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Fm(n)p Fn(\()p Fm(k)45 b Fn(\))434 b(when)f Fm(k)479 b Fn(is)434 b(a)g(m)-36 b(ultiple)433 b(of)h(10)g(up)f(to)h(120.)p Black Black 14490 58770 7860 45 v 14468 60375 45 1606 v 15155 59894 a Fm(k)p 17097 60375 V 1953 w(n)p Fn(\()p Fm(k)45 b Fn(\))p 22328 60375 V 14490 60420 7860 45 v 14490 60641 V 14468 62246 45 1606 v 15155 61765 a(10)p 17097 62246 V 1329 w(90)p 22328 62246 V 14468 63851 V 15155 63370 a(20)p 17097 63851 V 1329 w(7560)p 22328 63851 V 14468 65456 V 15155 64975 a(30)p 17097 65456 V 1329 w(28680)p 22328 65456 V 14468 67061 V 15155 66580 a(40)p 17097 67061 V 1329 w(119772)p 22328 67061 V 14468 68666 V 15155 68185 a(50)p 17097 68666 V 1329 w(154876)p 22328 68666 V 14468 70272 V 15155 69790 a(60)p 17097 70272 V 1329 w(297864)p 22328 70272 V 14490 70316 7860 45 v 28649 58770 10461 45 v 28627 60375 45 1606 v 29313 59894 a Fm(k)p 31906 60375 V 2604 w(n)p Fn(\()p Fm(k)g Fn(\))p 39088 60375 V 28649 60420 10461 45 v 28649 60641 V 28627 62246 45 1606 v 29313 61765 a(70)p 31906 62246 V 1980 w(968760)p 39088 62246 V 28627 63851 V 29313 63370 a(80)p 31906 63851 V 1980 w(7989168)p 39088 63851 V 28627 65456 V 29313 64975 a(90)p 31906 65456 V 1980 w(2879865)p 39088 65456 V 28627 67061 V 29313 66580 a(100)p 31906 67061 V 1330 w(87699842)p 39088 67061 V 28627 68666 V 29313 68185 a(110)p 31906 68666 V 1330 w(497975920)p 39088 68666 V 28627 70272 V 29313 69790 a(120)p 31906 70272 V 1330 w(179888904)p 39088 70272 V 28649 70316 10461 45 v Black 26475 74617 a(2)p Black eop %%Page: 3 3 3 2 bop Black 0 TeXcolorgray Black Black 2751 1424 a Fn(W)-108 b(e)352 b(shall)g(no)-36 b(w)352 b(sho)-36 b(w)352 b(ho)-36 b(w)352 b(one)f(can)h(construct)f(arbitrary)h(large)h (in)-36 b(terv)-72 b(als)352 b(free)g(of)g(Niv)-36 b(en)352 b(n)-36 b(um)g(b)36 b(ers,)800 3029 y(sa)-36 b(y)434 b(in)-36 b(terv)-72 b(als)434 b(of)h(length)e Fm(`)p Fn(,)g(and)g(thereb)-36 b(y)433 b(establish)h(the)f(pro)36 b(of)434 b(of)g(Theorem)g(1.)2751 4634 y(First,)g(giv)-36 b(en)434 b(a)g(p)36 b(ositiv)-36 b(e)434 b(in)-36 b(teger)434 b Fm(m)p Fn(,)f(set)18727 8258 y Fm(t)368 b Fn(=)h Fm(t)p Fn(\()p Fm(m)p Fn(\))f(=)25314 6385 y Fd(\271)26222 7359 y Fn(log)19 b(\(18)p Fm(m)p Fn(\))p 26222 7952 5131 54 v 27186 9169 a(log)241 b(10)31485 6385 y Fd(\272)32555 8258 y Fn(+)295 b(1)p Fm(;)16265 b Fn(\(3\))800 11955 y(where)588 b Fk(b)p Fm(y)48 b Fk(c)588 b Fn(stands)f(for)h(the)g (largest)g(in)-36 b(teger)588 b Fk(\267)631 b Fm(y)48 b Fn(,)627 b(and)587 b(let)h Fm(m)f Fn(b)36 b(e)588 b(the)f(smallest)i (p)36 b(ositiv)-36 b(e)588 b(in)-36 b(teger)800 13560 y(satisfying)22069 14785 y(9)p Fm(m)295 b Fk(\241)h Fn(9)p Fm(t)f Fk(\241)g Fn(1)p 22069 15378 6806 54 v 23786 16595 a(9)p Fm(t)g Fn(+)g(2)29376 15684 y Fm(>)369 b(`:)19474 b Fn(\(4\))800 18449 y(Then)406 b(consider)h(in)-36 b(tegers)406 b Fm(n)h Fn(whic)-36 b(h)406 b(can)h(b)36 b(e)406 b(written)g(as)h(the) f(concatenation)h(of)g(the)f(n)-36 b(um)g(b)36 b(ers)405 b(10)49748 17967 y Fh(m)50877 18449 y Fk(\241)240 b Fn(1)800 20054 y(and)433 b Fm(d)p Fn(,)h(where)f Fm(d)h Fn(is)g(a)f Fm(t)h Fn(digit)g(n)-36 b(um)g(b)36 b(er)432 b(y)-36 b(et)433 b(to)h(b)36 b(e)433 b(determined,)g(that)g(is)h(the)f Fm(m)295 b Fn(+)g Fm(t)433 b Fn(digit)h(n)-36 b(um)g(b)36 b(er)16161 22988 y Fm(n)369 b Fn(=)g Fk(h)p Fn(99)221 b Fk(\242)g(\242)g(\242)h Fn(9)19204 23479 y Fd(|)p 19802 23479 777 160 v 777 w({z)p 21775 23479 V 777 w(})20759 24608 y Fh(m)23147 22988 y Fm(;)f(d)p Fk(i)369 b Fn(=)f(\(10)28477 22439 y Fh(m)29661 22988 y Fk(\241)295 b Fn(1\))g Fk(\242)h Fn(10)34405 22439 y Fh(t)35096 22988 y Fn(+)e Fm(d:)13699 b Fn(\(5\))2751 27103 y(Then)433 b(let)h Fm(b)295 b Fk(\242)g Fn(9)p Fm(m)434 b Fn(b)36 b(e)433 b(the)g(smallest)i(m)-36 b(ultiple)433 b(of)h(9)p Fm(m)g Fn(lo)36 b(cated)434 b(in)f(the)g(in)-36 b(terv)-72 b(al)18293 30036 y Fm(I)473 b Fn(=)369 b([99)221 b Fk(\242)g(\242)g(\242)j Fn(9)21082 30528 y Fd(|)p 21680 30528 V 777 w({z)p 23653 30528 V 777 w(})22638 31657 y Fh(m)25247 30036 y Fn(00)d Fk(\242)g(\242)g(\242) i Fn(0)25247 30528 y Fd(|)p 25845 30528 V 777 w({z)p 27818 30528 V 777 w(})27049 31657 y Fh(t)29624 30036 y Fm(;)656 b Fn(99)221 b Fk(\242)g(\242)g(\242)i Fn(9)30641 30528 y Fd(|)p 31239 30528 V 777 w({z)p 33212 30528 V 777 w(})31660 31657 y Fh(m)p Fj(+)p Fh(t)34584 30036 y Fn(])p Fm(:)800 34347 y Fn(Note)474 b(that)f(at)h(least)g(t)-36 b(w)g(o)474 b(suc)-36 b(h)472 b(m)-36 b(ultiples)474 b(of)g(9)p Fm(m)g Fn(b)36 b(elong)473 b(to)h Fm(I)578 b Fn(since)473 b(the)h(length)f(of)h Fm(I)578 b Fn(is)474 b(10)48272 33865 y Fh(t)48667 34347 y Fn(,)484 