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Fk(b)3463 33559 y Fs(m)p Fq(\000)p Fp(1)6001 33360 y Fm(con)-36 b(tributes)446 b(and)i Fk(b)15875 33559 y Fs(m)17156 33360 y Fj(\024)393 b Fk(b)19135 33559 y Fs(m)p Fq(\000)p Fp(1)21225 33360 y Fm(.)621 b(Th)-36 b(us)447 b(whether)g(or)h(not)g(a) g(digit)g(con)-36 b(tributes)447 b(is)h(completely)800 34965 y(determined)432 b(b)-36 b(y)434 b(the)f(b)36 b(eha)-36 b(vior)434 b(of)g(the)f(digit)h(to)g(its)g(immediate)f(righ)-36 b(t.)800 35223 y SDict begin H.S end 800 35223 a 800 35223 a SDict begin 14.5 H.A end 800 35223 a 800 35223 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.2) cvn H.B /DEST pdfmark end 800 35223 a Black 2454 x Fl(Example)488 b(2.)p Black 2271 w Fm(Here)424 b(is)g(an)g(example)g(of)h(a)f(string,)j Fk(\015)440 b Fm(=)369 b(201120100121,)430 b(with)424 b(drops)g(indicated)f(b)-36 b(y)800 39282 y(arro)g(ws)434 b(and)f(con)-36 b(tributing)433 b(digits)h(underlined.)1464 43204 y Fk(\015)441 b Fm(:)2879 b(2)1439 b(0)p 7909 43416 651 54 v 1439 w(1)p 9998 43416 V 1440 w(1)p 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42081 V 29522 42081 V 29489 42081 V 29456 42081 V 29423 42082 V 29391 42084 V 29358 42086 V 29326 42088 V 29294 42091 V 29261 42094 V 29229 42098 V 29197 42101 V 29165 42106 V 29133 42111 V 29102 42116 V 29070 42121 V 29038 42127 V 29007 42133 V 28976 42140 V 28944 42146 V 28913 42154 V 28882 42161 V 28851 42169 V 28821 42177 V 33120 43204 a Fm(1)p Black Black 2751 47189 a(Note)344 b(that)g(w)-36 b(e)344 b(ha)-36 b(v)g(e)345 b(not)e(indicated)h(drops)g(within)g(con)-36 b(tributing)343 b(substrings)g(since)h(the)g(imp)36 b(ortan)-36 b(t)800 48794 y(c)g(haracteristic)434 b(is)g(whether)f(a)h(digit)f(con) -36 b(tributes.)800 49052 y SDict begin H.S end 800 49052 a 800 49052 a SDict begin 14.5 H.A end 800 49052 a 800 49052 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.3) cvn H.B /DEST pdfmark end 800 49052 a Black 2454 x Fl(De\014nition)571 b(3.)p Black 2381 w Fm(F)-108 b(or)496 b(a)f(giv)-36 b(en)497 b(base-)p Fk(a)d Fm(string)i Fk(\014)74 b(;)496 b Fm(de\014ne)e Fk(z)59 b Fm(\()p 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begin HPSdict begin 4 3 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.4) cvn H.B /DEST pdfmark end 800 -4000 a Black Black Black 1328 x Fi(Pr)-66 b(o)g(of.)p Black 649 w Fm(If)352 b Fk(j)444 b Fm(=)368 b Fk(k)396 b Fm(or)352 b Fk(j)443 b Fm(=)369 b(0)351 b(the)g(result)g(is)g(clear.) 552 b(Supp)36 b(ose)350 b Fk(k)414 b(>)368 b(j)444 b(>)369 b Fm(0.)551 b(The)351 b(assumption)g(on)g Fk(b)47981 -2473 y Fs(j)48819 -2672 y Fm(implies)800 -1067 y(that)494 b(either)g Fk(b)7978 -868 y Fs(j)8960 -1067 y Fm(do)36 b(es)494 b(not)g(con)-36 b(tribute)494 b(to)g Fk(\014)74 b Fm(,)510 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Fm(and)433 b Fk(b)36056 3948 y Fs(j)36912 3749 y Fk(<)368 b(b)38845 3948 y Fs(j)51 b Fq(\000)p Fp(1)40535 3749 y Fm(.)2751 5354 y(It)586 b(su\016ces)g(to)g(pro)-36 b(v)g(e)586 b(that)g(eac)-36 b(h)586 b(digit)g(of)h Fk(\014)660 b Fm(that)586 b(con)-36 b(tributes)585 b(to)h Fk(\014)660 b Fm(also)587 b(con)-36 b(tributes)585 b(to)i(the)800 6959 y(sum)409 b Fk(z)59 b Fm(\()p Fk(b)5251 7158 y Fs(k)6042 6959 y Fj(\001)221 b(\001)g(\001)h Fk(b)8366 7158 y Fs(j)8853 6959 y Fm(\))246 b(+)h Fk(z)59 b Fm(\()p Fk(b)12586 7158 y Fs(j)13294 6959 y Fj(\001)221 b(\001)g(\001)h Fk(b)15618 7158 y Fp(0)16144 6959 y Fm(\))410 b(once)f(and)h(only)g(once.)571 b(In)409 b(case)i(\(i\),)j Fk(b)35827 7158 y Fs(j)36724 6959 y Fm(con)-36 b(tributes)409 b(to)h Fk(b)45583 7158 y Fs(j)46069 6959 y Fk(b)46622 7158 y Fs(j)51 b Fq(\000)p Fp(1)48533 6959 y Fk(:)221 b(:)g(:)i(b)50834 7158 y Fp(1)51360 6959 y Fk(b)51913 7158 y Fp(0)52439 6959 y Fm(;)800 8564 y(ho)-36 b(w)g(ev)g(er,)645 b(it)602 b(cannot)g(con)-36 b(tribute)601 b(to)h Fk(b)20989 8763 y Fs(k)21780 8564 y Fj(\001)221 b(\001)g(\001)h Fk(b)24104 8763 y Fs(j)51 b Fp(+1)25793 8564 y Fk(b)26346 8763 y Fs(j)27435 8564 y Fm(as)602 b(it)h(cannot)f(follo)-36 b(w)604 b(a)e(drop.)1084 b(Th)-36 b(us)601 b(the)h(digit)800 10169 y Fk(b)1353 10368 y Fs(j)2419 10169 y Fm(con)-36 b(tributes)577 b(once)i(to)g(the)f (sum.)1014 b(The)578 b(digits)h(in)g(the)f(substring)g Fk(b)37158 10368 y Fs(j)37645 10169 y Fk(b)38198 10368 y Fs(j)51 b Fq(\000)p Fp(1)40109 10169 y Fk(:)221 b(:)g(:)i(b)42410 10368 y Fp(1)43514 10169 y Fm(are)579 b(con)-36 b(tributing)800 11774 y(if)484 b(and)g(only)g(if)g(they)g(con)-36 b(tribute)482 b(to)i Fk(\014)74 b Fm(.)729 b(Since)483 b Fk(b)25582 11973 y Fs(j)51 b Fp(+1)27755 11774 y Fm(con)-36 b(tributes)482 b(to)i Fk(\014)74 b Fm(,)496 b(the)483 b(digits)h(of)h(the)e(substring) 800 13379 y Fk(b)1353 13578 y Fs(k)2143 13379 y Fj(\001)221 b(\001)g(\001)h Fk(b)4467 13578 y Fs(j)51 b Fp(+1)6506 13379 y Fm(con)-36 b(tribute)349 b(to)h Fk(b)14732 13578 y Fs(k)15522 13379 y Fj(\001)221 b(\001)g(\001)h Fk(b)17846 13578 y Fs(j)51 b Fp(+1)19535 13379 y Fk(b)20088 13578 y Fs(j)20925 13379 y Fm(if)350 b(and)f(only)i(if)f(they)g(con)-36 b(tribute)348 b(to)i Fk(\014)74 b Fm(.)550 b(The)350 b(pro)36 b(of)350 b(for)h(case)f(\(ii\))800 14984 y(is)328 b(analogous)h(except)e(that)g Fk(b)15142 15183 y Fs(j)51 b Fp(+1)17159 14984 y Fm(do)36 b(es)328 b(not)f(con)-36 b(tribute)327 b(to)g Fk(\014)402 b Fm(and)327 b(do)36 b(es)328 b(not)f(con)-36 b(tribute)327 b(to)g Fk(b)46595 15183 y Fs(k)47385 14984 y Fj(\001)221 b(\001)g(\001)i Fk(b)49710 15183 y Fs(j)51 b Fp(+1)51399 14984 y Fk(b)51952 15183 y Fs(j)52439 14984 y Fm(,)800 16589 y(but)433 b(all)h(con)-36 b(tributions)433 b(from)h(the)f(left)h(of)g Fk(b)22532 16788 y Fs(j)51 b Fp(+1)24655 16589 y Fm(are)434 b(the)f(same)h(in)f(b) 36 b(oth)433 b(strings.)p 51860 16589 45 878 v 51905 15756 781 45 v 51905 16589 V 52684 16589 45 878 v 2751 19191 a(Giv)-36 b(en)435 b(strings)h Fk(\013)444 b Fm(and)435 b Fk(\014)74 b Fm(,)436 b(their)f(concatenation)h(will)h(b)36 b(e)435 b(denoted)g(b)-36 b(y)435 b Fk(\013)8 b(\014)74 b Fm(.)585 b(Lastly)-108 b(,)436 b(w)-36 b(e)436 b(de\014ne)f(the)800 20796 y(\\star")f(notation.)800 20796 y SDict begin H.S end 800 20796 a 800 20796 a SDict