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21239 y(in)g(teger)472 b Fn(g)520 b Fs(in)472 b(the)g(sequence)h Fn(D)36 b Fs(.)694 b(Th)-36 b(us,)482 b(the)472 b(gap)h(of)g(4)f(follo) -36 b(wing)475 b(7)e(is)f(a)h(\257rst)f(o)36 b(ccurrence,)482 b(while)473 b(the)800 22788 y(gap)399 b(of)g(4)g(follo)-36 b(wing)400 b(13)f(is)g(not.)566 b(Note)399 b(that)f(this)g(usage)h(of)g (the)f(comp)36 b(ound)397 b(adjectiv)-36 b(e)400 b Fl(\257rst)432 b(o)-66 b(c)g(curr)g(enc)g(e)800 24338 y Fs(carries)633 b(no)f(implication)h(whatso)36 b(ev)-36 b(er)633 b(regarding)f (historical)h(precedence)f(of)g(disco)-36 b(v)g(ery)-108 b(.)1175 b(Multiple)800 25888 y(instances)504 b(of)h(gaps)g(of)g(1048)h (are)e(kno)-36 b(wn,)523 b(but)503 b(none)h(is)g(y)-36 b(et)505 b(kno)-36 b(wn)504 b(to)h(b)36 b(e)504 b(a)h(\257rst)e(o)36 b(ccurrence,)522 b(ev)-36 b(en)800 27438 y(though)474 b(one)h(of)g(them)f(b)36 b(ears)475 b(an)f(earliest)i(historical)f (date)g(of)g(disco)-36 b(v)g(ery)-108 b(.)702 b(This)475 b(terminology)h(follo)-36 b(ws)800 28987 y(that)592 b(of)i(Y)-108 b(oung)592 b(and)g(P)-36 b(otler)593 b([)p 0 1 0 0 TeXcolorcmyk(20)p (#cite.YP) [[222 456 234 468] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(],)633 b(and)593 b(pro)36 b(duces)591 b(more)i(concise)h (phrasing)e(than)g(some)h(past)g(and)800 30537 y(presen)-36 b(t)432 b(alternativ)-36 b(e)435 b(nomenclature.)2128 32087 y(A)350 b(prime)g(gap)g(of)h(measure)f Fn(g)397 b Fs(is)350 b(titled)g Fl(maximal)f Fs(if)i(it)f(strictly)h(exceeds)f (all)h(preceding)e(gaps,)368 b(i.e.,)g(the)800 33637 y(di\256erence)f(b)36 b(et)-36 b(w)g(een)366 b(an)-36 b(y)368 b(t)-36 b(w)g(o)367 b(consecutiv)-36 b(e)368 b(smaller)g(primes)e(is)407 b Fn(<)369 b(g)48 b Fs(,)380 b(so)367 b(that)g Fn(g)415 b Fs(exceeds)367 b(all)i(preceding)800 35186 y(elemen)-36 b(ts)562 b(of)h Fn(D)36 b Fs(.)965 b(Th)-36 b(us)561 b(the)h(gap)h(of)g(6)f(follo)-36 b(wing)565 b(the)c(prime)h(23)h(is)g(a)f(maximal)i(prime)e(gap,)595 b(since)800 36736 y(eac)-36 b(h)430 b(and)g(ev)-36 b(ery)431 b(smaller)g(prime)f(is)g(follo)-36 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163 234 175] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(],)546 b(as)524 b(part)e(of)i(a)g(pro)72 b(ject)524 b(generating)f(n)-36 b(umerical)523 b(evidence)h(for)g(the)800 63082 y(Goldbac)-36 b(h)433 b(conjecture.)p Black Black eop %%Page: 3 3 3 2 bop Black 0 TeXcolorgray Black 52288 -3114 a Fj(3)p Black 17117 -458 a Fs(2.)651 b Fo(Comput)-89 b(a)g(tional)503 b(Technique)2128 1867 y Fs(The)322 b(calculations)h(w)-36 b(ere)322 b(carried)g(out)f(o)-36 b(v)g(er)323 b(a)f(p)36 b(erio)g(d)321 b(of)i(y)-36 b(ears,)345 b(distributed)320 b(async)-36 b(hronously)323 b(among)800 3417 y(n)-36 b(umerous)350 b(p)36 b(ersonal)352 b(computers,)368 b(taking)352 b(adv)-72 b(an)-36 b(tage)352 b(of)g(otherwise)g(idle)g(CPU)g(time.)551 