b(whic)-36 b(h)800 35953 y(itself)435 b(is)e(larger)i(than)e(18)p Fm(m)g Fn(b)36 b(ecause)434 b(of)g(\()p 0 1 0 0 TeXcolorcmyk(3)p (#equation.3) [[267 393 273 405] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\).)2751 37558 y(W)-108 b(e)434 b(no)-36 b(w)433 b(coun)-36 b(t)433 b(the)g(n)-36 b(um)g(b)36 b(er)432 b(of)i(Niv)-36 b(en)434 b(n)-36 b(um)g(b)36 b(ers)432 b(b)36 b(elonging)434 b(to)g(the)f(in)-36 b(terv)-72 b(al)18518 40491 y Fm(J)492 b Fn(:=)369 b([)p Fm(b)295 b Fk(\242)g Fn(9)p Fm(m;)221 b Fn(\()p Fm(b)296 b Fn(+)f(1\))g Fk(\242)g Fn(9)p Fm(m)p Fn([)p Fk(\275)370 b Fm(I)104 b(:)800 43425 y Fn(F)-108 b(or)422 b(an)-36 b(y)422 b(p)36 b(ositiv)-36 b(e)423 b(in)-36 b(teger)422 b Fm(n)h Fn(of)f(the)g(form)h (\()p 0 1 0 0 TeXcolorcmyk(5)p (#equation.5) [[280 326 286 338] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\),)h(it)e(is)h(clear)g(that)e Fm(n)370 b Fk(2)e Fm(I)526 b Fn(and)422 b(th)-36 b(us)421 b(that)h Fm(s)p Fn(\()p Fm(n)p Fn(\))f(can)i(tak)-36 b(e)800 45030 y(at)386 b(most)g(9)p Fm(t)198 b Fn(+)g(1)384 b(v)-72 b(alues)387 b(ranging)f(from)g(9)p Fm(m)g Fn(to)g(9)p Fm(m)198 b Fn(+)g(9)p Fm(t)p Fn(.)561 b(It)386 b(follo)-36 b(ws)387 b(that)f(for)g(an)-36 b(y)386 b(\257xed)f(v)-72 b(alue)387 b(of)f Fm(s)p Fn(\()p Fm(n)p Fn(\),)800 46635 y(there)g(is)g(at)h(most) f(one)g(m)-36 b(ultiple)386 b(of)h Fm(s)p Fn(\()p Fm(n)p Fn(\))f(in)h(the)e(in)-36 b(terv)-72 b(al)387 b Fm(J)123 b Fn(,)396 b(and)386 b(therefore)g(that)g(there)g(exist)h(at)f(most)800 48240 y(9)p Fm(t)295 b Fn(+)g(1)434 b(Niv)-36 b(en)434 b(n)-36 b(um)g(b)36 b(ers)432 b(in)h Fm(J)123 b Fn(.)2751 49845 y(W)-108 b(e)336 b(ha)-36 b(v)g(e)337 b(th)-36 b(us)335 b(created)h(an)g(in)-36 b(terv)-72 b(al)336 b Fm(J)459 b Fn(of)337 b(length)f(9)p Fm(m)g Fn(con)-36 b(taining)337 b(at)f(most)g(9)p Fm(t)96 b Fn(+)g(1)337 b(Niv)-36 b(en)336 b(n)-36 b(um)g(b)36 b(ers,)800 51450 y(and)525 b(therefore,)549 b(b)-36 b(y)526 b(a)g(pigeon-hole)g(argumen) -36 b(t,)548 b(con)-36 b(taining)526 b(a)g(subin)-36 b(terv)-72 b(al)525 b(free)h(of)h(Niv)-36 b(en)526 b(n)-36 b(um)g(b)36 b(ers)800 53574 y(and)415 b(of)i(length)f(at)g(least)13461 52675 y(9)p Fm(m)295 b Fk(\241)g Fn(9)p Fm(t)h Fk(\241)f Fn(1)p 13461 53268 6806 54 v 15177 54485 a(9)p Fm(t)h Fn(+)e(2)20399 53574 y(,)420 b(whic)-36 b(h)415 b(is)i(larger)f(than)f Fm(`)h Fn(b)-36 b(y)416 b(condition)f(\()p 0 1 0 0 TeXcolorcmyk(4)p (#equation.4) [[448 235 454 247] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\),)420 b(th)-36 b(us)415 b(completing)800 55620 y(our)433 b(task)i(of)f(constructing)f(arbitrarily)h(large)h(in)-36 b(terv)-72 b(als)433 b(free)h(of)h(Niv)-36 b(en)434 b(n)-36 b(um)g(b)36 b(ers.)2751 57225 y(F)-108 b(or)413 b(example,)418 b(if)c(w)-36 b(e)413 b(require)g(gaps)h(of)g(width)e Fm(`)369 b Fn(=)g(100,)418 b(200)c(and)f(300)h(resp)36 b(ectiv)-36 b(ely)-108 b(,)418 b(free)c(of)g(Niv)-36 b(en)800 58830 y(n)g(um)g(b)36 b(ers,)436 b(here)g(is)h(a)f(table)h (sho)-36 b(wing)437 b(the)e(corresp)36 b(onding)436 b(v)-72 b(alues)437 b(of)h Fm(m)e Fn(and)g Fm(t)p Fn(,)h(as)f(w)-36 b(ell)438 b(as)e(the)g(length)800 60435 y(of)e(the)f(in)-36 b(terv)-72 b(al)434 b Fm(J)557 b Fn(whic)-36 b(h)433 b(needs)g(to)h(b)36 b(e)433 b(in)-36 b(v)g(estigated)434 b(to)g(\257nd)e(the)h(prop)36 b(er)433 b(gap.)p Black Black 18393 62287 16814 45 v 18371 63892 45 1606 v 19760 63411 a Fm(`)p 21650 63892 V 2763 w(m)p 25580 63892 V 2150 w(t)p 27559 63892 V 1418 w Fn(length)g(of)i Fm(J)p 35185 63892 V 18393 63936 16814 45 v 18393 64158 V 18371 65763 45 1606 v 19057 65281 a Fn(100)p 21650 65763 V 1655 w(416)p 25580 65763 V 1654 w(4)p 27559 65763 V 3177 w(3744)p 35185 65763 V 18371 67368 V 19057 66887 a(200)p 21650 67368 V 1330 w(1028)p 25580 67368 V 1329 w(5)p 27559 67368 V 3177 w(9252)p 35185 67368 V 18371 68973 V 19057 68492 a(300)p 21650 68973 V 1330 w(1539)p 25580 68973 V 1329 w(5)p 27559 68973 V 2852 w(13851)p 35185 68973 V 18393 69017 16814 45 v Black 26475 74617 a(3)p Black eop %%Page: 4 4 4 3 bop Black 0 TeXcolorgray Black Black 800 1424 a Fn(The)411 b(go)36 b(o)g(d)411 b(news)f(ab)36 b(out)411 b(this)f(algorithm)h(is)g (that)f(b)-36 b(y)410 b(scanning)h(relativ)-36 b(ely)412 b(small)f(in)-36 b(terv)-72 b(als)411 b(\(of)g(length)800 3029 y Fm(O)36 b Fn(\()p Fm(`)221 b Fn(log)241 b Fm(`)p Fn(\)\),)408 b(w)-36 b(e)403 b(are)f(guaran)-36 b(teed)402 b(arbitrary)g(large)i(gaps.)568 b(The)402 b(bad)g(news)g(is)h(that)f (gaps)h(this)f(large)h(are)800 4634 y(more)437 b(lik)-36 b(ely)439 b(to)e(o)36 b(ccur)437 b(m)-36 b(uc)g(h)436 b(so)36 b(oner,)438 b(as)f(is)h(sho)-36 b(wn,)438 b(for)f(instance,)h (in)f(the)f(\257rst)g(table)h(of)h(this)f(section)800 6239 y(in)458 b(the)f(case)h Fm(`)410 b Fn(=)f(100.)652 b(Nev)-36 b(ertheless,)464 b(our)458 b(algorithm)g(pro)-36 b(vides)458 b(a)g(non)f(trivial)i(b)36 b(ound)457 b(on)g Fm(n)48247 6438 y Fh(`)48687 6239 y Fn(,)464 b(whic)-36 b(h)800 7844 y(is)418 b(precisely)h(the)e(ob)72 b(ject)418 b(of)h(Theorem)f(1)g(whic)-36 b(h)418 b(no)-36 b(w)418 b(b)36 b(ecomes)418 b(easy)g(to)g(pro)-36 b(v)g(es.)574 b(Indeed,)420 b(if)e Fm(`)g Fn(is)g(large)800 9450 y(enough,)433 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)434 b(in)f(view)i(of)g(\()p 0 1 0 0 TeXcolorcmyk(4)p (#equation.4) [[226 632 231 644] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))22080 10748 y(9)p Fm(m)295 b Fk(\241)g Fn(9)p Fm(t)p 22080 11342 4532 54 v 23786 12558 a Fn(9)p Fm(t)27113 11647 y(<)369 b(`)295 b Fn(+)g(1)p Fm(;)800 14205 y Fn(so)434 b(that)22200 15810 y Fm(m)369 b(<)25220 14911 y(m)p 25220 15504 1138 54 v 25554 16721 a(t)26860 15810 y(<)f(`)295 b Fn(+)g(2)p Fm(:)800 18441 y Fn(Therefore,)434 b(using)g(\()p 0 1 0 0 TeXcolorcmyk(3)p (#equation.3) [[171 551 176 563] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))f(and)g(\()p 0 1 0 0 TeXcolorcmyk(5)p (#equation.5) [[212 551 218 563] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\),)h(w)-36 b(e)434 b(ha)-36 b(v)g(e)3500 21530 y Fm(n)4276 21729 y Fh(`)5823 21530 y Fm(<)1106 b Fn(10)9241 20981 y Fh(m)p Fj(+)p Fh(t)11570 21530 y Fn(=)369 b(10)14384 20608 y Fc(m)p Fb(+)p Fc(t)p 14384 20780 1665 40 v 15058 21328 a(t)16181 20981 y Fi(\242)p Fh(t)17207 21530 y Fn(=)g(10)19888 21054 y(\()20527 20582 y Fc(m)p 20527 20743 727 40 v 20732 21291 a(t)21386 20944 y Fj(+1)22588 21055 y Fn(\))23094 20944 y Fi(\242)p Fh(t)24120 21530 y Fm(<)f Fn(10)26800 20981 y Fj(\()p Fh(`)p Fj(+3\))p Fi(\242)p Fh(t)30145 21530 y Fm(<)h Fn(10)32826 20910 y Fj(\()p Fh(`)p Fj(+3\)\()35686 20500 y Fb(log)152 b Fc(m)p 35644 20709 2010 40 v 35644 21257 a Fb(log)f(10)37786 20910 y Fj(+2\))39778 21530 y Fm(<)369 b Fn(10)42459 20910 y Fj(\()p Fh(`)p Fj(+3\)\()45276 20463 y Fb(log)11 b(\()p Fc(`)p Fb(+2\))p 45277 20709 3068 40 v 45806 21257 a(log)151 b(10)48477 20910 y Fj(+2\))5823 23467 y Fn(=)1106 b Fm(e)8544 22918 y Fj(log)15 b(\()p Fh(`)p Fj(+2\))12078 22606 y Fc(`)p Fb(+3)13855 23467 y Fk(\242)296 b Fn(10)15820 22918 y Fj(2\()p Fh(`)p Fj(+3\))19033 23467 y Fn(=)369 b(\()p Fm(`)295 b Fn(+)g(2\))24224 22918 y Fh(`)p Fj(+3)26161 23467 y Fk(\242)g Fn(10)28125 22918 y Fj(2\()p Fh(`)p Fj(+3\))31339 23467 y Fn(=)369 b(\(100\()p Fm(`)295 b Fn(+)g(2\)\))39492 22821 y Fh(`)p Fj(+3)41356 23467 y Fm(;)800 26152 y Fn(whic)-36 b(h)433 b(pro)-36 b(v)g(es)434 b(Theorem)g(1.)800 30802 y Fo(4)2152 b(Small)717 b(gaps)f(b)60 b(et)-60 b(w)g(een)717 b(consecutiv)-60 b(e)716 b(Niv)-60 b(en)716 b(n)-60 b(um)g(b)60 b(ers)800 33722 y Fn(It)525 b(follo)-36 b(ws)528 b(from)e(\()p 0 1 0 0 TeXcolorcmyk(1)p (#equation.1) [[165 414 171 426] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))f(that)g(the)g(sum)g(of)h(the)e(recipro)36 b(cals)527 b(of)f(the)f(Niv)-36 b(en)525 b(n)-36 b(um)g(b)36 b(ers)524 b(div)-36 b(erges,)549 b(and)525 b(in)800 35327 y(fact)434 b(that)19042 35931 y Fd(X)19194 38611 y Fc(n)p Fa(\267)p Fc(x)17031 39363 y(n)g Fb(Niv)-22 b(en)277 b(n)-22 b(um)g(b)22 b(er)23521 36294 y Fn(1)p 23458 36887 777 54 v 23458 38104 a Fm(n)24736 37193 y Fn(=)369 b(\()p Fm(c)295 b Fn(+)f Fm(o)p Fn(\(1\)\))221 b(log)241 b(log)g Fm(x;)800 41217 y Fn(where)434 b Fm(c)f Fn(is)h(giv)-36 b(en)434 b(b)-36 b(y)433 b(\()p 0 1 0 0 TeXcolorcmyk(2)p (#equation.2) [[185 346 191 358] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\).)2751 43375 y(W)-108 b(e)483 b(shall)h(call)g Fl(twin)510 b(Niven)f(numb)-66 b(ers)482 b Fn(those)h(pairs)h(\()p Fm(n;)221 b(n)330 b Fn(+)f(1\),)496 b(suc)-36 b(h)482 b(as)i(\(20,21\))g(and)f(\(152,153\),)800 44980 y(b)36 b(oth)413 b(mem)-36 b(b)36 b(ers)412 b(of)i(whic)-36 b(h)413 b(are)g(Niv)-36 b(en)413 b(n)-36 b(um)g(b)36 b(ers.)570 b(W)-108 b(e)413 b(can)g(sho)-36 b(w)413 b(that)g(the)g(sum) f(of)i(the)e(recipro)36 b(cals)414 b(of)800 46585 y(t)-36 b(win)426 b(Niv)-36 b(en)427 b(n)-36 b(um)g(b)36 b(ers)425 b(con)-36 b(v)g(erges.)576 b(In)427 b(fact,)h(w)-36 b(e)427 b(can)f(establish)h(that)f(if)h Fm(T)607 b Fn(stands)426 b(for)h(the)f(set)g(of)h(t)-36 b(win)800 48190 y(Niv)g(en)434 b(n)-36 b(um)g(b)36 b(ers)432 b(\()p Fm(n;)221 b(n)296 b Fn(+)f(1\))434 b(and)16835 50876 y Fm(T)181 b Fn(\()p Fm(x)p Fn(\))369 b(:=)g(#)p Fk(f)p Fm(n)g Fk(\267)g Fm(x)g Fn(:)h(\()p Fm(n;)221 b(n)296 b Fn(+)f(1\))369 b Fk(2)g Fm(T)181 b Fk(g)p Fm(;)800 53562 y Fn(then)19799 