begin 14.5 H.A end 800 20796 a 800 20796 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.5) cvn H.B /DEST pdfmark end 800 20796 a Black 2712 x Fl(De\014nition)500 b(5.)p Black 2288 w Fm(F)-108 b(or)433 b(nonempt)-36 b(y)433 b(strings)g Fk(\013)442 b Fm(and)433 b Fk(\014)74 b(;)434 b Fm(w)-36 b(e)434 b(de\014ne)f(the)g(relation)h Fj(\003)f Fm(b)-36 b(y)21415 26441 y Fk(\013)304 b Fj(\003)295 b Fk(\014)443 b Fj(,)369 b(j)p Fk(\013)8 b Fj(j)370 b Fk(<)e(z)59 b Fm(\()p Fk(\014)74 b Fm(\))800 28102 y SDict begin H.S end 800 28102 a 800 28102 a SDict begin 14.5 H.A end 800 28102 a 800 28102 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.6) cvn H.B /DEST pdfmark end 800 28102 a Black 1273 x Fl(Example)474 b(6.)p Black 2249 w Fm(Consider)412 b(the)g(ternary)f(string)h(1211111201.)575 b(If)412 b Fk(\013)378 b Fm(=)368 b(12)413 b(and)e Fk(\014)443 b Fm(=)369 b(11111201,)419 b(then)800 30980 y Fk(z)59 b Fm(\()p Fk(\014)74 b Fm(\))568 b(=)g(6)551 b(and)f Fj(j)p Fk(\013)8 b Fj(j)569 b Fm(=)f(5.)930 b(In)551 b(this)f(case,)581 b(12)375 b Fj(\003)g Fm(11111201)554 b(holds;)609 b(note)550 b(that)h(len\()p Fk(\014)74 b Fm(\))567 b(=)h(8)h(=)f(7)375 b(+)f(1)p Fk(:)800 32585 y Fm(Ho)-36 b(w)g(ev)g(er,)435 b(121)296 b Fj(\003)f Fm(1111201)436 b(do)36 b(es)434 b(not)f(hold)g(as)h(16)370 b Fj(6)p Fk(<)f Fm(5.)2751 35297 y(The)467 b(follo)-36 b(wing)469 b(theorem)d(is)h(one)g(of)h(the) e(main)h(results)f(of)i(this)e(pap)36 b(er.)678 b(In)467 b(Section)f(3)h(w)-36 b(e)467 b(giv)-36 b(e)468 b(an)800 36902 y(algorithmic)435 b(description)e(of)h(this)f(theorem)g(and)h (brie\015y)f(discuss)g(its)h(complexit)-36 b(y)-108 b(.)800 37160 y SDict begin H.S end 800 37160 a 800 37160 a SDict begin 14.5 H.A end 800 37160 a 800 37160 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.7) cvn H.B /DEST pdfmark end 800 37160 a Black 2454 x Fl(Theorem)562 b(7.)p Black 582 w Fi(L)-66 b(et)514 b Fk(n)462 b Fj(2)f Fh(N)p Fi(.)748 b(Then)515 b(the)g(F)-100 b(r)-66 b(ob)g(enius)514 b(level)g(of)h Fk(n;)h(f)34312 39813 y Fs(a)34867 39614 y Fm(\()p Fk(n)p Fm(\))p Fk(;)g Fi(is)f(the)f(smal)66 b(lest)516 b Fk(k)560 b Fi(for)515 b(which)800 41219 y(we)465 b(c)-66 b(an)464 b(write)h Fk(n)369 b Fm(=)g Fj(j)p Fk(\013)8 b(\014)74 b Fj(j)465 b Fi(with)g(len)o Fm(\()p Fk(\014)74 b Fm(\))369 b(=)f Fk(k)341 b Fm(+)295 b(1)465 b Fi(and)g Fk(\013)304 b Fj(\003)295 b Fk(\014)74 b Fi(.)2751 43931 y Fm(The)541 b(previous)g(example)h(sho)-36 b(ws)541 b(that)g(the)f(F)-108 b(rob)36 b(enius)540 b(lev)-36 b(el)543 b(of)e Fk(n)553 b Fm(=)e(36091)j(=)d Fj(j)p Fm(1211111201)p Fj(j)545 b Fm(is)800 45536 y Fk(f)1441 45735 y Fp(3)1967 45536 y Fm(\(36091\))370 b(=)f(7)p Fk(:)2751 47142 y Fm(Theorem)p 0 .5 0 TeXcolorrgb 8326 47142 a SDict begin H.S end 8326 47142 a 0 .5 0 TeXcolorrgb Fm(7)p 0 .5 0 TeXcolorrgb 8977 46286 a SDict begin H.R end 8977 46286 a 8977 47142 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.7) cvn H.B /ANN pdfmark end 8977 47142 a Black 517 w Fm(reduces)517 b(to)g(the)g(results)g (of)h(Johnson)f(and)g(Merzel)g(when)g Fk(a)511 b Fm(=)h(2.)829 b(In)518 b(the)e(Johnson)800 48747 y(and)405 b(Merzel)h(pap)36 b(er)405 b Fk(z)59 b Fm(\()p Fk(\014)74 b Fm(\))405 b(is)h(de\014ned)e (as)i(the)f(n)-36 b(um)g(b)36 b(er)404 b(of)i(non-trailing)g(zeros)g (in)f Fk(\014)480 b Fm(and)405 b(our)g(de\014nition)800 50352 y(of)434 b Fk(z)59 b Fm(\()p Fk(\014)74 b Fm(\))434 b(reduces)f(to)g(the)g(Johnson-Merzel)g(de\014nition)g(in)g(the)h(case) g Fk(a)368 b Fm(=)h(2.)800 52686 y SDict begin H.S end 800 52686 a 800 52686 a SDict begin 14.5 H.A end 800 52686 a 800 52686 a SDict begin [ /View [/XYZ H.V] /Dest (section.2) cvn H.B /DEST pdfmark end 800 52686 a 2103 x Fn(2)2152 b(Pro)60 b(of)716 b(of)h(Theorem)g(1)800 57709 y Fm(The)519 b(pro)36 b(of)519 b(of)h(Theorem)p 0 .5 0 TeXcolorrgb 14184 57709 a SDict begin H.S end 14184 57709 a 0 .5 0 TeXcolorrgb Fm(7)p 0 .5 0 TeXcolorrgb 14834 56853 a SDict begin H.R end 14834 56853 a 14834 57709 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.7) cvn H.B /ANN pdfmark end 14834 57709 a Black 519 w Fm(is)f(organized)h(as)f(follo)-36 b(ws:)751 b(Lemma)p 0 .5 0 TeXcolorrgb 34041 57709 a SDict begin H.S end 34041 57709 a 0 .5 0 TeXcolorrgb Fm(8)p 0 .5 0 TeXcolorrgb 34692 56853 a SDict begin H.R end 34692 56853 a 34692 57709 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.8) cvn H.B /ANN pdfmark end 34692 57709 a Black 519 w Fm(giv)-36 b(es)520 b(a)f(particularly)g(useful)g(w)-36 b(a)g(y)800 59314 y(to)448 b(represen)-36 b(t)446 b(in)-36 b(tegers)447 b(that)g(are)h(not)f(in)h Fk(F)22565 59513 y Fs(a)23120 59314 y Fm(\()p Fk(k)45 b Fm(\).)619 b(Lemmas)p 0 .5 0 TeXcolorrgb 31003 59314 a SDict begin H.S end 31003 59314 a 0 .5 0 TeXcolorrgb Fm(9)p 0 .5 0 TeXcolorrgb 31653 58458 a SDict begin H.R end 31653 58458 a 31653 59314 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.9) cvn H.B /ANN pdfmark end 31653 59314 a Black 448 w Fm(and)p 0 .5 0 TeXcolorrgb 34644 59314 a SDict begin H.S end 34644 59314 a 0 .5 0 TeXcolorrgb Fm(10)p 0 .5 0 TeXcolorrgb 35944 58458 a SDict begin H.R end 35944 58458 a 35944 59314 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.10) cvn H.B /ANN pdfmark end 35944 59314 a Black 448 w Fm(sho)-36 b(w)447 b(that)g(the)g(sets)h Fk(F)48114 59513 y Fs(a)48669 59314 y Fm(\()p Fk(k)45 b Fm(\))447 b(can)800 60920 y(b)36 b(e)502 b(describ)36 b(ed)501 b(recursiv)-36 b(ely)-108 b(.)784 b(Lemmas)p 0 .5 0 TeXcolorrgb 20760 60920 a SDict begin H.S end 20760 60920 a 0 .5 0 TeXcolorrgb Fm(13)p 0 .5 0 TeXcolorrgb 22060 60063 a SDict begin H.R end 22060 60063 a 22060 60920 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.13) cvn H.B /ANN pdfmark end 22060 60920 a Black 502 w Fm(and)p 0 .5 0 TeXcolorrgb 25160 60920 a SDict begin H.S end 25160 60920 a 0 .5 0 TeXcolorrgb Fm(14)p 0 .5 0 TeXcolorrgb 26460 60063 a SDict begin H.R end 26460 60063 a 26460 60920 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.14) cvn H.B /ANN pdfmark end 26460 60920 a Black 502 w Fm(set)502 b(up)f(tec)-36 b(hnical)502 b(details)h(to)f(assist)g(in)g(the)f(pro)36 b(of)800 62525 y(of)444 b(Theorem)p 