b(Nyman)352 b(accom-)800 4966 y(plished)292 b(the)h(bulk)f(of)i(the)e (computations;)340 b(emplo)-36 b(ying)293 b(as)h(man)-36 b(y)293 b(as)g(eigh)-36 b(t)g(y)293 b(systems)g(from)g(1998)h(to)f (2002,)800 6516 y(he)387 b(accoun)-36 b(ted)386 b(for)i(the)e(surv)-36 b(ey)387 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b(en)377 b(to)g(guard)g(against) g(errors)g(\(whether)f(originating)i(in)f(logic,)390 b(soft)-36 b(w)g(are,)800 25113 y(or)577 b(hardw)-36 b(are\),)612 b(the)576 b(coun)-36 b(t)576 b Fn(\274)48 b Fs(\()p Fn(x)p Fs(\))575 b(of)j(primes)e(w)-36 b(as)577 b(main)-36 b(tained)576 b(and)g(c)-36 b(hec)g(k)g(ed)576 b(p)36 b(erio)g(dically)578 b(against)800 26663 y(kno)-36 b(wn)565 b(v)-72 b(alues,)598 b(suc)-36 b(h)564 b(as)h(those)f (published)g(b)-36 b(y)564 b(Riesel)i([)p 0 1 0 0 TeXcolorcmyk(14)p (#cite.Ri94) [[342 477 354 489] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(],)598 b(and)564 b(esp)36 b(ecially)566 b(the)e(extensiv)-36 b(e)566 b(v)-72 b(alues)800 28213 y(computed)509 b(recen)-36 b(tly)511 b(b)-36 b(y)510 b(Silv)-72 b(a)511 b([)p 0 1 0 0 TeXcolorcmyk(17)p (#cite.Silva) [[228 463 240 475] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(].)809 b(In)511 b(addition,)529 b(Nicely)512 b(has)e(since)h(duplicated)e(Nyman's)i(results)800 29762 y(through)433 b(4)p Fn(:)p Fs(5)296 b Fq(\243)f Fs(10)10299 29280 y Fr(15)11296 29762 y Fs(.)18011 33426 y(3.)651 b Fo(Comput)-89 b(a)g(tional)503 b(Resul)-89 b(ts)2128 35751 y Fs(T)-108 b(able)p 0 1 0 0 TeXcolorcmyk 476 w(3)p (#section.3) [[124 395 130 407] [1 1 1 [3 3]] [0 0 1]] pdfm Black 476 w(lists)476 b(the)f(newly)h(disco)-36 b(v)g(ered)476 b(\257rst)f(o)36 b(ccurrence)475 b(prime)g(gaps)h(resulting)f(from)h (the)f(presen)-36 b(t)800 37301 y(study;)412 b(maximal)403 b(gaps)f(are)g(indicated)f(b)-36 b(y)402 b(a)g(double)f(dagger)h(\()p Fq(z)p Fs(\).)568 b(Eac)-36 b(h)401 b(table)h(en)-36 b(try)401 b(sho)-36 b(ws)402 b(the)f(mea-)800 38851 y(sure)409 b Fn(g)457 b Fs(of)410 b(the)f(gap)g(and)g(the)g(initial)h(prime)f Fn(p)23482 39050 y Fr(1)24008 38851 y Fs(.)570 b(The)409 b(\257fteen)g(gaps)h(b)36 b(et)-36 b(w)g(een)408 b(10)40745 38369 y Fr(15)42152 38851 y Fs(and)h(1)p Fn(:)p Fs(598508912)249 b Fq(\243)800 40400 y Fs(10)2100 39918 y Fr(15)3531 40400 y Fs(are)433 b(due)g(to)h(Nicely;)h(all)f(the)f(rest)h(w)-36 b(ere)433 b(disco)-36 b(v)g(ered)434 b(b)-36 b(y)434 b(Nyman.)21569 44064 y(4.)651 b Fo(Obser)-118 b(v)g(a)-89 b(tions)2128 46389 y Fs(As)340 b(a)f(collateral)i(result)e(of)h(his)f (calculations,)359 b(Nyman)339 b(has)h(computed)e(for)h(the)g(coun)-36 b(t)338 b(of)i(t)-36 b(win)339 b(primes)800 47939 y(the)329 b(v)-72 b(alue)330 b Fn(\274)6930 48138 y Fr(2)7455 47939 y Fs(\(5)82 b Fq(\243)g Fs(10)11108 47457 y Fr(16)12106 47939 y Fs(\))369 b(=)g(47177404870103,)355 b(the)329 b(maxim)-36 b(um)329 b(argumen)-36 b(t)329 