55599 y Fm(T)g Fn(\()p Fm(x)p Fn(\))369 b(=)g Fm(O)25494 53726 y Fd(\265)26605 54700 y Fm(x)221 b Fn(log)241 b(log)f Fm(x)p 26605 55293 5503 54 v 27267 56510 a Fn(\(log)g Fm(x)p Fn(\))30919 56126 y Fj(2)32240 53726 y Fd(\266)33439 55599 y Fm(;)800 58507 y Fn(whic)-36 b(h)433 b(is)h(precisely)h(the)e (statemen)-36 b(t)432 b(of)j(Theorem)e(2)h(and)f(whic)-36 b(h)434 b(implies)g(that)22984 60450 y Fd(X)23687 62949 y Fc(n)21899 63702 y Fb(\()p Fc(n;n)p Fb(+1\))p Fa(2)p Fc(T)26537 60813 y Fn(1)p 26474 61406 777 54 v 26474 62623 a Fm(n)27752 61712 y(<)369 b Fn(+)p Fk(1)p Fm(:)19304 b Fn(\(6\))800 66115 y(Indeed,)433 b(using)g(Theorem)h(2,)g(one)g(can)f (write)h(that)11388 68228 y Fd(X)11540 70909 y Fc(n)p Fa(\267)p Fc(x)10304 71661 y Fb(\()p Fc(n;n)p Fb(+1\))p Fa(2)p Fc(T)14942 68592 y Fn(1)p 14878 69185 V 14878 70401 a Fm(n)16157 69490 y Fn(=)18755 68228 y Fd(X)19458 70728 y Fc(n)17670 71481 y Fb(\()p Fc(n;n)p Fb(+1\))p Fa(2)p Fc(T)22308 68592 y Fn(1)p 22245 69185 V 22245 70401 a Fm(n)23449 69490 y Fk(\241)25995 68228 y Fd(X)26147 70833 y Fc(n>x)24911 71585 y Fb(\()p Fc(n;n)p Fb(+1\))p Fa(2)p Fc(T)29548 68592 y Fn(1)p 29485 69185 V 29485 70401 a Fm(n)30764 69490 y Fn(=)368 b Fm(c)32704 69689 y Fj(2)33525 69490 y Fn(+)295 b Fm(O)36083 67617 y Fd(\265)37194 68592 y Fn(log)240 b(log)h Fm(x)p 37194 69185 4542 54 v 38145 70401 a Fn(log)f Fm(x)41869 67617 y Fd(\266)43068 69490 y Fm(;)p Black 26475 74617 a Fn(4)p Black eop %%Page: 5 5 5 4 bop Black 0 TeXcolorgray Black Black 800 1424 a Fn(where)800 4618 y Fm(c)1360 4817 y Fj(2)2255 4618 y Fn(=)368 b(1)73 b(+)5576 3719 y(1)p 5576 4312 651 54 v 5576 5529 a(2)6431 4618 y(+)7649 3719 y(1)p 7649 4312 V 7649 5529 a(3)8504 4618 y(+)g Fk(\242)221 b(\242)g(\242)73 b Fn(+)12429 3719 y(1)p 12429 4312 V 12429 5529 a(9)13284 4618 y(+)14827 3719 y(1)p 14502 4312 1301 54 v 14502 5529 a(20)16008 4618 y(+)17550 3719 y(1)p 17226 4312 V 17226 5529 a(80)18731 4618 y(+)20599 3719 y(1)p 19949 4312 1951 54 v 19949 5529 a(110)22105 4618 y(+)23972 3719 y(1)p 23323 4312 V 23323 5529 a(111)25479 4618 y(+)27346 3719 y(1)p 26697 4312 V 26697 5529 a(132)28852 4618 y(+)30720 3719 y(1)p 30070 4312 V 30070 5529 a(152)32226 4618 y(+)34094 3719 y(1)p 33444 4312 V 33444 5529 a(200)35600 4618 y(+)37467 3719 y(1)p 36818 4312 V 36818 5529 a(209)38974 4618 y(+)40841 3719 y(1)p 40192 4312 V 40192 5529 a(224)42347 4618 y(+)44215 3719 y(1)p 43565 4312 V 43565 5529 a(399)45721 4618 y(+)g Fk(\242)221 b(\242)g(\242)369 b(\274)g Fn(3)p Fm(:)p Fn(07)p Fm(;)800 7964 y Fn(whic)-36 b(h)433 b(in)h(particular)f (implies)h(\()p 0 1 0 0 TeXcolorcmyk(6)p (#equation.6) [[225 645 230 657] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\).)2751 10676 y(Before)476 b(w)-36 b(e)476 b(pro)-36 b(v)g(e)475 b(Theorem)h(2,)487 b(note)475 b(that)g(it)g(is)h(easy)g(to) g(sho)-36 b(w)476 b(that)f Fm(T)656 b Fn(is)476 b(an)f(in\257nite)g (set.)704 b(This)800 12281 y(can)357 b(b)36 b(e)357 b(established)f(b) -36 b(y)357 b(observing)g(that)g(2)g(divides)g(2)138 b Fk(\242)g Fn(10)29510 11799 y Fh(k)30438 12281 y Fn(and)356 b(3)h(divides)g(2)138 b Fk(\242)g Fn(10)40794 11799 y Fh(k)41504 12281 y Fn(+)g(1)357 b(for)g(eac)-36 b(h)357 b(p)36 b(ositiv)-36 b(e)800 13887 y(in)g(teger)433 b Fm(k)45 b Fn(.)2751 15492 y(On)433 b(the)g(other)g(hand,)g(using)g(a)h (computer,)f(one)h(can)f(obtain)h(the)f(follo)-36 b(wing)436 b(table:)p Black Black 10422 17344 7952 45 v 10400 18949 45 1606 v 11086 18468 a Fm(x)p 14330 18949 V 3191 w(T)181 b Fn(\()p Fm(x)p Fn(\))p 18352 18949 V 10422 18993 7952 45 v 10400 20598 45 1606 v 11086 20117 a(10)p 14330 20598 V 2630 w(9)p 18352 20598 V 10400 22204 V 11086 21722 a(100)p 14330 22204 V 1980 w(11)p 18352 22204 V 10400 23809 V 11086 23327 a(1000)p 14330 23809 V 1330 w(32)p 18352 23809 V 10422 23853 7952 45 v 22391 17344 7302 45 v 22369 18949 45 1606 v 23055 18468 a Fm(x)p 25523 18949 V 2416 w(T)g Fn(\()p Fm(x)p Fn(\))p 29670 18949 V 22391 18993 7302 45 v 22369 20598 45 1606 v 23055 20117 a(10)24355 19635 y Fj(4)p 25523 20598 V 26210 20117 a Fn(145)p 29670 20598 V 22369 22204 V 23055 21722 a(10)24355 21240 y Fj(5)p 25523 22204 V 26210 21722 a Fn(904)p 29670 22204 V 22369 23809 V 23055 23327 a(10)24355 22845 y Fj(6)p 25523 23809 V 26210 23327 a Fn(6)217 b(191)p 29670 23809 V 22391 23853 7302 45 v 33709 17344 9470 45 v 33687 18949 45 1606 v 34373 18468 a Fm(x)p 36841 18949 V 2416 w(T)181 b Fn(\()p Fm(x)p Fn(\))p 43156 18949 V 33709 18993 9470 45 v 33687 20598 45 1606 v 34373 20117 a(10)35673 19635 y Fj(7)p 36841 20598 V 37528 20117 a Fn(44)217 b(742)p 43156 20598 V 33687 22204 V 34373 21722 a(10)35673 