0 .5 0 TeXcolorrgb 7791 62525 a SDict begin H.S end 7791 62525 a 0 .5 0 TeXcolorrgb Fm(15)p 0 .5 0 TeXcolorrgb 9092 61669 a SDict begin H.R end 9092 61669 a 9092 62525 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.15) cvn H.B /ANN pdfmark end 9092 62525 a Black 443 w Fm(b)-36 b(y)442 b(induction.)605 b(Theorem)p 0 .5 0 TeXcolorrgb 23165 62525 a SDict begin H.S end 23165 62525 a 0 .5 0 TeXcolorrgb Fm(7)p 0 .5 0 TeXcolorrgb 23816 61669 a SDict begin H.R end 23816 61669 a 23816 62525 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.7) cvn H.B /ANN pdfmark end 23816 62525 a Black 442 w Fm(is)443 b(then)f(a)h(corollary)i(of)e(Theorem)p 0 .5 0 TeXcolorrgb 42024 62525 a SDict begin H.S end 42024 62525 a 0 .5 0 TeXcolorrgb Fm(15)p 0 .5 0 TeXcolorrgb 43325 61669 a SDict begin H.R end 43325 61669 a 43325 62525 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.15) cvn H.B /ANN pdfmark end 43325 62525 a Black Fm(.)606 b(Along)443 b(the)f(w)-36 b(a)g(y)-108 b(,)800 64130 y(Theorem)p 0 .5 0 TeXcolorrgb 6279 64130 a SDict begin H.S end 6279 64130 a 0 .5 0 TeXcolorrgb Fm(11)p 0 .5 0 TeXcolorrgb 7579 63274 a SDict begin H.R end 7579 63274 a 7579 64130 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.11) cvn H.B /ANN pdfmark end 7579 64130 a Black 421 w Fm(giv)-36 b(es)421 b(an)f(explicit)i(form)-36 b(ula)420 b(for)h(the)f(F)-108 b(rob)36 b(enius)419 b(n)-36 b(um)g(b)36 b(er)419 b(of)i Fk(G)39685 64329 y Fs(a)40240 64130 y Fm(\()p Fk(k)45 b Fm(\))420 b(whic)-36 b(h)421 b(corresp)36 b(onds)800 65735 y(to)434 b(the)f(w)-36 b(ell-kno)g(wn)434 b(results)f(of)i(Nijenh)-36 b(uis)433 b(and)g(Wilf)i([)p 0 .5 0 TeXcolorrgb 28532 65735 a SDict begin H.S end 28532 65735 a 0 .5 0 TeXcolorrgb Fm(8)p 0 .5 0 TeXcolorrgb 29183 64879 a SDict begin H.R end 29183 64879 a 29183 65735 a SDict begin [ /Color [0 1 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (cite.NW) cvn H.B /ANN pdfmark end 29183 65735 a Black Fm(].)800 66067 y SDict begin H.S end 800 66067 a 800 66067 a SDict begin 14.5 H.A end 800 66067 a 800 66067 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.8) cvn H.B /DEST pdfmark end 800 66067 a Black 26475 70521 a Fm(4)p Black eop end end %%Page: 5 5 TeXDict begin HPSdict begin 5 4 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.5) cvn H.B /DEST pdfmark end 800 -4000 a Black Black Black 1328 x Fl(Lemma)551 b(8.)p Black 577 w Fi(If)506 b Fk(n)446 b Fj(62)g Fk(F)12688 -2473 y Fs(a)13243 -2672 y Fm(\()p Fk(k)45 b Fm(\))p Fi(,)516 b(then)506 b(ther)-66 b(e)505 b(exist)h Fk(c)25725 -2473 y Fp(1)26696 -2672 y Fj(2)446 b Fh(Z)28914 -2473 y Fq(\025)p Fp(0)30171 -2672 y Fi(,)517 b Fk(c)31647 -2473 y Fp(2)32172 -2672 y Fk(;)221 b(:)g(:)g(:)j(;)d(c)35645 -2473 y Fs(k)24 b Fp(+1)37863 -2672 y Fj(2)445 b(f)p Fm(0)p Fk(;)221 b(:)g(:)g(:)j(;)d(a)327 b Fj(\000)f Fm(1)p Fj(g)507 b Fi(such)g(that)800 -1067 y Fk(n)369 b Fm(=)g Fk(c)3886 -868 y Fp(1)4412 -1067 y Fk(a)5095 -1549 y Fs(k)5958 -1067 y Fm(+)295 b Fk(c)7825 -868 y Fp(2)8351 -1067 y Fm(\()p Fk(a)9540 -1549 y Fs(k)10403 -1067 y Fm(+)g Fk(a)12393 -1549 y Fs(k)24 b Fq(\000)p Fp(1)14164 -1067 y Fm(\))295 b(+)g Fj(\001)221 b(\001)g(\001)296 b Fm(+)f Fk(c)19984 -868 y Fs(k)24 b Fp(+1)21755 -1067 y Fm(\()p Fk(a)22944 -1549 y Fs(k)23807 -1067 y Fm(+)295 b Fk(a)25797 -1549 y Fs(k)24 b Fq(\000)p Fp(1)27863 -1067 y Fm(+)295 b Fj(\001)221 b(\001)g(\001)296 b Fm(+)f Fk(a)g Fm(+)g(1\))p Fk(:)p Black 800 1405 a Fi(Pr)-66 b(o)g(of.)p Black 649 w Fm(Supp)36 b(ose)433 b Fk(n)369 b Fj(62)g Fk(F)13216 1604 y Fs(a)13771 1405 y Fm(\()p Fk(k)45 b Fm(\).)578 b(Then)433 b(there)g(exist)h(co)36 b(e\016cien)-36 b(ts)434 b Fk(w)33754 1604 y Fs(i)34129 1405 y Fm(,)g(1)369 b Fj(\024)h Fk(i)e Fj(\024)h Fk(k)341 b Fm(+)295 b(1,)434 b(suc)-36 b(h)432 b(that)8041 4059 y Fk(n)369 b Fm(=)g Fk(w)11497 4258 y Fp(1)12023 4059 y Fk(a)12706 3510 y Fs(k)13569 4059 y Fm(+)295 b Fk(w)15806 4258 y Fp(2)16332 4059 y Fm(\()p Fk(a)17521 3510 y Fs(k)18384 4059 y Fm(+)g Fk(a)20374 3510 y Fs(k)24 b Fq(\000)p Fp(1)22145 4059 y Fm(\))295 b(+)g Fj(\001)221 b(\001)g(\001)296 b Fm(+)f Fk(w)28335 4258 y Fs(k)24 b Fp(+1)30106 4059 y Fm(\()p Fk(a)31295 3510 y Fs(k)32158 4059 y Fm(+)295 b Fk(a)34148 3510 y Fs(k)24 b Fq(\000)p Fp(1)36214 4059 y Fm(+)295 b Fj(\001)221 b(\001)g(\001)296 b Fm(+)f Fk(a)41356 3510 y Fp(1)42177 4059 y Fm(+)f Fk(a)44166 3510 y Fp(0)44692 4059 y Fm(\))p Fk(:)800 6712 y Fm(If)467 b(the)f(co)36 b(e\016cien)-36 b(ts)466 b Fk(w)11951 6911 y Fs(i)12793 6712 y Fm(satisfy)i(the)e(conditions)g(of)h(the)f(lemma)g(then)g(w)-36 b(e)466 b(are)h(done;)483 b(otherwise,)475 b(let)466 b Fk(j)800 8317 y Fm(b)36 b(e)426 b(the)g(largest)h(subscript)f(for)h (whic)-36 b(h)426 b Fk(w)21183 8516 y Fs(j)22039 8317 y Fj(\025)369 b Fk(a)p Fm(.)576 b(Using)426 b(the)g(division)i (algorithm,)h(write)e Fk(w)46276 8516 y Fs(j)47131 8317 y Fm(=)369 b Fk(aq)328 b Fm(+)280 b Fk(c)51952 8516 y Fs(j)52439 8317 y Fk(;)800 9922 y Fm(where)434 b(0)369 b Fj(\024)g Fk(c)7539 10121 y Fs(j)8395 9922 y Fk(<)f(a:)434 b Fm(Substitution)e(giv)-36 b(es)6890 14180 y Fk(w)7820 14379 y Fs(j)8307 14180 y Fm(\()p Fk(a)9496 13632 y Fs(k)10360 14180 y Fm(+)294 b Fk(a)12349 13632 y Fs(k)24 b Fq(\000)p Fp(1)14416 14180 y Fm(+)294 b Fj(\001)221 b(\001)g(\001)296 b Fm(+)f Fk(a)19557 13632 y Fs(k)24 b Fq(\000)p Fs(j)51 b Fp(+1)22491 14180 y Fm(\))1107 b(=)g(\()p Fk(aq)342 b Fm(+)295 b Fk(c)30198 14379 y Fs(j)30684 14180 y Fm(\)\()p Fk(a)32379 13632 y Fs(k)33243 14180 y Fm(+)g Fk(a)35233 13632 y Fs(k)24 b Fq(\000)p Fp(1)37299 14180 y Fm(+)295 b Fj(\001)221 b(\001)g(\001)296 b Fm(+)e Fk(a)42440 13632 y Fs(k)24 b Fq(\000)p Fs(j)51 b Fp(+1)45375 14180 y Fm(\))24104 16118 y(=)1107 b Fk(aq)48 b Fm(\()p Fk(a)28720 15569 y Fs(k)29583 16118 y Fm(+)294 b Fk(a)31572 15569 y Fs(k)24 b Fq(\000)p Fp(1)33638 16118 y Fm(+)295 b Fj(\001)221 b(\001)g(\001)296 b Fm(+)f Fk(a)38780 15569 y Fs(k)24 b Fq(\000)p Fs(j)51 b Fp(+1)41714 16118 y Fm(\))26223 18055 y(+)p Fk(c)27795 18254 y Fs(j)28281 18055 y Fm(\()p Fk(a)29470 17506 y Fs(k)30334 18055 y Fm(+)295 b Fj(\001)221 b(\001)g(\001)295 b Fm(+)g Fk(a)35475 17506 y Fs(k)24 b Fq(\000)p Fs(j)51 