b(for)h(whic)-36 b(h)329 b(this)g(function)g(has)800 49489 y(b)36 b(een)345 b(ev)-72 b(aluated.)549 b(Nyman)346 b(also)h(obtained)e Fn(\274)48 b Fs(\(5)116 b Fq(\243)g Fs(10)26827 49007 y Fr(16)27823 49489 y Fs(\))369 b(=)f(1336094767763971)351 b(for)346 b(the)f(corresp)36 b(onding)800 51038 y(coun)-36 b(t)442 b(of)i(primes;)k(this)443 b(is)g(the)g(largest)h(v)-72 b(alue)443 b(of)h Fn(x)f Fs(for)h(whic)-36 b(h)443 b Fn(\274)48 b Fs(\()p Fn(x)p Fs(\))442 b(has)h(b)36 b(een)443 b(determined)e(b)-36 b(y)443 b(direct)800 52588 y(en)-36 b(umeration,)518 b(and)501 b(con\257rms)g(the)g(v)-72 b(alue)501 b(previously)i(obtained)e(b)-36 b(y)501 b(Del)-36 b(\266)-614 b(eglise)503 b(and)e(Riv)-72 b(at)503 b([)p 0 1 0 0 TeXcolorcmyk(5)p (#cite.DR) [[503 244 509 256] [1 1 1 [3 3]] [0 0 1]] pdfm Black(],)519 b(using)800 54138 y(indirect)411 b(sieving)i(metho)36 b(ds.)570 b(Nyman)412 b(has)f(also)i(generated)e(frequency)h(tables)g (for)g(the)f(distribution)f(of)800 55688 y(all)435 b(prime)e(gaps)h(b) 36 b(elo)-36 b(w)434 b(5)295 b Fq(\243)h Fs(10)16521 55206 y Fr(16)17518 55688 y Fs(.)2128 57237 y(Listings)500 b(of)h(the)e(423)i(previously)g(kno)-36 b(wn)500 b(\257rst)f(o)36 b(ccurrence)499 b(prime)h(gaps)g(\(including)f(61)i(maximal)800 58787 y(gaps\),)350 b(those)329 b(b)36 b(elo)-36 b(w)330 b(10)12733 58305 y Fr(15)13730 58787 y Fs(,)350 b(ha)-36 b(v)g(e)329 b(b)36 b(een)328 b(published)g(collectiv)-36 b(ely)331 b(b)-36 b(y)329 b(Y)-108 b(oung)329 b(and)f(P)-36 b(otler)329 b([)p 0 1 0 0 TeXcolorcmyk(20)p (#cite.YP) [[476 188 487 200] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])g(and)g(Nicely)800 60337 y([)p 0 1 0 0 TeXcolorcmyk(12)p (#cite.Ni99) [[82 174 94 186] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(],)434 b(and)f(are)h(herein)f(omitted)g(for)h(brevit)-36 b(y)-108 b(.)2128 61887 y(A)366 b(comprehensiv)-36 b(e)365 b(listing)h(of)g(\257rst)e(o)36 b(ccurrence)365 b(and)g(maximal)i (prime)e(gaps,)380 b(annotated)364 b(with)i(addi-)800 63436 y(tional)415 b(information,)k(is)c(a)-36 b(v)-72 b(ailable)415 b(at)g(Nicely's)g(URL.)g(Nicely)g(also)g(main)-36 b(tains)415 b(at)f(his)g(URL)g(extensiv)-36 b(e)800 64986 y(lists)400 b(of)h(\257rst)e(kno)-36 b(wn)400 b(o)36 b(ccurrence)399 b(prime)g(gaps,)407 b(lying)401 b(b)36 b(ey)-36 b(ond)399 b(the)g(presen)-36 b(t)399 b(upp)36 b(er)398 b(b)36 b(ound)399 b(of)h(exhaus-)800 66536 y(tiv)-36 b(e)368 b(computation,)381 b(and)366 b(disco)-36 b(v)g(ered)368 b(mostly)g(b)-36 b(y)368 b(third)e(parties,)381 b(notably)368 b(Harv)-36 b(ey)368 b(Dubner)f([)p 0 1 0 0 TeXcolorcmyk(6)p (#cite.Du) [[500 118 506 130] [1 1 1 [3 3]] [0 0 1]] pdfm Black(].)557 b(These)800 68086 y(lists)536 b(exhibit)g(sp)36 b(eci\257c)535 b(gaps)h(for)g(ev)-36 b(ery)536 b(ev)-36 b(en)536 b(p)36 b(ositiv)-36 b(e)536 b(in)-36 b(teger)536 b(up)e(to)i(10884,)563 b(as)536 b(w)-36 b(ell)536 