fg(Theorem)488 b(3.3.)p Black 546 w Ff(The)455 b(p)-66 b(ositive)455 b(inte)-66 b(ger)453 b Fj(m)i Ff(is)h(such)g(that)f Fk(7)370 b Fb(-)f Fj(m)455 b Ff(and)h Fk(9)369 b Fb(-)g Fj(m)455 b Ff(if)g(and)g(only)h(if)e Fj(\261)50 b Fk(\()p Fj(m)p Fk(\))369 b(=)f(1)p Ff(.)800 11608 y(If)458 b Fk(7)369 b Fh(j)g Fj(m)459 b Ff(and)g Fk(9)369 b Fb(-)g Fj(m)459 b Ff(then)f Fj(\261)50 b Fk(\()p Fj(m)p Fk(\))369 b(=)f(5)p Fj(=)p Fk(7)p Ff(.)597 b(If)458 b Fk(9)369 b Fh(j)g Fj(m)459 b Ff(and)g Fk(7)369 b Fb(-)g Fj(m)459 b Ff(then)f Fj(\261)50 b Fk(\()p Fj(m)p Fk(\))368 b(=)h(5)p Fj(=)p Fk(9)p Ff(.)596 b(If)458 b Fk(7)369 b Fh(j)g Fj(m)459 b Ff(and)g Fk(9)369 b Fh(j)g Fj(m)800 13213 y Ff(then)464 b Fj(\261)50 b Fk(\()p Fj(m)p Fk(\))368 b(=)h(25)p Fj(=)p Fk(63)p Ff(.)p Black 800 15925 a(Pr)-66 b(o)g(of.)p Black 649 w Fk(The)511 b(\\if")i(direction)e(follo)-36 b(ws)513 b(directly)e(from)g(the)g (example)h(at)f(the)f(start)h(of)h(this)f(section,)531 b(so)800 17530 y(assume)434 b Fj(m)f Fk(is)h(suc)-36 b(h)432 b(that)i(7)369 b Fb(-)g Fj(m)433 b Fk(and)g(9)369 b Fb(-)g Fj(m)p Fk(.)2751 19135 y(By)455 b([)p 0 1 0 0 TeXcolorcmyk(9)p (#cite.vosper) [[119 545 124 557] [1 1 1 [3 3]] [0 0 1]] pdfm Black(],)460 b Fj(\261)50 b Fk(\()p Fj(p)p Fk(\))404 b(=)g(1)454 b(for)h Fj(p)404 b Fh(6)p Fk(=)g(2)p Fj(;)221 b Fk(3)p Fj(;)g Fk(7.)642 b(Simple)454 b(computations)g(lead)h(to)f (the)g(v)-72 b(alues)455 b Fj(\261)50 b Fk(\(2\))404 b(=)f(1)p Fj(;)221 b(\261)50 b Fk(\(3\))405 b(=)800 20741 y(1)p Fj(;)221 b(\261)50 b Fk(\(9\))370 b(=)e(5)p Fj(=)p Fk(9,)435 b Fj(\261)50 b Fk(\(7\))369 b(=)g(5)p Fj(=)p Fk(7.)2751 22346 y(By)397 b(the)f(Chinese)g(remainder)f(theorem)h(and)g (the)g(de\257nition)f(of)i(addition)f(and)g(m)-36 b(ultiplication)397 b(in)f(a)800 23951 y(pro)36 b(duct)403 b(ring,)411 b(if)404 b(\()p Fj(N)66 b(;)221 b(M)139 b Fk(\))370 b(=)e(1,)411 b(then)403 b Fj(\261)50 b Fk(\()p Fj(M)139 b(N)g Fk(\))368 b(=)h Fj(\261)50 b Fk(\()p Fj(M)139 b Fk(\))p Fj(\261)50 b Fk(\()p Fj(N)139 b Fk(\).)567 b(Hence)404 b(w)-36 b(e)404 b(need)g(only)h(consider)f(v)-72 b(alues)800 25556 y(of)434 b Fj(m)g Fk(whic)-36 b(h)433 b(are)h(prime)f(p)36 b(o)-36 b(w)g(ers.)2751 27161 y(By)518 b(Lemma)g(1,)540 b(if)518 b Fj(p)512 b Fh(6)p Fk(=)h(3,)539 b Fj(\261)50 b Fk(\()p Fj(p)18455 26679 y Fc(\256)19114 27161 y Fk(\))513 b(=)f Fj(\261)50 b Fk(\()p Fj(p)p Fk(\))517 