b Fp(+1)38410 18055 y Fm(\))24104 19992 y(=)1107 b Fk(aq)48 b Fm(\()p Fk(a)28720 19443 y Fs(k)29287 19992 y Fm(\))295 b(+)g Fk(q)48 b Fm(\()p Fk(a)33209 19443 y Fs(k)34072 19992 y Fm(+)295 b Fk(a)36062 19443 y Fs(k)24 b Fq(\000)p Fp(1)38128 19992 y Fm(+)295 b Fj(\001)221 b(\001)g(\001)296 b Fm(+)f Fk(a)43270 19443 y Fs(k)24 b Fq(\000)p Fs(j)51 b Fp(+2)46204 19992 y Fm(\))26223 21929 y(+)p Fk(c)27795 22128 y Fs(j)28281 21929 y Fm(\()p Fk(a)29470 21381 y Fs(k)30334 21929 y Fm(+)295 b Fj(\001)221 b(\001)g(\001)295 b Fm(+)g Fk(a)35475 21381 y Fs(k)24 b Fq(\000)p Fs(j)51 b Fp(+1)38410 21929 y Fm(\))p Fk(:)800 24582 y Fm(Next,)434 b(de\014ne)f Fk(c)8658 24781 y Fs(m)9914 24582 y Fm(:=)369 b Fk(w)12586 24781 y Fs(m)13907 24582 y Fm(for)434 b(all)h Fk(j)443 b(<)369 b(m)g Fj(\024)g Fk(k)340 b Fm(+)295 b(1.)579 b(Th)-36 b(us)433 b Fk(n)h Fm(can)f(b)36 b(e)433 b(written)h(as)5156 27236 y Fk(n)1107 b Fm(=)g(\()p Fk(w)10594 27435 y Fp(1)11414 27236 y Fm(+)295 b Fk(aq)48 b Fm(\))p Fk(a)15218 26687 y Fs(k)16081 27236 y Fm(+)295 b Fk(w)18318 27435 y Fp(2)18843 27236 y Fm(\()p Fk(a)20032 26687 y Fs(k)20896 27236 y Fm(+)g Fk(a)22886 26687 y Fs(k)24 b Fq(\000)p Fp(1)24657 27236 y Fm(\))295 b(+)f Fj(\001)221 b(\001)g(\001)296 b Fm(+)f Fk(w)30846 27435 y Fs(j)51 b Fq(\000)p Fp(2)32535 27236 y Fm(\()p Fk(a)33724 26687 y Fs(k)34588 27236 y Fm(+)295 b Fk(a)36578 26687 y Fs(k)24 b Fq(\000)p Fp(1)38644 27236 y Fm(+)294 b Fj(\001)221 b(\001)g(\001)296 b Fm(+)f Fk(a)43785 26687 y Fs(k)24 b Fq(\000)p Fs(j)51 b Fp(+3)46720 27236 y Fm(\))9158 29173 y(+\()p Fk(w)11606 29372 y Fs(j)g Fq(\000)p Fp(1)13589 29173 y Fm(+)295 b Fk(q)48 b Fm(\)\()p Fk(a)17216 28624 y Fs(k)18079 29173 y Fm(+)295 b Fk(a)20069 28624 y Fs(k)24 b Fq(\000)p Fp(1)22135 29173 y Fm(+)295 b Fj(\001)221 b(\001)g(\001)296 b Fm(+)e Fk(a)27276 28624 y Fs(k)24 b Fq(\000)p Fs(j)51 b Fp(+2)30211 29173 y Fm(\))295 b(+)g Fk(c)32879 29372 y Fs(j)33365 29173 y Fm(\()p Fk(a)34554 28624 y Fs(k)35418 29173 y Fm(+)g Fk(a)37408 28624 y Fs(k)24 b Fq(\000)p Fp(1)39474 29173 y Fm(+)295 b Fj(\001)221 b(\001)g(\001)296 b Fm(+)e Fk(a)44615 28624 y Fs(k)24 b Fq(\000)p Fs(j)51 b Fp(+1)47550 29173 y Fm(\))9158 31110 y(+)p Fk(c)10730 31309 y Fs(j)g Fp(+1)12418 31110 y Fm(\()p Fk(a)13607 30562 y Fs(k)14471 31110 y Fm(+)295 b Fk(a)16461 30562 y Fs(k)24 b Fq(\000)p Fp(1)18527 31110 y Fm(+)295 b Fj(\001)221 b(\001)g(\001)296 b Fm(+)e Fk(a)23668 30562 y Fs(k)24 b Fq(\000)p Fs(j)25400 31110 y Fm(\))295 b(+)g Fj(\001)221 b(\001)g(\001)296 b Fm(+)f Fk(c)31220 31309 y Fs(k)24 b Fp(+1)32991 31110 y Fm(\()p Fk(a)34180 30562 y Fs(k)35044 31110 y Fm(+)294 b Fk(a)37033 30562 y Fs(k)24 b Fq(\000)p Fp(1)39100 31110 y Fm(+)294 b Fj(\001)221 b(\001)g(\001)296 b Fm(+)f Fk(a)44241 30562 y Fp(1)45062 31110 y Fm(+)g Fk(a)47052 30562 y Fp(0)47577 31110 y Fm(\))p Fk(:)800 33763 y Fm(No)-36 b(w)431 b Fk(c)4319 33962 y Fs(j)4806 33763 y Fk(;)221 b(c)5948 33962 y Fs(j)51 b Fp(+1)7638 33763 y Fk(;)221 b(:::;)g(c)10445 33962 y Fs(k)24 b Fp(+1)12587 33763 y Fj(2)368 b(f)p Fm(0)p Fk(;)221 b Fm(1)p Fk(;)g Fm(2)p Fk(;)g(:)g(:)g(:)227 b(;)221 b(a)289 b Fj(\000)h Fm(1)p Fj(g)p Fm(,)432 b(and)e(rep)36 b(eating)431 b(the)f(pro)36 b(cedure)429 b(ab)36 b(o)-36 b(v)g(e)432 b(at)f(most)f Fk(j)364 b Fj(\000)290 b Fm(2)800 35368 y(times)434 b(giv)-36 b(es)434 b(the)f(co)36 b(e\016cien)-36 b(ts)434 b Fk(c)16845 35567 y Fs(i)17654 35368 y Fm(in)g(the)f(desired) g(range)g(for)i Fk(i)368 b Fm(=)h(2)p Fk(;)221 b Fm(3)p Fk(;)g(:::;)g(k)344 b Fm(+)295 b(1.)p 51860 35368 45 878 v 51905 34535 781 45 v 51905 35368 V 52684 35368 45 878 v 800 36543 a SDict begin H.S end 800 36543 a 800 36543 a SDict begin 14.5 H.A end 800 36543 a 800 36543 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.9) cvn H.B /DEST pdfmark end 800 36543 a Black 1297 x Fl(Lemma)549 b(9.)p Black 576 w Fi(L)-66 b(et)504 b Fk(n)444 b Fj(2)f Fh(N)p Fk(;)506 b Fi(and)f(let)g Fk(q)553 b Fi(and)505 b Fk(r)541 b Fi(b)-66 b(e)504 b(the)h(unique)g(inte)-66 b(gers)503 b(such)j(that)f Fk(n)444 b Fm(=)f Fk(aq)372 b Fm(+)324 b Fk(r)36 b Fi(,)515 b(wher)-66 b(e)800 39445 y Fm(0)369 b Fj(\024)h Fk(r)405 b(<)368 b(a)p Fi(.)597 b(L)-66 b(et)464 b Fk(R)380 b Fm(=)369 b Fk(r)13021 38922 y Fs(a)13521 38610 y Fg(k)14 b Ff(+1)15056 38922 y Fq(\000)p Fp(1)p 13021 39140 3238 54 v 13788 39903 a Fs(a)p Fq(\000)p Fp(1)16391 39445 y Fi(.)597 b(Then)465 b Fk(n)369 b Fj(2)f Fk(F)24016 39644 y Fs(a)24572 39445 y Fm(\()p Fk(k)45 b Fm(\))464 b Fi(if)g(and)h(only)g(if)f Fk(n)369 b(<)g(R)476 b Fi(or)40441 38922 y Fs(n)p Fq(\000)p Fs(R)p 40441 39140 2017 54 v 41200 39903 a(a)42960 39445 y Fj(2)369 b Fk(F)45057 39644 y Fs(a)45612 39445 y Fm(\()p Fk(k)340 b Fj(\000)295 b Fm(1\))p Fi(.)p Black 800 42779 a(Pr)-66 b(o)g(of.)p Black 649 w Fm(W)-108 b(e)434 b(pro)-36 b(v)g(e)433 b(that)g Fk(n)524 b(=)-804 b Fj(2)368 b Fk(F)16676 42978 y Fs(a)17231 42779 y Fm(\()p Fk(k)45 b Fm(\))434 b(if)g(and)f(only)h(if)h Fk(n)369 b Fj(\025)g Fk(R)445 b Fm(and)33809 41881 y Fk(n)296 b Fj(\000)f Fk(R)p 33809 42474 3401 54 v 35169 43691 a(a)37867 42779 y(=)-805 b Fj(2)369 b Fk(F)39809 42978 y Fs(a)40364 42779 y Fm(\()p Fk(k)340 b Fj(\000)295 b Fm(1\))p Fk(:)2751 45608 y Fm(Supp)36 b(ose)382 b Fk(n)369 b Fj(\025)g Fk(R)394 b Fm(and)14344 45085 y Fs(n)p Fq(\000)p Fs(R)p 14344 45302 2017 54 v 15102 46066 a(a)16862 45608 y Fj(62)369 b Fk(F)18959 45807 y Fs(a)19514 45608 y Fm(\()p Fk(k)236 b Fj(\000)191 b Fm(1\).)562 b(Then)27713 44709 y Fk(n)295 b Fj(\000)h Fk(R)p 27713 45302 3401 54 v 29072 46519 a(a)31629 45608 y Fm(is)383 b(a)g(nonnegativ)-36 b(e-in)g(tegral)383 b(com)-36 b(bination)800 47626 y(of)434 b(the)f(elemen)-36 b(ts)434 b(of)g Fk(G)12347 47825 y Fs(a)12902 47626 y Fm(\()p Fk(k)340 b Fj(\000)295 b Fm(1\);)434 b(th)-36 b(us)7947 50025 y Fk(n)295 b Fj(\000)g Fk(R)p 7947 50618 V 9306 51834 a(a)11849 50923 y Fm(=)369 b Fk(c)13790 51122 y Fp(1)14315 50923 y Fk(a)14998 50375 y Fs(k)24 b Fq(\000)p Fp(1)17065 50923 y Fm(+)294 b Fk(c)18931 51122 y Fp(2)19457 50923 y Fm(\()p Fk(a)20646 50375 y Fs(k)24 b Fq(\000)p Fp(1)22712 50923 y Fm(+)295 b Fk(a)24702 50375 y Fs(k)24 b Fq(\000)p Fp(2)26473 50923 y Fm(\))295 