b(as)g(for)g(other) 800 69635 y(scattered)399 b(ev)-36 b(en)400 b(in)-36 b(tegers)400 b(up)f(to)h(233822;)413 b(for)400 b(some)g(of)h(the)e (gaps)h(exceeding)h(8000)g(in)e(magnitude,)407 b(the)p Black Black eop %%Page: 4 4 4 3 bop Black 0 TeXcolorgray Black 800 -3114 a Fj(4)p Black Black Black Black 3326 79 46949 45 v 3304 1518 45 1440 v 3525 1518 V 4227 1086 a Fi(Gap)p 6989 1518 V 1108 w(F)-101 b(ollo)-34 b(wing)404 b(the)h(prime)p 18655 1518 V 18877 1518 V 1522 w(Gap)p 22408 1518 V 1107 w(F)-101 b(ollo)-34 b(wing)405 b(the)f(prime)p 34073 1518 V 34295 1518 V 2061 w(Gap)p 38365 1518 V 1107 w(F)-101 b(ollo)-34 b(wing)405 b(the)g(prime)p 50031 1518 V 50253 1518 V 3326 1562 46949 45 v 3304 3002 45 1440 v 3526 3002 V 4640 2570 a(796)p 6989 3002 V 1969 w(1271309838631957)p 18655 3002 V 18877 3002 V 1935 w(928)p 22408 3002 V 1363 w(10244316228469423)p 34073 3002 V 34295 3002 V 1868 w(1010)p 38365 3002 V 1363 w(21743496643443551)p 50031 3002 V 50253 3002 V 3304 4441 V 3526 4441 V 4640 4009 a(812)p 6989 4441 V 1969 w(1710270958551941)p 18655 4441 V 18877 4441 V 1935 w(930)p 22408 4441 V 1969 w(3877048405466683)p 34073 4441 V 34295 4441 V 1868 w(1012)p 38365 4441 V 1363 w(22972837749135871)p 50031 4441 V 50253 4441 V 3304 5880 V 3526 5880 V 4640 5448 a(824)p 6989 5880 V 1969 w(1330854031506047)p 18655 5880 V 18877 5880 V 1935 w(932)p 22408 5880 V 1363 w(10676480515967939)p 34073 5880 V 34295 5880 V 1868 w(1014)p 38365 5880 V 1363 w (13206732046682519)p 50031 5880 V 50253 5880 V 3304 7319 V 3526 7319 V 4640 6887 a(838)p 6989 7319 V 1969 w(1384201395984013)p 18655 7319 V 18877 7319 V 1935 w(934)p 22408 7319 V 1969 w(8775815387922523)p 34073 7319 V 34295 7319 V 1868 w(1016)p 38365 7319 V 1363 w(25488154987300883)p 50031 7319 V 50253 7319 V 3304 8758 V 3526 8758 V 4640 8326 a(842)p 6989 8758 V 1969 w(1142191569235289)p 18655 8758 V 18877 8758 V 1935 w(936)p 22408 8758 V 1969 w(2053649128145117)p 34073 8758 V 34295 8758 V 1868 w(1018)p 38365 8758 V 1363 w(37967240836435909)p 50031 8758 V 50253 8758 V 3304 10197 V 3526 10197 V 4640 9765 a(846)p 6989 10197 V 1969 w(1045130023589621)p 18655 10197 V 18877 10197 V 1935 w(938)p 22408 10197 V 1969 w(3945256745730569)p 34073 10197 V 34295 10197 V 1868 w(1020)p 38365 10197 V 1363 w(24873160697653789)p 50031 10197 V 50253 10197 V 3304 11636 V 3526 11636 V 4640 11204 a(848)p 6989 11636 V 1969 w(2537070652896083)p 18655 11636 V 18877 11636 V 1935 w(940)p 22408 11636 V 1969 w(9438544090485889)p 34073 11636 V 34295 11636 V 1868 w(1022)p 38365 11636 V 1363 w(10501301105720969)p 50031 11636 V 50253 11636 V 3304 13075 V 3526 13075 V 4640 12643 a(850)p 6989 13075 V 1969 w(2441387599467679)p 18655 13075 V 18877 13075 V 1935 w(942)p 22408 13075 V 1363 w(10369943471405191)p 34073 13075 V 34295 13075 V 1868 w(1024)p 38365 13075 V 1363 w(22790428875364879)p 50031 13075 V 50253 13075 V 3304 