21240 y Fj(8)p 36841 22204 V 37528 21722 a Fn(332)g(037)p 43156 22204 V 33687 23809 V 34373 23327 a(10)35673 22845 y Fj(9)p 36841 23809 V 37528 23327 a Fn(2)g(551)g(917)p 43156 23809 V 33709 23853 9470 45 v 2751 27506 a(W)-108 b(e)434 b(no)-36 b(w)433 b(mo)-36 b(v)g(e)434 b(to)g(pro)-36 b(v)g(e)434 b(\()p 0 1 0 0 TeXcolorcmyk(6)p (#equation.6) [[222 469 228 481] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\).)2751 29111 y(Let)f(\()p Fm(n;)221 b(n)297 b Fn(+)d(1\))370 b Fk(2)e Fm(T)615 b Fn(and)433 b(assume)g(that)g Fm(n)h Fn(is)g(a)g Fm(K)95 b Fn(-digit)434 b(n)-36 b(um)g(b)36 b(er,)432 b(with)h Fm(K)464 b Fk(\270)370 b Fn(2.)578 b(W)-108 b(rite)434 b Fm(n)g Fn(as)17192 32044 y Fm(n)370 b Fn(=)e Fm(a)295 b Fk(\242)h Fn(10)22661 31496 y Fh(R)5 b Fj(+)p Fh(B)25211 32044 y Fn(+)295 b Fm(b)g Fk(\242)g Fn(10)29330 31496 y Fh(R)30396 32044 y Fn(+)f(10)33002 31496 y Fh(R)34068 32044 y Fk(\241)h Fn(1)p Fm(;)14731 b Fn(\(7\))800 34978 y(where)437 b Fm(a)g Fn(is)h(a)f Fm(K)393 b Fk(\241)298 b Fm(B)365 b Fk(\241)298 b Fm(R)448 b Fn(digit)437 b(n)-36 b(um)g(b)36 b(er)436 b(and)h Fm(b)g Fn(is)h(a)f Fm(B)505 b Fn(digit)437 b(n)-36 b(um)g(b)36 b(er)436 b(not)h(ending)f(with)i(a)f(9.)590 b(Here)800 36583 y Fm(B)537 b Fn(and)469 b Fm(R)480 b Fn(are)470 b(non)f(negativ)-36 b(e)470 b(in)-36 b(tegers;)488 b(later,)479 b(w)-36 b(e)470 b(shall)g(set)f Fm(B)537 b Fn(as)470 b(a)g(function)f(of)h Fm(K)95 b Fn(.)686 b(No)-36 b(w,)480 b(observ)-36 b(e)800 38188 y(that)455 b Fm(s)p Fn(\()p Fm(n)p Fn(\))405 b(=)g Fm(s)p Fn(\()p Fm(a)p Fn(\))309 b(+)g Fm(s)p Fn(\()p Fm(b)p Fn(\))g(+)h(9)p Fm(R)466 b Fn(and)454 b(that)h Fm(s)p Fn(\()p Fm(n)310 b Fn(+)f(1\))406 b(=)f Fm(s)p Fn(\()p Fm(a)p Fn(\))308 b(+)i Fm(s)p Fn(\()p Fm(b)p Fn(\))f(+)g(1.)643 b(Since)455 b(\()p Fm(n;)221 b(n)311 b Fn(+)e(1\))406 b Fk(2)f Fm(T)181 b Fn(,)461 b(w)-36 b(e)800 39793 y(ha)g(v)g(e)4750 42727 y Fm(s)p Fn(\()p Fm(n)p Fn(\))p Fk(j)p Fm(a)295 b Fk(\242)g Fn(10)10462 42178 y Fh(R)5 b Fj(+)p Fh(B)13013 42727 y Fn(+)295 b Fm(b)g Fk(\242)g Fn(10)17132 42178 y Fh(R)18197 42727 y Fn(+)g(10)20804 42178 y Fh(R)21870 42727 y Fk(\241)g Fn(1)1734 b(and)g Fm(s)p Fn(\()p Fm(n)295 b Fn(+)g(1\))p Fk(j)p Fm(a)g Fk(\242)g Fn(10)37376 42178 y Fh(R)5 b Fj(+)p Fh(B)39927 42727 y Fn(+)295 b Fm(b)g Fk(\242)g Fn(10)44046 42178 y Fh(R)45111 42727 y Fn(+)g(10)47718 42178 y Fh(R)48488 42727 y Fm(:)800 45660 y Fn(W)-108 b(e)434 b(therefore)f(ha)-36 b(v)g(e)12898 47163 y Fd(\275)14448 48223 y Fm(b)295 b Fk(\242)g Fn(10)17260 47741 y Fh(R)18399 48223 y Fk(\264)369 b(\241)p Fm(a)296 b Fk(\242)f Fn(10)23777 47741 y Fh(R)5 b Fj(+)p Fh(B)26327 48223 y Fk(\241)296 b Fn(10)28956 47741 y Fh(R)30021 48223 y Fn(+)f(1)1328 b(\(mo)36 b(d)443 b Fm(s)p Fn(\()p Fm(n)p Fn(\)\))p Fm(;)14448 49828 y(b)295 b Fk(\242)g Fn(10)17260 49346 y Fh(R)18399 49828 y Fk(\264)369 b(\241)p Fm(a)296 b Fk(\242)f Fn(10)23777 49346 y Fh(R)5 b Fj(+)p Fh(B)26327 49828 y Fk(\241)296 b Fn(10)28956 49346 y Fh(R)31054 49828 y Fn(\(mo)36 b(d)443 b Fm(s)p Fn(\()p Fm(n)295 b Fn(+)g(1\)\))p Fm(:)800 52745 y Fn(Since)500 b Fm(s)p Fn(\()p Fm(n)p Fn(\))h(and)f Fm(s)p Fn(\()p Fm(n)341 b Fn(+)f(1\))501 b(are)g(relativ)-36 b(ely)502 b(prime,)518 b(w)-36 b(e)501 b(can)f(use)h(the)f(Chinese)g (Remainder)h(Theorem)800 54350 y(to)451 b(state)g(that)f(there)g (exists)h(one)g(\(and)f(only)h(one\))f(non)h(negativ)-36 b(e)451 b(in)-36 b(teger)451 b Fm(m)397 b(<)h(s)p Fn(\()p Fm(n)p Fn(\))p Fm(s)p Fn(\()p Fm(n)306 b Fn(+)h(1\),)455 b(where)800 55955 y Fm(m)369 b Fn(=)f Fm(m)p Fn(\()p Fm(a;)221 b(R)11 b(;)221 b(B)67 b(;)221 b(s)p Fn(\()p Fm(n)p Fn(\))p Fm(;)g(s)p Fn(\()p Fm(n)297 b Fn(+)e(1\)\),)434 b(suc)-36 b(h)433 b(that)17209 58888 y Fm(b)295 b Fk(\242)g Fn(10)20021 58340 y Fh(R)21161 58888 y Fk(\264)369 b Fm(m)1328 b Fn(\(mo)36 b(d)442 b Fm(s)p Fn(\()p Fm(n)p Fn(\))p Fm(s)p Fn(\()p Fm(n)295 b Fn(+)g(1\)\))p Fm(:)14747 b Fn(\(8\))800 61822 y(Observing)401 b(that)g(\()p Fm(n;)221 b Fn(10)12903 61340 y Fh(R)13674 61822 y Fn(\))369 b(=)g(1)401 b(and)g Fm(s)p Fn(\()p Fm(n)p Fn(\))p Fk(j)p Fm(n)p Fn(,)409 b(w)-36 b(e)401 b(ha)-36 b(v)g(e)402 b(that)f(\()p Fm(s)p Fn(\()p Fm(n)p Fn(\))p Fm(;)221 b Fn(10)36218 61340 y Fh(R)36988 61822 y Fn(\))369 b(=)g(1.)567 b(Hence)402 b(it)f(follo)-36 b(ws)404 b(from)800 64159 y(\()p 0 1 0 0 TeXcolorcmyk(8)p (#equation.8) [[84 140 90 152] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))527 b(that)g(there)g(exists)h(one)f(\(and)g(only)h(one\))f (non)g(negativ)-36 b(e)528 b(in)-36 b(teger)527 b Fm(m)37921 63611 y Fi(0)38600 64159 y Fm(<)40746 63261 y(s)p Fn(\()p Fm(n)p Fn(\))p Fm(s)p Fn(\()p Fm(n)294 