b(+)f Fj(\001)221 b(\001)g(\001)296 b Fm(+)f Fk(c)32292 51122 y Fs(k)32861 50923 y Fm(\()p Fk(a)34050 50375 y Fs(k)24 b Fq(\000)p Fp(1)36116 50923 y Fm(+)295 b Fk(a)38106 50375 y Fs(k)24 b Fq(\000)p Fp(2)40172 50923 y Fm(+)294 b Fj(\001)221 b(\001)g(\001)296 b Fm(+)f(1\))800 53916 y(for)434 b(some)g Fk(c)6606 54115 y Fp(1)7132 53916 y Fk(;)221 b(:)g(:)g(:)i(;)e(c)10604 54115 y Fs(k)11543 53916 y Fj(2)368 b Fh(Z)13683 54115 y Fq(\025)p Fp(0)14940 53916 y Fk(:)434 b Fm(Therefore)9564 56569 y Fk(n)369 b Fm(=)g Fk(c)12650 56768 y Fp(1)13175 56569 y Fk(a)13858 56020 y Fs(k)14722 56569 y Fm(+)295 b Fk(c)16589 56768 y Fp(2)17115 56569 y Fm(\()p Fk(a)18304 56020 y Fs(k)19167 56569 y Fm(+)g Fk(a)21157 56020 y Fs(k)24 b Fq(\000)p Fp(1)22928 56569 y Fm(\))295 b(+)g Fj(\001)221 b(\001)g(\001)296 b Fm(+)f Fk(c)28748 56768 y Fs(k)29316 56569 y Fm(\()p Fk(a)30505 56020 y Fs(k)31369 56569 y Fm(+)g Fk(a)33359 56020 y Fs(k)24 b Fq(\000)p Fp(1)35425 56569 y Fm(+)295 b Fj(\001)221 b(\001)g(\001)296 b Fm(+)e Fk(a)p Fm(\))h(+)g Fk(R)11 b(;)800 59588 y Fm(where)364 b Fk(c)5048 59787 y Fp(1)5574 59588 y Fm(,)378 b Fk(c)6873 59787 y Fp(2)7399 59588 y Fm(,)g(...,)h Fk(c)10521 59787 y Fs(k)11459 59588 y Fj(2)369 b Fh(Z)13600 59787 y Fq(\025)p Fp(0)14857 59588 y Fm(.)555 b(Because)365 b Fk(R)380 b Fm(=)368 b Fk(r)24273 58113 y Fc(\020)25199 59064 y Fs(a)25699 58752 y Fg(k)14 b Ff(+1)27234 59064 y Fq(\000)p Fp(1)p 25199 59282 3238 54 v 25967 60046 a Fs(a)p Fq(\000)p Fp(1)28569 58113 y Fc(\021)29731 59588 y Fm(=)369 b Fk(r)31956 58512 y Fc(\000)32564 59588 y Fk(a)33247 59105 y Fs(k)34111 59588 y Fm(+)295 b Fk(a)36101 59105 y Fs(k)24 b Fq(\000)p Fp(1)38167 59588 y Fm(+)295 b Fj(\001)221 b(\001)g(\001)296 b Fm(+)f Fk(a)g Fm(+)f(1)45560 58512 y Fc(\001)46170 59588 y Fm(,)378 b Fk(n)369 b Fj(62)g Fk(F)50151 59787 y Fs(a)50706 59588 y Fm(\()p Fk(k)45 b Fm(\).)2751 61558 y(Con)-36 b(v)g(ersely)-108 b(,)450 b(supp)36 b(ose)445 b Fk(n)390 b Fj(62)f Fk(F)17789 61757 y Fs(a)18344 61558 y Fm(\()p Fk(k)45 b Fm(\).)614 b(By)446 b(Lemma)p 0 .5 0 TeXcolorrgb 27760 61558 a SDict begin H.S end 27760 61558 a 0 .5 0 TeXcolorrgb Fm(8)p 0 .5 0 TeXcolorrgb 28410 60702 a SDict begin H.R end 28410 60702 a 28410 61558 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.8) cvn H.B /ANN pdfmark end 28410 61558 a Black Fm(,)j(there)c(exist)i Fk(c)36207 61757 y Fp(1)37122 61558 y Fj(2)389 b Fh(Z)39283 61757 y Fq(\025)p Fp(0)40986 61558 y Fm(and)445 b Fk(c)44087 61757 y Fp(2)44612 61558 y Fk(;)221 b(c)45754 61757 y Fp(3)46281 61558 y Fk(;)g(:)g(:)g(:)i(;)e(c)49753 61757 y Fs(k)24 b Fp(+1)51914 61558 y Fj(2)800 63163 y(f)p Fm(0)p Fk(;)221 b(:)g(:)g(:)j(;)d(a)296 b 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V 15079 41635 V 15117 41632 V 15156 41628 V 15195 41625 V 15234 41622 V 15272 41619 V 15311 41616 V 16629 41616 V 16668 41619 V 16707 41622 V 16746 41625 V 16785 41628 V 16823 41632 V 16862 41635 V 16901 41639 V 16940 41643 V 16978 41647 V 17017 41651 V 17056 41656 V 17095 41661 V 17134 41665 V 17172 41670 V 17211 41675 V 17250 41681 V 17289 41686 V 17327 41692 V 17366 41697 V 17405 41703 V 17444 41709 V 17482 41716 V 17521 41722 V 17560 41729 V 17599 41735 V 17638 41742 V 17676 41749 V 17715 41757 V 17754 41764 V 17793 41771 V 17831 41779 V 17870 41787 V 17909 41795 V 17948 41803 V 17987 41812 V 18025 41820 V 18064 41829 V 18103 41838 V 18142 41847 V 18180 41856 V 18219 41865 V 18258 41875 V 18297 41884 V 18335 41894 V 18374 41904 V 18413 41914 V 18452 41924 V 18491 41935 V 18529 41946 V 18568 41956 V 18607 41967 V 18646 41978 V 18684 41990 V 18723 42001 V 18762 42013 V 18801 42024 V 18840 42036 V 18878 42048 V 18917 42061 V 18956 42073 V 18995 42086 V 19033 42098 V 19072 42111 V 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34429 42279 V 34462 42294 V 34496 42309 V 34529 42324 V 34563 42340 V 34596 42355 V 34630 42371 V 34663 42387 V 34697 42403 V 34730 42419 V 34764 42436 V 34797 42452 V 34831 42469 V 34864 42486 V 34898 42503 V 34931 42521 V 34965 42538 V 34998 42556 V 35032 42573 V 35065 42591 V 29385 42964 a Fk(:)g(:)g(:)31355 43066 y(b)31908 43265 y Fs(l)32189 43388 y Ff(1)p 31355 43601 1351 54 v 30923 43654 a Fb(BC)p 32008 43654 45 1 v 27080 45867 a(G)q(F)p 28187 44783 2736 45 v 27080 46310 a Fe(\017)p Fd(\017)p 27058 46310 45 444 v 33148 42964 a Fk(:)g(:)g(:)35117 43260 y(b)35670 43459 y Fp(0)800 47422 y Fk(\014)369 b Fj(\000)296 b Fk(\016)418 b Fm(:)9007 47233 y Fk(b)9560 47432 y Fs(j)9941 47543 y Fg(m)11073 47233 y Fj(\000)296 b Fm(2)p 9007 47817 4045 54 v 12166 w Fk(b)25771 47432 y Fs(j)26152 47555 y Ff(1)26964 47233 y Fj(\000)f Fm(2)p 25218 47817 3724 54 v 2751 49586 a(The)644 b Fk(\014)6369 49785 y Fs(i)7390 49586 y Fm(satisfy)h(the)f(h)-36 b(yp)36 b(otheses)644 b(of)h(Lemma)p 0 .5 0 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Fm(has)444 b(a)f(drop,)j(w)-36 b(e)444 b(ha)-36 b(v)g(e)443 b Fk(n)387 b(<)e(a)21921 43061 y Fs(k)24 b Fp(+1)23994 43543 y Fj(\000)302 b Fm(1)p Fk(:)444 b Fm(\(The)g(base-)p Fk(a)f Fm(digits)g(of)i Fk(\014)517 b Fm(cannot)443 b(all)i(equal)f Fk(a)302 b Fj(\000)g Fm(1)6913 45148 y(since)434 b Fk(\014)507 b Fm(has)434 b(a)g(drop.\))577 b(There)434 b(are)f(t)-36 b(w)g(o)434 b(cases.)p Black 7033 47306 a(\(1\))p Black 650 w(If)295 b Fj(j)p Fk(\014)74 b Fj(j)369 b Fk(<)f(a)14484 46824 y Fs(k)15062 47306 y Fm(+)10 b Fj(\001)221 b(\001)g(\001)10 b Fm(+)g Fk(a)g Fm(+)g(1,)320 b(then)293 b Fk(n)369 b(<)g(R)304 b Fm(\()p Fk(n)370 b Fj(6\021)f Fm(0)590 b(\(mo)36 b(d)443 b Fk(a)p Fm(\))738 b(=)-221 b Fj(\))737 b Fk(R)380 b Fj(\025)369 b Fk(a)44328 46824 y Fs(k)44906 47306 y Fm(+)10 b Fk(:)221 b(:)g(:)11 b Fm(+)f Fk(a)49169 46824 y Fp(1)49703 47306 y Fm(+)g Fk(a)51408 46824 y Fp(0)51933 47306 y Fm(\).)9345 48912 y(Th)-36 b(us,)495 b(b)-36 b(y)483 b(Lemma)p 0 .5 0 TeXcolorrgb 19610 48912 a SDict begin H.S end 19610 48912 a 0 .5 0 TeXcolorrgb Fm(9)p 0 .5 0 TeXcolorrgb 20261 48055 a SDict begin H.R end 20261 48055 a 20261 48912 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.9) cvn H.B /ANN pdfmark end 20261 48912 a Black Fm(,)495 b Fk(n)453 b Fj(2)f Fk(F)24526 49111 y Fs(a)25081 