14514 V 3526 14514 V 4640 14082 a(852)p 6989 14514 V 1969 w(1432204101894959)p 18655 14514 V 18877 14514 V 1935 w(944)p 22408 14514 V 1969 w(4698198022874969)p 34073 14514 V 34295 14514 V 1868 w(1026)p 38365 14514 V 1363 w(14337646064564951)p 50031 14514 V 50253 14514 V 3304 15953 V 3526 15953 V 4640 15521 a(854)p 6989 15953 V 1969 w(1361832741886937)p 18655 15953 V 18877 15953 V 1935 w(946)p 22408 15953 V 1969 w(8445899254653313)p 34073 15953 V 34295 15953 V 1868 w(1028)p 38365 15953 V 1363 w(16608210365179331)p 50031 15953 V 50253 15953 V 3304 17392 V 3526 17392 V 4640 16960 a(856)p 6989 17392 V 1969 w(1392892713537313)p 18655 17392 V 18877 17392 V 1935 w(948)p 22408 17392 V 1969 w(5806170698601659)p 34073 17392 V 34295 17392 V 1868 w(1030)p 38365 17392 V 1363 w(21028354658071549)p 50031 17392 V 50253 17392 V 3304 18831 V 3526 18831 V 4640 18399 a(858)p 6989 18831 V 1969 w(1464551007952943)p 18655 18831 V 18877 18831 V 1935 w(950)p 22408 18831 V 1969 w(5000793739812263)p 34073 18831 V 34295 18831 V 1868 w(1032)p 38365 18831 V 1363 w(19449190302424919)p 50031 18831 V 50253 18831 V 3304 20270 V 3526 20270 V 4640 19838 a(864)p 6989 20270 V 1969 w(2298355839009413)p 18655 20270 V 18877 20270 V 1935 w(952)p 22408 20270 V 1969 w(3441724070563411)p 34073 20270 V 34295 20270 V 1868 w(1034)p 38365 20270 V 1363 w(11453766801670289)p 50031 20270 V 50253 20270 V 3304 21709 V 3526 21709 V 4640 21277 a(866)p 6989 21709 V 1969 w(2759317684446707)p 18655 21709 V 18877 21709 V 1935 w(954)p 22408 21709 V 1969 w(8909512917643439)p 34073 21709 V 34295 21709 V 1868 w(1036)p 38365 21709 V 1363 w(36077433695182153)p 50031 21709 V 50253 21709 V 3304 23148 V 3526 23148 V 4640 22716 a(868)p 6989 23148 V 1969 w(1420178764273021)p 18655 23148 V 18877 23148 V 1935 w(956)p 22408 23148 V 1969 w(7664508840731297)p 34073 23148 V 34295 23148 V 1868 w(1038)p 38365 23148 V 1363 w(28269785077311409)p 50031 23148 V 50253 23148 V 3304 24587 V 3526 24587 V 4640 24156 a(870)p 6989 24587 V 1969 w(1598729274799313)p 18655 24587 V 18877 24587 V 1935 w(958)p 22408 24587 V 1969 w(6074186033971933)p 34073 24587 V 34295 24587 V 1868 w(1040)p 38365 24587 V 1363 w(46246848392875127)p 50031 24587 V 50253 24587 V 3304 26026 V 3526 26026 V 4640 25595 a(874)p 6989 26026 V 1969 w(1466977528790023)p 18655 26026 V 18877 26026 V 1935 w(960)p 22408 26026 V 1969 w(5146835719824811)p 34073 26026 V 34295 26026 V 1868 w(1042)p 38365 26026 V 1363 w(33215047653774409)p 50031 26026 V 50253 26026 V 3304 27465 V 3526 27465 V 4640 27034 a(876)p 6989 27465 V 1969 w(1125406185245561)p 18655 27465 V 18877 27465 V 1935 w(962)p 22408 27465 V 1969 w(9492966874626647)p 34073 27465 V 34295 27465 V 1868 w(1044)p 38365 27465 V 1969 w(7123663452896833)p 50031 27465 V 50253 27465 V 3304 28904 V 3526 28904 V 4640 28473 a(878)p 6989 28904 V 1969 w(2705074880971613)p 18655 28904 V 