b Fn(+)h(1\))p 40114 63854 8318 54 v 40114 65070 a(\()p Fm(s)p Fn(\()p Fm(n)g Fn(+)f(1\))p Fm(;)221 b Fn(10)47154 64687 y Fh(R)47925 65070 y Fn(\))48564 64159 y(,)551 b(where)800 66510 y Fm(m)1938 66027 y Fi(0)2617 66510 y Fn(=)369 b Fm(m)5136 66027 y Fi(0)5446 66510 y Fn(\()p Fm(a;)221 b(R)11 b(;)221 b(B)67 b(;)221 b(s)p Fn(\()p Fm(n)p Fn(\))p Fm(;)g(s)p Fn(\()p Fm(n)297 b Fn(+)e(1\)\),)434 b(satisfying)h(the)e(congruence) 17804 70176 y Fm(b)369 b Fk(\264)g Fm(m)21266 69627 y Fi(0)22905 70176 y Fn(\(mo)36 b(d)27111 69277 y Fm(s)p Fn(\()p Fm(n)p Fn(\))p Fm(s)p Fn(\()p Fm(n)295 b Fn(+)f(1\))p 26479 69870 V 26479 71087 a(\()p Fm(s)p Fn(\()p Fm(n)h Fn(+)g(1\))p Fm(;)221 b Fn(10)33520 70703 y Fh(R)34291 71087 y Fn(\))34929 70176 y(\))p Fm(:)15342 b Fn(\(9\))p Black 26475 74617 a(5)p Black eop %%Page: 6 6 6 5 bop Black 0 TeXcolorgray Black Black 2751 1424 a Fn(Assume)627 b(for)h(no)-36 b(w)627 b(that)g(the)g(in)-36 b(tegers)627 b Fm(K)95 b Fn(,)676 b Fm(R)11 b Fn(,)676 b Fm(a)p Fn(,)g Fm(B)694 b Fn(and)627 b Fm(s)p Fn(\()p Fm(b)p Fn(\))g(are)g(all)i(\257xed.)1159 b(Since)627 b Fm(s)p Fn(\()p Fm(n)p Fn(\))698 b(=)800 3029 y Fm(s)p Fn(\()p Fm(a)p Fn(\))271 b(+)h Fm(s)p Fn(\()p Fm(b)p Fn(\))f(+)h(9)p Fm(R)434 b Fn(and)421 b Fm(s)p Fn(\()p Fm(n)272 b Fn(+)g(1\))369 b(=)g Fm(s)p Fn(\()p Fm(a)p Fn(\))271 b(+)h Fm(s)p Fn(\()p Fm(b)p Fn(\))f(+)h(1,)425 b(the)d(n)-36 b(um)g(b)36 b(er)421 b(of)i Fm(b)p Fn('s)f(satisfying)i (\()p 0 1 0 0 TeXcolorcmyk(9)p (#equation.9) [[476 690 482 702] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))e(is)h(less)g(than)20063 5888 y(10)21363 5405 y Fh(B)22606 5888 y Fn(\()p Fm(s)p Fn(\()p Fm(n)295 b Fn(+)g(1\))p Fm(;)221 b Fn(10)29647 5405 y Fh(R)30418 5888 y Fn(\))p 20063 6481 10861 54 v 21966 7697 a Fm(s)p Fn(\()p Fm(n)p Fn(\))p Fm(s)p Fn(\()p Fm(n)295 b Fn(+)g(1\))31351 6786 y(+)g(1)p Fm(:)800 10465 y Fn(No)-36 b(w,)430 b(as)f(the)f(v)-72 b(alue)429 b(of)h Fm(s)p Fn(\()p Fm(b)p Fn(\))e(v)-72 b(aries)429 b(from)g(0)g(to)f(9)p Fm(B)353 b Fk(\241)285 b Fn(1,)430 b(the)e(n)-36 b(um)g(b)36 b(er)427 b(\241)369 b(=)f(\241\()p Fm(K)23 b(;)221 b(R)11 b(;)221 b(a;)g(B)67 b Fn(\))430 b(of)f(suitable)800 12070 y Fm(b)p Fn('s)434 b(\()f(that)g(is,)i(those)e(satisfying)i(\()p 0 1 0 0 TeXcolorcmyk(9)p (#equation.9) [[232 608 238 620] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\),)f(for)g Fm(K)95 b Fn(,)434 b Fm(R)11 b Fn(,)434 b Fm(a)f Fn(and)g Fm(B)501 b Fn(\257xed,)433 b(satis\257es)12848 15827 y(\241)369 b Fk(\267)17437 14565 y Fd(X)15432 17443 y Fj(0)p Fi(\267)p Fh(s)p Fj(\()p Fh(b)p Fj(\))p Fi(\267)p Fj(9)p Fh(B)44 b Fi(\241)p Fj(1)21583 13954 y Fd(\265)22694 14928 y Fn(10)23994 14446 y Fh(B)24803 14928 y Fn(\()p Fm(s)p Fn(\()p Fm(b)p Fn(\))294 b(+)h Fm(s)p Fn(\()p Fm(a)p Fn(\))f(+)h(1)p Fm(;)221 b Fn(10)35529 14446 y Fh(R)36300 14928 y Fn(\))p 22694 15521 14113 54 v 23678 16738 a(\()p Fm(s)p Fn(\()p Fm(a)p Fn(\))295 b(+)f(9)p Fm(R)11 b Fn(\)\()p Fm(s)p Fn(\()p Fm(a)p Fn(\))295 b(+)f(1\))37234 15827 y(+)h(1)39191 13954 y Fd(\266)40391 15827 y Fm(:)800 20233 y Fn(W)-108 b(e)434 b(then)e(ha)-36 b(v)g(e,)434 b(letting)g Fm(k)414 b Fn(=)369 b Fm(s)p Fn(\()p Fm(b)p Fn(\),)11877 23989 y(\241)f Fk(\267)14679 22728 y Fd(X)14460 25647 y Fh(d)p Fi(j)p Fj(10)16145 25395 y Fc(R)19711 22728 y Fd(X)18552 25408 y Fb(0)p Fa(\267)p Fc(k)14 b Fa(\267)p Fb(9)p Fc(B)32 b Fa(\241)p Fb(1)17171 26306 y(\()p Fc(k)14 b Fb(+)p Fc(s)p Fb(\()p Fc(a)p Fb(\)+1)p Fc(;)p Fb(10)22132 26115 y Fc(R)22802 26306 y Fb(\)=)p Fc(d)24524 22116 y Fd(\265)29834 23091 y Fn(10)31134 22609 y Fh(B)32239 23091 y Fk(\242)295 b Fm(d)p 25635 23684 12143 54 v 25635 24901 a Fn(\()p Fm(s)p Fn(\()p Fm(a)p Fn(\))f(+)h(9)p Fm(R)11 b Fn(\)\()p Fm(s)p Fn(\()p Fm(a)p Fn(\))294 b(+)h(1\))38206 23989 y(+)g(1)40163 22116 y Fd(\266)41362 23989 y Fm(:)800 28966 y Fn(It)434 b(follo)-36 b(ws)435 b(that)5241 32391 y(\241)1106 b Fk(\267)9520 31129 y Fd(X)9301 34049 y Fh(d)p Fi(j)p Fj(10)10986 33797 y Fc(R)11879 30518 y Fd(\265)12989 31493 y Fn(9)p Fm(B)p 12989 32086 1706 54 v 13504 33302 a(d)15123 32391 y Fn(+)295 b(1)17080 30518 y Fd(\266)221 b(\265)23589 31493 y Fn(10)24889 31011 y Fh(B)25994 31493 y Fk(\242)295 b Fm(d)p 19390 32086 12143 54 v 19390 33302 a Fn(\()p Fm(s)p Fn(\()p Fm(a)p Fn(\))f(+)h(9)p Fm(R)11 b Fn(\)\()p Fm(s)p Fn(\()p Fm(a)p Fn(\))294 b(+)h(1\))31961 32391 y(+)g(1)33918 30518 y Fd(\266)7160 36832 y Fk(\267)10243 35933 y Fn(\()p Fm(R)306 b Fn(+)295 b(1\))14508 35451 y Fj(2)15329 35933 y Fk(\242)g Fn(9)p Fm(B)362 b Fk(\242)296 b Fn(10)19958 35451 y Fh(B)p 9433 36527 V 9433 37743 a Fn(\()p Fm(s)p Fn(\()p Fm(a)p