48912 y Fm(\()p Fk(k)45 b Fm(\))p Fk(;)483 b Fm(and)f(therefore)g Fk(\013)338 b Fj(\003)328 b Fk(\014)557 b Fm(and)482 b Fk(n)453 b Fj(2)f Fk(F)45175 49111 y Fs(a)45730 48912 y Fm(\()p Fk(k)45 b Fm(\))483 b(are)g(b)36 b(oth)9345 50517 y(true.)p Black 7033 52398 a(\(2\))p Black 650 w(Again)317 b(let)f Fk(\016)366 b Fm(b)36 b(e)316 b(a)g(string)g(of)g Fk(k)100 b Fm(+)55 b(1)316 b(ones)g(in)g(base)g Fk(a;)g Fm(and)f(assume)h(that)g Fk(a)44582 51916 y Fs(k)45205 52398 y Fm(+)55 b Fj(\001)221 b(\001)g(\001)56 b Fm(+)f Fk(a)g Fm(+)g(1)368 b Fj(\024)9345 54004 y(j)p Fk(\014)74 b Fj(j)369 b Fk(<)g(a)13324 53521 y Fs(k)24 b Fp(+1)15390 54004 y Fj(\000)295 b Fm(1.)579 b(Since)433 b Fj(j)p Fk(\013)8 b Fj(j)370 b Fm(=)f(0)p Fk(;)434 b Fm(w)-36 b(e)434 b(ma)-36 b(y)434 b(apply)g(Lemma)p 0 .5 0 TeXcolorrgb 39425 54004 a SDict begin H.S end 39425 54004 a 0 .5 0 TeXcolorrgb Fm(9)p 0 .5 0 TeXcolorrgb 40075 53147 a SDict begin H.R end 40075 53147 a 40075 54004 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.9) cvn H.B /ANN pdfmark end 40075 54004 a Black 434 w Fm(to)g(obtain)9345 56937 y Fk(\013)185 b Fj(\003)177 b Fk(\014)444 b Fj(,)369 b Fk(\013)185 b Fj(\003)177 b Fm(\()p Fk(\014)251 b Fj(\000)177 b Fk(\016)50 b Fm(\))369 b Fj(,)g(j)p Fk(\013)8 b(\014)74 b Fj(j)177 b(\000)g(j)p Fk(\016)50 b Fj(j)370 b Fm(=)e Fk(n)177 b Fj(\000)g Fm(\()p Fk(a)32059 56389 y Fs(k)32805 56937 y Fm(+)g Fk(:)221 b(:)g(:)178 b Fm(+)f Fk(a)g Fm(+)g(1\))367 b Fj(2)i Fk(F)42555 57136 y Fs(a)43110 56937 y Fm(\()p Fk(k)45 b Fm(\))369 b Fj(,)g Fk(n)g Fj(2)g Fk(F)50151 57136 y Fs(a)50706 56937 y Fm(\()p Fk(k)45 b Fm(\))p Fk(;)9345 59870 y Fm(where)408 b(it)f(is)h(understo)36 b(o)g(d)406 b(that)h(len\()p Fk(\014)316 b Fj(\000)242 b Fk(\016)50 b Fm(\))369 b(=)f(len\()p Fk(\014)74 b Fm(\))p Fk(:)407 b Fm(The)h(\014rst)f(equiv)-72 b(alence)408 b(follo)-36 b(ws)9345 61476 y(from)415 b(Lemma)p 0 .5 0 TeXcolorrgb 17020 61476 a SDict begin H.S end 17020 61476 a 0 .5 0 TeXcolorrgb Fm(14)p 0 .5 0 TeXcolorrgb 18321 60619 a SDict begin H.R end 18321 60619 a 18321 61476 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.14) cvn H.B /ANN pdfmark end 18321 61476 a Black 414 w Fm(\(recall)f(that)g(the)g(h) -36 b(yp)36 b(othesis)413 b Fj(j)p Fk(\013)8 b Fj(j)370 b Fk(>)f Fm(0)414 b(w)-36 b(as)415 b(unnecessary)f(for)g(Case)9345 63081 y(\(i\)\),)425 b(the)d(second)g(from)h(the)f(induction)g(h)-36 b(yp)36 b(othesis,)425 b(and)d(the)g(last)h(from)g(Lemma)p 0 .5 0 TeXcolorrgb 51138 63081 a SDict begin H.S end 51138 63081 a 0 .5 0 TeXcolorrgb Fm(10)p 0 .5 0 TeXcolorrgb 52439 62225 a SDict begin H.R end 52439 62225 a 52439 63081 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.10) cvn H.B /ANN pdfmark end 52439 63081 a Black Fm(.)p Black 26150 70521 a(11)p Black eop end end %%Page: 12 12 TeXDict begin HPSdict begin 12 11 bop 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a Black 0 TeXcolorgray 800 -4000 a SDict begin H.S end 800 -4000 a 0 TeXcolorgray 0 TeXcolorgray 800 -4000 a SDict begin H.R end 800 -4000 a 800 -4000 a SDict begin [ /View [/XYZ H.V] /Dest (page.12) cvn H.B /DEST pdfmark end 800 -4000 a Black Black Black 4673 -2672 a Fm(\(c\))p Black 650 w(Supp)36 b(ose)433 b Fk(z)59 b Fm(\()p Fk(\014)74 b Fm(\))369 b Fk(>)f Fm(0)434 b(and)f Fj(j)p Fk(\013)8 b Fj(j)370 b Fk(>)e Fm(0;)435 b(then)d(b)-36 b(y)434 b(Lemma)p 0 .5 0 TeXcolorrgb 34040 -2672 a SDict begin H.S end 34040 -2672 a 0 .5 0 TeXcolorrgb Fm(14)p 0 .5 0 TeXcolorrgb 35341 -3528 a SDict begin H.R end 35341 -3528 a 35341 -2672 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.14) cvn H.B /ANN pdfmark end 35341 -2672 a Black 17322 38 a Fk(\013)304 b Fj(\003)296 b Fk(\014)442 b Fj(,)370 b Fk(\013)304 b Fj(\003)295 b Fm(\()p Fk(\014)369 b Fj(\000)295 b Fk(\016)50 b Fm(\))434 b(or)655 b([)p Fj(j)p Fk(\013)8 b Fj(j)296 b(\000)f Fm(1])730 b Fj(\003)295 b Fm(\([1])p Fk(\014)370 b Fj(\000)295 b Fk(\016)50 b Fm(\))20589 2126 y Fj(,)370 b Fk(n)295 b Fj(\000)h Fm(\()p Fk(a)25876 1577 y Fs(k)26739 2126 y Fm(+)f Fj(\001)221 b(\001)g(\001)296 b Fm(+)f Fk(a)g Fm(+)g(1\))369 b Fj(2)f Fk(F)37104 2325 y Fs(a)37660 2126 y Fm(\()p Fk(k)45 b Fm(\))20589 4063 y Fj(,)370 b Fk(n)f Fj(2)g Fk(F)25529 4262 y Fs(a)26084 4063 y Fm(\()p Fk(k)45 b Fm(\))p Fk(;)6913 6773 y Fm(where)363 b(the)g(\014rst)f(equiv)-72 b(alence)364 b(follo)-36 b(ws)365 b(from)f(Lemma)p 0 .5 0 TeXcolorrgb 33946 6773 a SDict begin H.S end 33946 6773 a 0 .5 0 TeXcolorrgb Fm(14)p 0 .5 0 TeXcolorrgb 35247 5917 a SDict begin H.R end 35247 5917 a 35247 6773 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.14) cvn H.B /ANN pdfmark end 35247 6773 a Black Fm(,)378 b(the)362 b(second)h(from)g(the)g (induction)6913 8378 y(h)-36 b(yp)36 b(othesis,)434 b(and)f(the)g(last) h(from)g(Lemma)p 0 .5 0 TeXcolorrgb 28630 8378 a SDict begin H.S end 28630 8378 a 0 .5 0 TeXcolorrgb Fm(10)p 0 .5 0 TeXcolorrgb 29931 7522 a SDict begin H.R end 29931 7522 a 29931 8378 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.10) cvn H.B /ANN pdfmark end 29931 8378 a Black Fm(.)p 51860 11294 45 878 v 51905 10460 781 45 v 51905 11294 V 52684 11294 45 878 v 2751 13831 a(Theorem)p 0 .5 0 TeXcolorrgb 8336 13831 a SDict begin H.S end 8336 13831 a 0 .5 0 TeXcolorrgb Fm(7)p 0 .5 0 TeXcolorrgb 8986 12975 a SDict begin H.R end 8986 12975 a 8986 13831 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.7) cvn H.B /ANN pdfmark end 8986 13831 a Black 527 w Fm(is)527 b(actually)h(a)f(corollary)i (of)e(Theorem)p 0 .5 0 TeXcolorrgb 29770 13831 a SDict begin H.S end 29770 13831 a 0 .5 0 TeXcolorrgb Fm(15)p 0 .5 0 TeXcolorrgb 31071 12975 a SDict begin H.R end 31071 12975 a 31071 13831 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.15) cvn H.B /ANN pdfmark end 31071 13831 a Black Fm(.)857 b(One)527 b(can)f(easily)i(compute)f Fk(f)47293 14030 y Fs(a)47848 13831 y Fm(\()p Fk(n)p Fm(\))g(from)800 15437 y(Theorem)p 0 .5 0 TeXcolorrgb 6805 15437 a SDict begin H.S end 6805 15437 a 0 .5 0 TeXcolorrgb Fm(7)p 0 .5 0 TeXcolorrgb 7455 14580 a SDict begin H.R end 7455 14580 a 7455 15437 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.7) cvn H.B /ANN pdfmark