18877 28904 V 1935 w(964)p 22408 28904 V 1969 w(5241451254010087)p 34073 28904 V 34295 28904 V 1868 w(1046)p 38365 28904 V 1363 w(25702173876611591)p 50031 28904 V 50253 28904 V 3304 30343 V 3526 30343 V 4640 29912 a(882)p 6989 30343 V 1969 w(3371055452381147)p 18655 30343 V 18877 30343 V 1935 w(966)p 22408 30343 V 1969 w(5158509484643071)p 34073 30343 V 34295 30343 V 1868 w(1050)p 38365 30343 V 1363 w(13893290219203981)p 50031 30343 V 50253 30343 V 3304 31782 V 3526 31782 V 4640 31351 a(884)p 6989 31782 V 1969 w(1385684246418833)p 18655 31782 V 18877 31782 V 1935 w(968)p 22408 31782 V 1363 w(19124990244992669)p 34073 31782 V 34295 31782 V 1868 w(1054)p 38365 31782 V 1363 w(26014156620917407)p 50031 31782 V 50253 31782 V 3304 33222 V 3526 33222 V 4640 32790 a(886)p 6989 33222 V 1969 w(4127074165753081)p 18655 33222 V 18877 33222 V 1935 w(970)p 22408 33222 V 1363 w(10048813989052669)p 34073 33222 V 34295 33222 V 1868 w(1056)p 38365 33222 V 1363 w(11765987635602143)p 50031 33222 V 50253 33222 V 3304 34661 V 3526 34661 V 4640 34229 a(888)p 6989 34661 V 1969 w(2389167248757889)p 18655 34661 V 18877 34661 V 1935 w(972)p 22408 34661 V 1969 w(4452510040366189)p 34073 34661 V 34295 34661 V 1868 w(1058)p 38365 34661 V 1363 w(28642379760272723)p 50031 34661 V 50253 34661 V 3304 36100 V 3526 36100 V 4640 35668 a(890)p 6989 36100 V 1969 w(3346735005760637)p 18655 36100 V 18877 36100 V 1935 w(974)p 22408 36100 V 1363 w(10773850897499933)p 34073 36100 V 34295 36100 V 1868 w(1060)p 38365 36100 V 1363 w(15114558265244791)p 50031 36100 V 50253 36100 V 3304 37539 V 3526 37539 V 4640 37107 a(892)p 6989 37539 V 1969 w(2606748800671237)p 18655 37539 V 18877 37539 V 1935 w(976)p 22408 37539 V 1363 w(14954841632404033)p 34073 37539 V 34295 37539 V 1868 w(1062)p 38365 37539 V 1363 w(15500910867678727)p 50031 37539 V 50253 37539 V 3304 38978 V 3526 38978 V 4640 38546 a(894)p 6989 38978 V 1969 w(2508853349189969)p 18655 38978 V 18877 38978 V 1935 w(978)p 22408 38978 V 1363 w(12040807275386881)p 34073 38978 V 34295 38978 V 1868 w(1064)p 38365 38978 V 1363 w(43614652195746623)p 50031 38978 V 50253 38978 V 3304 40417 V 3526 40417 V 4640 39985 a(896)p 6989 40417 V 1969 w(3720181237979117)p 18655 40417 V 18877 40417 V 1935 w(980)p 22408 40417 V 1363 w(19403684901755939)p 34073 40417 V 34295 40417 V 1868 w(1068)p 38365 40417 V 1363 w(23900175352205171)p 50031 40417 V 50253 40417 V 3304 41856 V 3526 41856 V 4640 41424 a(898)p 6989 41856 V 1969 w(4198168149492463)p 18655 41856 V 18877 41856 V 1935 w(982)p 22408 41856 V 1363 w(18730085806290949)p 34073 41856 V 34295 41856 V 1868 w(1072)p 38365 41856 V 1363 w(40433690575714297)p 50031 41856 V 50253 41856 V 3304 43295 V 3526 43295 V 4640 42863 a(900)p 6989 43295 V 1969 w(2069461000669981)p 18655 43295 V 18877 43295 V 1935 w(984)p 22408 43295 V 1363 w(11666708491143997)p 34073 43295 V 34295 