Fn(\))e(+)h(9)p Fm(R)11 b Fn(\)\()p Fm(s)p Fn(\()p Fm(a)p Fn(\))294 b(+)h(1\))22004 36832 y(+)23444 35933 y(5)p 23444 36527 651 54 v 23444 37743 a(2)27862 35933 y(10)29162 35451 y Fh(R)30227 35933 y Fk(\242)h Fn(10)32192 35451 y Fh(B)p 24360 36527 12143 54 v 24360 37743 a Fn(\()p Fm(s)p Fn(\()p Fm(a)p Fn(\))e(+)h(9)p Fm(R)11 b Fn(\)\()p Fm(s)p Fn(\()p Fm(a)p Fn(\))294 b(+)h(1\))36931 36832 y(+)g(\(9)p Fm(B)362 b Fn(+)295 b(1\)\()p Fm(R)306 b Fn(+)295 b(1\))47472 36284 y Fj(2)47998 36832 y Fm(:)800 40602 y Fn(Set)413 b Fm(B)436 b Fn(:=)369 b Fk(b)p Fn(log)240 b Fm(K)95 b Fk(c)p Fn(.)572 b(If)414 b Fm(R)380 b(>)369 b Fn(2)221 b(log)241 b Fm(K)95 b Fn(,)417 b(the)c(n)-36 b(um)g(b)36 b(er)411 b(of)j Fm(n)p Fn('s)g(satisfying)h(\()p 0 1 0 0 TeXcolorcmyk(7)p (#equation.7) [[404 352 410 364] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))e(is)g Fm(O)36 b Fn(\(10)42616 40120 y Fh(K)43528 40602 y Fm(=K)45380 40120 y Fj(2)45906 40602 y Fn(\).)571 b(Therefore)800 42207 y(w)-36 b(e)440 b(ma)-36 b(y)440 b(also)g(assume)f(that)g Fm(R)390 b Fk(\267)379 b Fn(2)221 b(log)242 b Fm(K)95 b Fn(.)595 b(If)440 b Fm(s)p Fn(\()p Fm(a)p Fn(\))378 b Fk(\267)i Fm(K)23 b(=)p Fn(2,)441 b(one)e(can)h(sho)-36 b(w)439 b(that)g(the)g(n)-36 b(um)g(b)36 b(er)438 b(of)i Fm(n)p Fn('s)800 43812 y(satisfying)459 b(\()p 0 1 0 0 TeXcolorcmyk(7)p (#equation.7) [[136 323 142 335] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))e(is)g Fm(O)36 b Fn(\(10)12905 43330 y Fh(K)63 b(\264)14321 43812 y Fn(\))457 b(for)h(some)g(p)36 b(ositiv)-36 b(e)458 b Fm(\264)e(<)409 b Fn(1.)650 b(Hence)457 b(w)-36 b(e)457 b(can)g(mak)-36 b(e)458 b(the)f(assumption)g(that)800 45417 y Fm(s)p Fn(\()p Fm(a)p Fn(\))368 b Fm(>)h(K)23 b(=)p Fn(2.)578 b(W)-108 b(e)434 b(then)f(get)g(that,)h(if)g Fm(B)i Fk(\270)369 b Fn(1,)16461 49174 y(\241)f Fk(\267)19177 48275 y Fn(5)296 b Fk(\242)f Fn(10)22087 47793 y Fh(B)22896 48275 y Fn(\(\()p Fm(R)306 b Fn(+)295 b(1\))27667 47793 y Fj(2)28488 48275 y Fk(\242)g Fn(9)p Fm(B)363 b Fn(+)295 b(2)g Fk(\242)g Fn(10)35369 47793 y Fh(R)36140 48275 y Fn(\))p 19177 48868 17469 54 v 27047 50085 a Fm(K)28249 49701 y Fj(2)36778 49174 y Fm(:)800 52447 y Fn(F)-108 b(rom)433 b(these)g(observ)-72 b(ations,)435 b(it)e(follo)-36 b(ws)436 b(that)9907 56130 y Fm(T)181 b Fn(\(10)12656 55581 y Fh(K)13568 56130 y Fn(\))1254 b Fk(\267)18807 54868 y Fd(X)17616 57697 y Fh(R)5 b Fi(\267)p Fj(2)157 b(log)171 b Fh(K)22943 55231 y Fn(10)24243 54749 y Fh(K)p 22271 55824 3556 54 v 22271 57041 a Fn(10)23571 56657 y Fh(R)5 b Fj(+)p Fh(B)26092 55231 y Fn(5)296 b Fk(\242)f Fn(10)29002 54749 y Fh(B)p 26092 55824 3720 54 v 27088 57041 a Fm(K)28290 56657 y Fj(2)29944 56130 y Fn(\(\()p Fm(R)306 b Fn(+)295 b(1\))34715 55581 y Fj(2)35536 56130 y Fk(\242)g Fn(9)p Fm(B)363 b Fn(+)295 b(2)g Fk(\242)g Fn(10)42417 55581 y Fh(R)43188 56130 y Fn(\))15339 60430 y(=)1265 b(5)18399 59532 y(10)19699 59050 y Fh(K)p 18399 60125 2213 54 v 18641 61342 a Fm(K)19843 60958 y Fj(2)22157 59168 y Fd(X)20965 61997 y Fh(R)5 b Fi(\267)p Fj(2)157 b(log)172 b Fh(K)25488 58557 y Fd(\265)26466 60430 y Fn(9)p Fm(B)28304 59532 y Fn(\()p Fm(R)306 b Fn(+)295 b(1\))32569 59050 y Fj(2)p 28304 60125 4791 54 v 29664 61342 a Fn(10)30964 60958 y Fh(R)33523 60430 y Fn(+)g(2)35480 58557 y Fd(\266)15180 64731 y Fk(\277)17749 63832 y Fn(10)19049 63350 y Fh(K)20182 63832 y Fn(log)240 b Fm(K)p 17749 64425 5537 54 v 19653 65642 a(K)20855 65258 y Fj(2)23418 64731 y Fm(:)800 69007 y Fn(Th)-36 b(us,)358 b(giv)-36 b(en)340 b(an)f(arbitrary)h(large)g Fm(x)p Fn(,)358 b(if)340 b(w)-36 b(e)340 b(c)-36 b(ho)36 b(ose)339 b Fm(K)464 b Fn(=)29274 67134 y Fd(\271)30462 68109 y Fn(log)241 b Fm(x)p 30182 68702 3203 54 v 30182 69918 a Fn(log)f(10)33517 67134 y Fd(\272)34292 69007 y Fn(,)358 b(Theorem)339 b(2)h(follo)-36 b(ws)341 b(immediately)-108 b(.)p Black 26475 74617 a(6)p Black eop %%Page: 7 7 7 6 bop Black 0 TeXcolorgray Black Black 800 1424 a Fo(5)2152 b(Final)716 b(remarks)800 4345 y Fn(Most)545 b(lik)-36 b(ely)-108 b(,)575 b(one)546 b(can)f(remo)-36 b(v)g(e)546 b(the)f(log)240 b(log)h Fm(x)545 b Fn(on)g(the)g(righ)-36 b(t)545 b(hand)g(side)g(of)h(\()p 0 1 0 0 TeXcolorcmyk(6)p (#equation.6) [[439 678 444 690] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\),)574 b(but)544 b(w)-36 b(e)546 b(could)f(not)800 5950 y(pro)-36 b(v)g(e)434 b(this.)2751 7555 y(On)394 b(the)h(other)f(hand,)403 b(it)395 b(has)g(b)36 b(een)394 b(sho)-36 b(wn)395 b(b)-36 b(y)395 b(Grundman)e([)p 0 1 0 0 TeXcolorcmyk(5)p (#cite.kn:g) [[371 649 376 661] [1 1 1 [3 3]] [0 0 1]] pdfm Black(])j(that,)402 b(giv)-36 b(en)396 b(an)f(in)-36 b(teger)395 