end 7455 15437 a Black Fm(.)699 b(Here)473 b(are)h(a)f(few)i(example)f (calculations.)699 b(Notice)474 b(that)f(to)h(apply)g(Theorem)p 0 .5 0 TeXcolorrgb 47951 15437 a SDict begin H.S end 47951 15437 a 0 .5 0 TeXcolorrgb Fm(7)p 0 .5 0 TeXcolorrgb 48601 14580 a SDict begin H.R end 48601 14580 a 48601 15437 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.7) cvn H.B /ANN pdfmark end 48601 15437 a Black 474 w Fm(it)f(ma)-36 b(y)800 17042 y(b)36 b(e)433 b(necessary)h(to)g(write)g(a)g(string)f (with)h(leading)g(zeros.)800 17300 y SDict begin H.S end 800 17300 a 800 17300 a SDict begin 14.5 H.A end 800 17300 a 800 17300 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.16) cvn H.B /DEST pdfmark end 800 17300 a Black 2262 x Fl(Corollary)560 b(16.)p Black 571 w Fi(L)-66 b(et)495 b Fk(n)427 b Fj(2)g Fh(Z)15654 19080 y Fp(+)16440 19562 y Fk(:)497 b Fi(Then)e Fk(n)427 b Fj(2)g Fk(F)24074 19761 y Fs(a)24629 19562 y Fm(\()p Fk(k)45 b Fm(\))496 b Fi(if)f(and)h(only)g (if)f(ther)-66 b(e)495 b(exist)g(b)-66 b(ase-)p Fk(a)496 b Fi(strings)f Fk(\013)505 b Fi(and)800 21167 y Fk(\014)539 b Fi(such)466 b(that)800 21819 y SDict begin H.S end 800 21819 a 800 21819 a SDict begin 14.5 H.A end 800 21819 a 800 21819 a SDict begin [ /View [/XYZ H.V] /Dest (Item.1) cvn H.B /DEST pdfmark end 800 21819 a Black 2338 23687 a Fi(1.)p Black 651 w Fj(j)p Fk(\013)8 b(\014)74 b Fj(j)369 b Fm(=)g Fk(n)p Fi(,)800 24541 y SDict begin H.S end 800 24541 a 800 24541 a SDict begin 14.5 H.A end 800 24541 a 800 24541 a SDict begin [ /View [/XYZ H.V] /Dest (Item.2) cvn H.B /DEST pdfmark end 800 24541 a Black 2338 26335 a Fi(2.)p Black 651 w Fk(\013)304 b Fj(\003)295 b Fk(\014)74 b(;)465 b Fi(and)800 27115 y SDict begin H.S end 800 27115 a 800 27115 a SDict begin 14.5 H.A end 800 27115 a 800 27115 a SDict begin [ /View [/XYZ H.V] /Dest (Item.3) cvn H.B /DEST pdfmark end 800 27115 a Black 2338 28983 a Fi(3.)p Black 651 w Fk(k)414 b Fm(=)368 b Fi(len)p Fm(\()p Fk(\014)74 b Fm(\))295 b Fj(\000)g Fm(1)p Fk(:)p Black 800 31504 a Fi(Pr)-66 b(o)g(of.)p Black 649 w Fm(If)504 b(suc)-36 b(h)503 b(strings)g Fk(\013)512 b Fm(and)503 b Fk(\014)577 b Fm(exist,)522 b(then)502 b Fk(n)489 b Fm(=)e Fk(\013)8 b(\014)563 b Fj(2)487 b Fk(F)32440 31703 y Fs(a)32995 31504 y Fm(\()p Fk(k)45 b Fm(\))503 b(b)-36 b(y)503 b(Theorem)p 0 .5 0 TeXcolorrgb 42669 31504 a SDict begin H.S end 42669 31504 a 0 .5 0 TeXcolorrgb Fm(15)p 0 .5 0 TeXcolorrgb 43970 30648 a SDict begin H.R end 43970 30648 a 43970 31504 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.15) cvn H.B /ANN pdfmark end 43970 31504 a Black Fm(.)787 b(Con)-36 b(v)g(ersely)-108 b(,)522 b(if)800 33109 y Fk(n)445 b Fj(2)f Fk(F)4193 33308 y Fs(a)4748 33109 y Fm(\()p Fk(k)45 b Fm(\))p Fk(;)478 b Fm(then)f(let)h Fk(\014)552 b Fm(b)36 b(e)477 b(the)h(last)g Fk(k)370 b Fm(+)325 b(1)479 b(digits)f(of)g(a)h(base-)p Fk(a)e Fm(represen)-36 b(tation)477 b(of)i Fk(n)p Fm(,)489 b(and)478 b(let)g Fk(\013)486 b Fm(b)36 b(e)800 34714 y(the)436 b(remaining)g(digits,)i(setting)e Fk(\013)383 b Fm(=)373 b(0)436 b(if)h(otherwise)g Fk(\013)445 b Fm(w)-36 b(ould)436 b(b)36 b(e)437 b(empt)-36 b(y)-108 b(.)585 b(This)437 b(giv)-36 b(es)437 b Fj(j)p Fk(\013)8 b(\014)74 b Fj(j)375 b Fm(=)e Fk(n)437 b Fm(and)800 36319 y Fk(k)414 b Fm(=)369 b(len\()p Fk(\014)74 b Fm(\))294 b Fj(\000)i Fm(1)434 b(directly)-108 b(.)578 b(F)-108 b(urthermore,)432 b(since)i Fj(j)p Fk(\013)8 b(\014)74 b Fj(j)369 b Fm(=)g Fk(n)g Fj(2)g Fk(F)33044 36518 y Fs(a)33599 36319 y Fm(\()p Fk(k)45 b Fm(\))p Fk(;)221 b(\013)304 b Fj(\003)296 b Fk(\014)507 b Fm(b)-36 b(y)434 b(Theorem)p 0 .5 0 TeXcolorrgb 46544 36319 a SDict begin H.S end 46544 36319 a 0 .5 0 TeXcolorrgb Fm(15)p 0 .5 0 TeXcolorrgb 47845 35463 a SDict begin H.R end 47845 35463 a 47845 36319 a SDict begin [ /Color [1 0 0] /H /I /Border [0 0 0] /Subtype /Link /Dest (theorem.15) cvn H.B /ANN pdfmark end 47845 36319 a Black Fm(.)p 51860 36319 45 878 v 51905 35485 781 45 v 51905 36319 V 52684 36319 45 878 v 800 37583 a SDict begin H.S end 800 37583 a 800 37583 a SDict begin 14.5 H.A end 800 37583 a 800 37583 a SDict begin [ /View [/XYZ H.V] /Dest (theorem.17) cvn H.B /DEST pdfmark end 800 37583 a Black 1256 x Fl(Example)499 b(17.)p Black 800 39759 a SDict begin H.S end 800 39759 a 800 39759 a SDict begin 14.5 H.A end 800 39759 a 800 39759 a SDict begin [ /View [/XYZ H.V] /Dest (Item.4) cvn H.B /DEST pdfmark end 800 39759 a Black 2389 41628 a Fm(1.)p Black 652 w(F)-108 b(or)492 b Fk(n)471 b Fm(=)f(24)h(=)f Fj(j)p Fm(11000)p Fj(j)i Fm(=)e Fj(j)p Fm(0011000)p Fj(j)495 b Fm(with)e(base)g Fk(a)470 b Fm(=)g(2)p Fk(;)494 b Fm(let)f Fk(\013)479 b Fm(=)470 b(0)493 b(and)g Fk(\014)544 b Fm(=)470 b(011000;)525 b(then)4052 43233 y Fj(j)p Fk(\013)8 b(\014)74 b Fj(j)436 b Fm(=)f(24)474 b(and)e(0)436 b(=)f Fj(j)p Fk(\013)8 b Fj(j)436 b Fk(<)g(z)59 b Fm(\()p Fk(\014)74 b Fm(\))435 b(=)g(1.)697 b(W)-108 b(e)472 b(see)h(that)f Fk(f)32639 43432 y Fp(2)33165 43233 y Fm(\(24\))436 b(=)f(len\()p Fk(\014)74 b Fm(\))322 b Fj(\000)g Fm(1)436 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[ /View [/XYZ H.V] /Dest (Item.18) cvn H.B /DEST pdfmark end 800 11300 a Black 2390 13200 a Fm(3.)p Black 651 w(If)434 b Fk(\013)6179 13399 y Fp(0)7000 13200 y Fj(\003)296 b Fk(\014)8694 13399 y Fp(0)9220 13200 y Fk(;)434 b Fm(then)e Fk(n)370 b Fj(2)e Fk(F)16219 13399 y Fs(a)16774 13200 y Fm(\(0\))p Fk(:)434 b Fm(STOP)-108 b(.)800 14086 y SDict begin H.S end 800 14086 a 800 14086 a SDict begin 14.5 H.A end 800 14086 a 800 14086 a SDict begin [ /View [/XYZ H.V] /Dest (Item.19) cvn H.B /DEST pdfmark end 800 14086 a Black 2390 15912 a Fm(4.)p Black 651 w(If)434 b(not)f Fk(\013)8491 16111 y Fp(0)9312 15912 y Fj(\003)296 b Fk(\014)11006 16111 y Fp(0)11532 15912 y Fk(;)434 b Fm(let)g Fk(l)398 b Fm(:=)369 b(1)p Fk(:)800 16724 y SDict begin H.S end 800 16724 a 800 16724 a SDict begin 14.5 H.A end 800 16724 a 800 16724 a SDict begin [ /View [/XYZ H.V] /Dest (Item.20) cvn H.B /DEST pdfmark end 800 16724 a Black 2390 18624 a Fm(5.)p Black 651 w(Let)433 b Fk(\013)7209 18823 y Fs(l)7925 18624 y Fm(:=)369 b Fk(c)10227 18823 y Fs(k)11017 