b(for)h(all)h Fj(\256)i Fh(\270)513 b Fk(1.)831 b(By)519 b(Lemma)e(2,)540 b Fj(\261)50 b Fk(\(3)43348 26679 y Fc(\256)44008 27161 y Fk(\))512 b(=)g Fj(\261)50 b Fk(\(9\))518 b(for)g(all)800 28766 y Fj(\256)378 b Fh(\270)369 b Fk(2)434 b(and)f(the)g(theorem)g(follo) -36 b(ws)436 b(directly)-108 b(.)p 51860 28766 V 51905 27932 781 45 v 51905 28766 V 52684 28766 45 878 v 800 33203 a Fl(4)2152 b(The)716 b(F)-179 b(unctions)715 b(Theta)i(and)f (Eta)p Black 800 36124 a Fg(De\257nition)579 b(4.1.)p Black 2393 w Fk(Let)503 b Fj(n)487 b Fh(2)g Fb(N)p Fk(.)787 b(Then)503 b Fj(\265)36 b Fk(\()p Fj(n)p Fk(\))503 b(is)g(the)f(least)i (p)36 b(ositiv)-36 b(e)504 b(in)-36 b(teger)503 b(suc)-36 b(h)502 b(that)g(the)h(Dio-)800 37729 y(phan)-36 b(tine)432 b(equation)22285 39334 y Fj(n\265)36 b Fk(\()p Fj(n)p Fk(\))369 b(=)g Fj(x)27980 38785 y Fi(3)28801 39334 y Fk(+)295 b Fj(y)30790 38785 y Fi(3)800 41658 y Fk(has)434 b(a)f(solution)h(with)g Fj(x)369 b Fh(\270)g Fk(0)434 b(and)f Fj(y)417 b Fh(\270)369 b Fk(0.)2751 44370 y(Because)17551 45976 y(\()p Fj(n)295 b Fk(+)g(1\))21591 45427 y Fi(3)22412 45976 y Fk(+)g(\()p Fj(n)g Fh(\241)h Fk(1\))27781 45427 y Fi(3)28676 45976 y Fk(=)368 b(2)p Fj(n)p Fk(\()p Fj(n)32764 45427 y Fi(2)33586 45976 y Fk(+)295 b(3\))800 48300 y(the)445 b(function)h Fj(\265)482 b Fk(is)447 b(w)-36 b(ell)447 b(de\257ned)d(and)h Fj(\265)36 b Fk(\()p Fj(n)p Fk(\))391 b Fh(\267)f Fk(2\()p Fj(n)26427 47818 y Fi(2)27257 48300 y Fk(+)303 b(3\).)616 b(The)446 b(p)36 b(ositiv)-36 b(e)447 b(in)-36 b(teger)446 b Fj(n)g Fk(is)g(expressible)h(as)800 49905 y(the)433 b(sum)g(of)h(t)-36 b(w)g(o)434 b(p)36 b(ositiv)-36 b(e)435 b(cub)36 b(es)433 b(if)h(and)f(only)h(if)h Fj(\265)36 b Fk(\()p Fj(n)p Fk(\))369 b(=)g(1.)2751 51510 y(Sometimes)427 b(a)h(distinction)f(is)g(made)h(b)36 b(et)-36 b(w)g(een)426 b(general)i(solutions)g(to)f(equations)h(lik)-36 b(e)428 b Fj(n)370 b 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b Fj(a)44860 30873 y Fi(1)45385 30674 y Fj(;)221 b Fh(\242)g(\242)g(\242)445 b Fj(;)221 b(a)49226 30873 y Fc(n)50271 30674 y Fk(with)12318 33608 y Fj(\265)36 b Fk(\()p Fj(a)14149 33807 y Fi(1)14674 33608 y Fk(\))369 b(=)g Fj(a)17613 33807 y Fi(2)18138 33608 y Fj(;)221 b(\265)36 b Fk(\()p Fj(a)20551 33807 y Fi(2)21078 33608 y Fk(\))368 b(=)h Fj(a)24016 33807 y Fi(3)24542 33608 y Fj(;)221 b Fh(\242)g(\242)g(\242)444 b Fj(;)221 