b Fm(`)p Fn(,)403 b(2)369 b Fk(\267)g Fm(`)g Fk(\267)800 9160 y Fn(20,)389 b(one)377 b(could)f(\257nd)g(an)g (in\257nite)g(n)-36 b(um)g(b)36 b(er)375 b(of)i(Niv)-36 b(en)377 b(n)-36 b(um)g(b)36 b(ers)376 b Fm(n)h Fn(suc)-36 b(h)375 b(that)i Fm(n)179 b Fn(+)g(1)p Fm(;)221 b(n)179 b Fn(+)g(2)p Fm(;)221 b(:)g(:)g(:)j(;)d(n)179 b Fn(+)g Fm(`)g Fk(\241)g Fn(1)800 10765 y(are)410 b(also)h(Niv)-36 b(en)410 b(n)-36 b(um)g(b)36 b(ers)408 b(\(and)h(that)g(there)g(do)36 b(es)410 b(not)f(exist)h(21)h(consecutiv)-36 b(e)410 b(n)-36 b(um)g(b)36 b(ers)408 b(whic)-36 b(h)409 b(are)h(all)800 12370 y(Niv)-36 b(en)536 b(n)-36 b(um)g(b)36 b(ers\).)884 b(F)-108 b(or)536 b(eac)-36 b(h)536 b(in)-36 b(teger)535 b Fm(`)543 b Fk(2)g Fn([2)p Fm(;)221 b Fn(20],)564 b(if)537 b(w)-36 b(e)536 b(denote)g(b)-36 b(y)536 b Fm(T)38025 12569 y Fh(`)38465 12370 y Fn(\()p Fm(x)p Fn(\))f(the)h(n)-36 b(um)g(b)36 b(er)534 b(of)j(Niv)-36 b(en)800 13975 y(n)g(um)g(b)36 b(ers)446 b Fm(n)392 b Fk(\267)h Fm(x)447 b Fn(suc)-36 b(h)446 b(that)h Fm(n)305 b Fn(+)f(1)p Fm(;)221 b(n)306 b Fn(+)e(2)p Fm(;)221 b(:)g(:)g(:)j(;)d(n)306 b Fn(+)e Fm(`)g Fk(\241)h Fn(1)447 b(are)h(also)g(Niv)-36 b(en)447 b(n)-36 b(um)g(b)36 b(ers,)450 b(w)-36 b(e)447 b(conjecture)800 15580 y(that)22437 17185 y Fm(T)23199 17384 y Fh(`)23639 17185 y Fn(\()p Fm(x)p Fn(\))369 b Fk(\277)28759 16287 y Fm(x)p 27589 16880 3081 54 v 27589 18226 a Fn(log)29269 17654 y Fh(`)29930 18226 y Fm(x)30802 17185 y(:)800 22645 y Fo(6)2152 b(Ac)-60 b(kno)g(wledgemen)g(ts)800 25565 y Fn(The)619 b(authors)f(w)-36 b(ould)619 b(lik)-36 b(e)620 b(to)f(thank)g(the)f(referee)h(for)g(sev)-36 b(eral)620 b(helpful)f(commen)-36 b(ts)618 b(whic)-36 b(h)619 b(greatly)800 27170 y(impro)-36 b(v)g(ed)433 b(the)g(qualit)-36 b(y)435 b(of)f(this)g(pap)36 b(er.)800 31607 y Fo(References)p Black 800 34528 a Fn([1])p Black 651 w(T.)354 b(Cai,)370 b(On)353 b(2-Niv)-36 b(en)353 b(n)-36 b(um)g(b)36 b(ers)351 b(and)i(3-Niv)-36 b(en)353 b(n)-36 b(um)g(b)36 b(ers,)368 b Fl(Fib)-66 b(onac)g(ci)389 b(Quart.)353 b Ff(34)g Fn(\(1996\),)370 b(118{120.)p Black 800 37240 a([2])p Black 651 w(C.)464 b(N.)f(Co)36 b(op)g(er)463 b(and)f(R.)h(E.)g(Kennedy)-108 b(,)469 b(On)462 b(consecutiv)-36 b(e)463 b(Niv)-36 b(en)463 b(n)-36 b(um)g(b)36 b(ers,)469 b Fl(Fib)-66 b(onac)g(ci)489 b(Quart.)463 b Ff(21)2823 38845 y Fn(\(1993\),)435 b(146-151.)p Black 800 41557 a([3])p Black 651 w(J.)500 b(M.)g(De)g(Koninc)-36 b(k)500 b(and)f(N.)h(Do)-36 b(y)g(on)500 b(On)f(the)g(n)-36 b(um)g(b)36 b(er)498 b(of)j(Niv)-36 b(en)500 b(n)-36 b(um)g(b)36 b(ers)498 b(up)g(to)i Fm(x)p Fn(,)517 b Fl(Fib)-66 b(onac)g(ci)2823 43162 y(Quart.)p Fn(,)434 b(to)g(app)36 b(ear.)p Black 800 45874 a([4])p Black 651 w(J.)608 b(M.)g(De)g(Koninc) -36 b(k,)652 b(N.)608 b(Do)-36 b(y)g(on,)652 b(and)607 b(I.)h(K\266)-650 b(atai,)652 b(On)607 b(the)g(coun)-36 b(ting)607 b(function)h(for)g(the)f(Niv)-36 b(en)2823 47479 y(n)g(um)g(b)36 b(ers,)433 b Fl(A)-66 b(cta)463 b(A)-33 b(rithmetic)-66 b(a)431 b Ff(106)j Fn(\(2003\),)h(265{275.)p Black 800 50191 a([5])p Black 651 w(H.)410 b(G.)g(Grundman,)j (Sequences)c(of)i(consecutiv)-36 b(e)410 b(Niv)-36 b(en)410 b(n)-36 b(um)g(b)36 b(ers,)413 b Fl(Fib)-66 b(onac)g(ci)441 b(Quart.)410 b Ff(32)g Fn(\(1994\),)2823 51797 y(174{175.)p Black 800 54509 a([6])p Black 651 w(I.)484 b(V)-108 b(ardi,)495 b(Niv)-36 b(en)483 b(n)-36 b(um)g(b)36 b(ers,)494 b Fk(x)p Fn(2.3)485 b(in)d Fe(Computational)i(Recreations)g(in)e(Mathematics)p Fn(,)496 b(Addison-)2823 56114 y(W)-108 b(esley)g(,)435 b(1991,)g(pp.)e(19)h(and)f(28{31.)p 800 59184 52000 45 v 800 61434 a(2000)i Fl(Mathematics)464 b(Subje)-66 b(ct)463 b(Classi\257c)-66 b(ation)p Fn(:)577 b(Primary:)h(11A63,)436 b(11A25.)800 64645 y Fl(Keywor)-66 b(ds:)597 b(Niven)464 b(numb)-66 b(ers.)p 800 66276 V 800 68600 a Fn(\(Concerned)433 b(with)h(sequence)p 0 1 0 0 TeXcolorcmyk 433 w(A005349)p 16090 68813 4878 54 v [[217 100 261 112] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A005349) pdfm Black 2 w(.\))p 800 70305 52000 45 v Black 26475 74617 a(7)p Black eop %%Page: 8 8 8 7 bop Black 0 TeXcolorgray Black Black 800 3147 a Fn(Receiv)-36 b(ed)442 b(Octob)36 b(er)441 b(21,)j(2002;)i(revised)c(v)-36 b(ersion)442 b(receiv)-36 b(ed)441 b(June)g(20,)j(2003.)603 b(Published)440 b(in)h Fl(Journal)472 b(of)800 4752 y(Inte)-66 b(ger)463 b(Se)-66 b(quenc)g(es)p Fn(,)431 b(July)k(9,)f(2003.)p 800 6383 52000 45 v 800 8633 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 [[133 639 338 651] [1 1 1 [3 3]] [0 0 1]] (http://www.math.uwaterloo.ca/JIS/) pdfm Black(.)p Black 26475 74617 a(8)p Black eop %%Trailer end end