18624 y Fj(\001)221 b(\001)g(\001)h Fk(c)13348 18823 y Fs(l)11 b Fp(+1)15330 18624 y Fm(and)433 b Fk(\014)18593 18823 y Fs(l)19309 18624 y Fm(:=)369 b Fk(c)21611 18823 y Fs(l)22179 18624 y Fj(\001)221 b(\001)g(\001)h Fk(c)24510 18823 y Fp(0)25036 18624 y Fk(:)800 19469 y SDict begin H.S end 800 19469 a 800 19469 a SDict begin 14.5 H.A end 800 19469 a 800 19469 a SDict begin [ /View [/XYZ H.V] /Dest (Item.21) cvn H.B /DEST pdfmark end 800 19469 a Black 2390 21336 a Fm(6.)p Black 651 w(If)434 b Fk(\013)6179 21535 y Fs(l)6821 21336 y Fj(\003)295 b Fk(\014)8514 21535 y Fs(l)8861 21336 y Fk(;)434 b Fm(then)f Fk(n)369 b Fj(2)g Fk(F)15861 21535 y Fs(a)16416 21336 y Fm(\()p Fk(l)29 b Fm(\))p Fk(:)434 b Fm(STOP)-108 b(.)800 22222 y SDict begin H.S end 800 22222 a 800 22222 a SDict begin 14.5 H.A end 800 22222 a 800 22222 a SDict begin [ /View [/XYZ H.V] /Dest (Item.22) cvn H.B /DEST pdfmark end 800 22222 a Black 2390 24049 a Fm(7.)p Black 651 w(If)434 b Fk(l)399 b(<)368 b(k)340 b Fj(\000)296 b Fm(1)p Fk(;)434 b Fm(then)f(put)f Fk(l)399 b Fm(:=)369 b Fk(l)324 b Fm(+)295 b(1)p Fk(:)434 b Fm(GOTO)f(step)g(5.)800 24860 y SDict begin H.S end 800 24860 a 800 24860 a SDict begin 14.5 H.A end 800 24860 a 800 24860 a SDict begin [ /View [/XYZ H.V] /Dest (Item.23) cvn H.B /DEST pdfmark end 800 24860 a Black 2390 26761 a Fm(8.)p Black 651 w(Let)g Fk(\013)7209 26960 y Fs(k)8147 26761 y Fm(:=)369 b(0)434 b(and)f Fk(\014)14236 26960 y Fs(k)15174 26761 y Fm(:=)369 b Fk(c)17476 26960 y Fs(k)18266 26761 y Fj(\001)221 b(\001)g(\001)h Fk(c)20597 26960 y Fp(0)21123 26761 y Fk(:)800 27572 y SDict begin H.S end 800 27572 a 800 27572 a SDict begin 14.5 H.A end 800 27572 a 800 27572 a SDict begin [ /View [/XYZ H.V] /Dest (Item.24) cvn H.B /DEST pdfmark end 800 27572 a Black 2390 29473 a Fm(9.)p Black 651 w(If)434 b Fk(\013)304 b Fj(\003)295 b Fk(\014)74 b(;)434 b Fm(then)f Fk(n)369 b Fj(2)g Fk(F)15250 29672 y Fs(a)15805 29473 y Fm(\()p Fk(k)45 b Fm(\))p Fk(:)433 b Fm(STOP)-108 b(.)800 30358 y SDict begin H.S end 800 30358 a 800 30358 a SDict begin 14.5 H.A end 800 30358 a 800 30358 a SDict begin [ /View [/XYZ H.V] /Dest (Item.25) cvn H.B /DEST pdfmark end 800 30358 a Black 1739 32185 a Fm(10.)p Black 652 w(Let)571 b Fk(\013)7347 32384 y Fs(k)24 b Fp(+1)9723 32185 y Fm(:=)605 b(0)572 b(and)f Fk(\014)16324 32384 y Fs(k)24 b Fp(+1)18700 32185 y Fm(=)605 b(0)p Fk(c)21527 32384 y Fs(k)22317 32185 y Fj(\001)221 b(\001)g(\001)h Fk(c)24648 32384 y Fp(0)25174 32185 y Fk(:)572 b Fm(Then)g Fj(j)p Fk(\013)8 b Fj(j)605 b Fm(=)f(0)572 b(and)g Fk(z)59 b Fm(\()p Fk(\014)39229 32384 y Fs(k)24 b Fp(+1)41000 32185 y Fm(\))605 b(=)f(1)p Fk(;)573 b Fm(so)f Fk(\013)398 b Fj(\003)390 b Fk(\014)646 b Fm(and)4052 33790 y Fk(n)369 b Fj(2)g Fk(F)7294 33989 y Fs(a)7849 33790 y Fm(\()p Fk(k)340 b Fm(+)295 b(1\))p Fk(:)434 b Fm(STOP)-108 b(.)2751 36502 y(It)414 b(is)g(clear)h(that)f (the)f(ab)36 b(o)-36 b(v)g(e)415 b(algorithm)g(terminates.)571 b(F)-108 b(urthermore,)417 b(since)d(the)f(algorithm)i(c)-36 b(hec)g(ks)800 38107 y(mem)g(b)36 b(ership)331 b(in)h Fk(F)10262 38306 y Fs(a)10817 38107 y Fm(\()p Fk(k)45 b Fm(\))332 b(for)h(eac)-36 b(h)332 b(v)-72 b(alue)332 b(of)h Fk(k)377 b Fm(sequen)-36 b(tially)334 b(b)36 b(eginning)332 b(with)g Fk(k)414 b Fm(=)369 b(0)p Fk(;)332 b Fm(it)h(m)-36 b(ust)331 b(determine)800 39712 y(the)433 b(least)h(v)-72 b(alue)434 b(of)h Fk(k)478 b Fm(suc)-36 b(h)433 b(that)g Fk(n)369 b Fj(2)g Fk(F)21072 39911 y Fs(a)21627 39712 y Fm(\()p Fk(k)45 b Fm(\))p Fk(;)434 b Fm(as)f(desired.)2751 41317 y(In)496 b(the)g(w)-36 b(orst)496 b(case,)513 b(steps)496 b(5)h(through)e(7)i(are)f(rep)36 b(eated)496 b(at)h(most)f(log)38294 41631 y Fs(a)38849 41317 y Fm(\()p Fk(n)338 b Fj(\000)g Fm(1\))g(+)g(1)497 b(times.)766 b(Eac)-36 b(h)800 42922 y(iteration)356 b(requires)f(ab)36 b(out)355 b(log)16126 43236 y Fs(a)16682 42922 y Fm(\()p Fk(n)p Fm(\))g(op)36 b(erations)356 b(\(mostly)g(from)f(computation)g(of)h Fk(z)59 b 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b(etersen,)3474 1854 y(and)510 b(C.)i(Starr,)530 b(Characterizing)512 b(a)g(generalized)g (in\014nite)e(family)j(of)f(F)-108 b(rob)36 b(enius)510 b(semigroups)h(b)-36 b(y)3474 3459 y(\014ltration,)433 b(in)h(preparation.)800 4263 y SDict begin H.S end 800 4263 a 800 4263 a SDict begin 14.5 H.A end 800 4263 a 800 4263 a SDict begin [ /View [/XYZ H.V] /Dest (cite.Cu) cvn H.B /DEST pdfmark end 800 4263 a Black 1450 6156 a Fm([2])p Black 652 w(F.)300 b(Curtis,)326 b(On)300 b(form)-36 b(ulas)300 b(for)g(the)g(F)-108 b(rob)36 b(enius)299 b(n)-36 b(um)g(b)36 b(er)298 b(of)i(a)h(n)-36 b(umerical)299 b(semigroup,)327 b Fi(Math.)342 b(Sc)-66 b(and.)3474 7761 y Fl(67)434 b Fm(\(1990\),)g(190{192.)800 8639 y SDict begin H.S end 800 8639 a 800 8639 a SDict begin 14.5 H.A end 800 8639 a 800 8639 a SDict begin [ /View [/XYZ H.V] /Dest (cite.D) cvn H.B /DEST pdfmark end 800 8639 a Black 1450 10458 a Fm([3])p Black 652 w(J.)457 b(L.)g(Da)-36 b(vison,)465 b(On)456 b(the)h(linear)g(Diophan)-36 b(tine)457 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b(gy,)3474 20667 y(Homotopy)464 b(Appl.)433 b Fl(9)h Fm(\(2007\),)h(185{191.)800 21545 y SDict begin H.S end 800 21545 a 800 21545 a SDict begin 14.5 H.A end 800 21545 a 800 21545 a SDict begin [ /View [/XYZ H.V] /Dest (cite.Kannan) cvn H.B /DEST pdfmark end 800 21545 a Black 1450 23364 a Fm([6])p Black 652 w(R.)389 b(Kannan,)398 b(Lattice)388 b(translates)h(of)h(a)f(p)36 b(olytop)g(e)390 b(and)e(the)g(F)-108 b(rob)36 b(enius)388 b(problem,)397 b Fi(Combinatoric)-66 b(a)3474 24969 y Fl(12)434 b Fm(\(1992\),)g(161{177.)800 25847 y SDict begin H.S end 800 25847 a 800 25847 a SDict begin 14.5 H.A end 800 25847 a 800 25847 a SDict begin [ /View [/XYZ H.V] /Dest (cite.MT) cvn H.B /DEST pdfmark end 800 25847 a Black 1450 27666 a Fm([7])p Black 652 w(R.)298 b(E.)g(Mosher)g(and)g(M.)g(C.) g(T)-108 b(angora,)327 b Fi(Cohomolo)-66 b(gy)340 b(Op)-66 b(er)g(ations)339 b(and)h(Applic)-66 b(ations)339 b(in)g(Homotopy)3474 29271 y(The)-66 b(ory)p Fm(,)433 b(Harp)36 b(er)433 b(&)h(Ro)-36 b(w,)434 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