b(\265)36 b Fk(\()p Fj(a)29530 33807 y Fc(n)p Fa(\241)p Fi(1)31728 33608 y Fk(=)368 b Fj(a)33791 33807 y Fc(n)34417 33608 y Fj(;)221 b(\265)36 b Fk(\()p Fj(a)36830 33807 y Fc(n)37457 33608 y Fk(\))369 b(=)g Fj(a)40396 33807 y Fi(1)40921 33608 y Fj(:)800 36541 y Fk(If)507 b Fj(n)g Fk(is)g(ev)-36 b(en)506 b Fj(a)8555 36740 y Fi(1)9573 36541 y Fh(\270)494 b Fj(a)11783 36740 y Fi(3)12801 36541 y Fh(\270)f Fj(a)15010 36740 y Fi(5)15536 36541 y Fj(;)221 b Fh(\242)g(\242)g(\242)494 b(\270)f Fj(a)20370 36740 y Fc(n)p Fa(\241)p Fi(1)22691 36541 y Fh(\270)g Fj(a)24900 36740 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a(Note)487 b(that)g(if)h(\()p Fj(x;)221 b(y)48 b Fk(\))487 b(is)h(the)f(closest)g(in)-36 b(tegral)488 b(p)36 b(oin)-36 b(t)487 b(on)g Fj(n)460 b Fk(=)g Fj(x)34953 45801 y Fi(2)35811 46283 y Fh(\241)332 b Fj(xy)379 b Fk(+)332 b Fj(y)40954 45801 y Fi(2)41967 46283 y Fk(to)487 b(the)f(line)i Fj(y)508 b Fk(=)460 b Fh(\241)p Fj(x)800 47888 y Fk(and)403 b(suc)-36 b(h)403 b(that)g Fj(x)234 b Fk(+)f Fj(y)417 b(>)368 b Fk(0)404 b(then)f Fj(\265)36 b Fk(\()p Fj(n)p Fk(\))369 b Fh(\267)g Fj(x)234 b Fk(+)g Fj(y)48 b Fk(.)568 b(Another)402 b(problem)i(is)f(to) h(c)-36 b(haracterize)404 b(those)f Fj(n)h Fk(suc)-36 b(h)800 49493 y(that)433 b Fj(\265)36 b Fk(\()p Fj(n)p Fk(\))369 b(=)g Fj(x)295 b Fk(+)g Fj(y)48 b Fk(,)434 b(this)f(minim)-36 b(um)433 b(p)36 b(ositiv)-36 b(e)434 b(v)-72 b(alue.)2751 51098 y(Note)578 b(also)h(that)e(a)h(function)g (lik)-36 b(e)579 b Fj(\265)614 b Fk(can)578 b(b)36 b(e)577 b(de\257ned)g(for)h(forms)g(with)g(appropriate)g(symmetry)800 52703 y(prop)36 b(erties,)433 b(e.g.)580 b Fj(f)142 b Fk(\()p Fj(x;)221 b(y)48 b Fk(\))369 b(=)f Fj(x)16090 52221 y Fc(k)16955 52703 y Fk(+)294 b Fj(y)18943 52221 y Fc(k)19946 52703 y Fk(for)434 b Fj(k)478 b Fk(o)36 b(dd.)2751 55636 y Fg(Ac)-42 b(kno)g(wledgemen)g(ts:)551 b Fk(The)371 b(supp)36 b(ort)370 b(of)i(the)e(Departmen)-36 b(t)370 b(of)i(Mathematics)f(of)h(the)e(Univ)-36 b(ersit)g(y)800 57241 y(of)307 b(W)-108 b(aik)-72 b(ato,)333 b(the)306 b(discussions)g(held)g(with)g(Ian)g(Ha)-36 b(wthorn)306 b(and)g(the)g(v)-72 b(aluable)306 b(con)-36 b(tributions,)331 b(esp)36 b(ecially)800 58847 y(to)434 b(shortening)f(the)g(pro)36 b(of)434 b(of)g(Theorem)g(3,)g(of)g(an)g(anon)-36 b(ymous)434 b(referee)f(are)h(w)-36 b(armly)435 b(ac)-36 b(kno)g(wledged.)800 63284 y Fl(References)p Black 800 66204 a Fk([1])p Black 651 w(T.)435 b(M.)e(Ap)36 b(ostol,)435 b Ff(Intr)-66 b(o)g(duction)462 b(to)j(A)-33 b(nalytic)463 b(Numb)-66 b(er)464 b(The)-66 b(ory)p Fk(,)433 b(Springer)g(V)-108 b(erlag,)434 b(1976.)p Black 800 68916 a([2])p Black 651 w(G.)343 b(H.)g(Hardy)h(and)e(E.)h(M.)g(W)-108 b(righ)-36 b(t,)361 b Ff(A)-33 b(n)381 b(Intr)-66 b(o)g(duction)379 b(to)j(the)f(The)-66 b(ory)381 b(of)h(Numb)-66 b(ers)p Fk(,)360 b(Fifth)343 b(Edition,)2823 70521 y(Oxford,)434 b(1979.)p Black 26475 74617 a(6)p Black eop %%Page: 7 7 7 6 bop Black 0 TeXcolorgray Black Black Black 800 1424 a Fk([3])p Black 651 w(K.)434 b(L.)g(Hua,)g Ff(Intr)-66 b(o)g(duction)462 b(to)j(Numb)-66 b(er)464 b(The)-66 b(ory)p Fk(,)433 b(Springer-V)-108 b(erlag,)433 b(1982.)p Black 800 4136 a([4])p Black 651 w(S.)419 b(Lang,)j(Old)c(and)h(new)g (conjectured)f(diophan)-36 b(tine)418 b(inequalities,)423 b Ff(Bul)66 b(l.)452 b(A)-33 b(mer.)450 b(Math.)h(So)-66 b(c.)p Fk(,)420 b Fg(23)p Fk(,)2823 5741 y(\(1990\),)435 b(37{75.)p Black 800 8453 a([5])p Black 651 w(L.)f(J.)g(Mordell,)g Ff(Diophantine)463 b(Equations)p Fk(,)434 b(Academic)g(Press,)g(1969.)p Black 800 11165 a([6])p Black 651 w(M.)408 b(B.)f(Nathanson,)412 b Ff(A)-66 b(dditive)439 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b(ct)466 b(Classi\257c)-66 b(ation)p Fk(:)584 b(Primary)438 b(11A07;)i(Secondary)e(11B13,)h(11B50,)h(11D25,)800 29511 y(11D79,)c(11P05.)800 31116 y Ff(Keywor)-66 b(ds:)597 b(sum)466 b(of)e(two)i(cub)-66 b(es;)464 b(diophantine)f(e)-66 b(quation)p 800 32747 V 800 35072 a Fk(\(Concerned)433 b(with)h(sequence)p 0 1 0 0 TeXcolorcmyk 433 w(A045980)p 16090 35285 4878 54 v [[217 401 261 413] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/ ~njas/sequences/eisA.cgi?Anum=A045980) pdfm Black 2 w(.\))p 800 36777 52000 45 v 800 39828 a(Receiv)-36 b(ed)463 b(Marc)-36 b(h)463 b(31)g(2003;)479 b(revised)463 b(v)-36 b(ersion)463 b(receiv)-36 b(ed)463 b(Decem)-36 b(b)36 b(er)462 b(18)i(2003.)666 b(Published)462 b(in)h Ff(Journal)800 41433 y(of)i(Inte)-66 b(ger)462 b(Se)-66 b(quenc)g(es)p Fk(,)432 b(Jan)-36 b(uary)434 b(14)g(2004.)p 800 43064 V 800 45314 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 [[133 309 338 321] [1 1 1 [3 3]] [0 0 1]] (http://www.math.uwaterloo.ca/JIS/) pdfm Black(.)p Black 26475 74617 a(7)p Black eop %%Trailer end end