43295 V 1868 w(1074)p 38365 43295 V 1363 w(33288359939765017)p 50031 43295 V 50253 43295 V 3304 44734 V 3526 44734 V 4640 44302 a(902)p 6989 44734 V 1969 w(1555616198548067)p 18655 44734 V 18877 44734 V 1935 w(986)p 22408 44734 V 1363 w(34847474118974633)p 34073 44734 V 34295 44734 V 1868 w(1076)p 38365 44734 V 1363 w(20931714475256591)p 50031 44734 V 50253 44734 V 3304 46173 V 3526 46173 V 4640 45741 a(904)p 6989 46173 V 1969 w(3182353047511543)p 18655 46173 V 18877 46173 V 1935 w(988)p 22408 46173 V 1363 w(11678629605932719)p 34073 46173 V 34295 46173 V 1868 w(1084)p 38365 46173 V 1363 w(41762363147589283)p 50031 46173 V 50253 46173 V 3304 47612 V 3526 47612 V 4640 47180 a(908)p 6989 47612 V 1969 w(2126985673135679)p 18655 47612 V 18877 47612 V 1935 w(990)p 22408 47612 V 1969 w(2764496039544377)p 34073 47612 V 34295 47612 V 1868 w(1098)p 38365 47612 V 1363 w(25016149672697549)p 50031 47612 V 50253 47612 V 3304 49051 V 3526 49051 V 4640 48619 a(910)p 6989 49051 V 1969 w(1744027311944761)p 18655 49051 V 18877 49051 V 1935 w(992)p 22408 49051 V 1969 w(4941033906441539)p 34073 49051 V 34295 49051 V 1868 w(1100)p 38365 49051 V 1363 w(21475286713974413)p 50031 49051 V 50253 49051 V 3304 50490 V 3526 50490 V 4640 50058 a(912)p 6989 50490 V 1969 w(2819939997576017)p 18655 50490 V 18877 50490 V 1935 w(994)p 22408 50490 V 1969 w(3614455901007619)p 34073 50490 V 34295 50490 V 1868 w(1102)p 38365 50490 V 1363 w(39793570504639117)p 50031 50490 V 50253 50490 V 3304 51929 V 3526 51929 V 4640 51497 a(914)p 6989 51929 V 1969 w(3780822371661509)p 18655 51929 V 18877 51929 V 1935 w(996)p 22408 51929 V 1363 w(14693181579822451)p 34073 51929 V 34295 51929 V 1868 w(1106)p 38365 51929 V 1363 w(29835422457878441)p 50031 51929 V 50253 51929 V 3304 53368 V 3526 53368 V 4101 52936 a Fh(z)p Fi(916)p 6989 53368 V 1969 w(1189459969825483)p 18655 53368 V 18877 53368 V 1935 w(998)p 22408 53368 V 1363 w(11813551133888459)p 34073 53368 V 34295 53368 V 1868 w(1108)p 38365 53368 V 1363 w(43986327184963729)p 50031 53368 V 50253 53368 V 3304 54807 V 3526 54807 V 4640 54376 a(918)p 6989 54807 V 1969 w(2406868929767921)p 18655 54807 V 18877 54807 V 1329 w(1000)p 22408 54807 V 1363 w(22439962446379651)p 34073 54807 V 34295 54807 V 1868 w(1120)p 38365 54807 V 1363 w(19182559946240569)p 50031 54807 V 50253 54807 V 3304 56246 V 3526 56246 V 4640 55815 a(920)p 6989 56246 V 1969 w(4020057623095403)p 18655 56246 V 18877 56246 V 1329 w(1002)p 22408 56246 V 1363 w(14595374896200821)p 34073 56246 V 34295 56246 V 1868 w(1122)p 38365 56246 V 1363 w(31068473876462989)p 50031 56246 V 50253 56246 V 3304 57685 V 3526 57685 V 4640 57254 a(922)p 6989 57685 V 1969 w(4286129201882221)p 18655 57685 V 18877 57685 V 1329 w(1004)p 22408 57685 V 1969 w(7548471163197917)p 34073 57685 V 34295 57685 V 1329 w Fh(z)p Fi(1132)p 38365 57685 V 1969 w(1693182318746371)p 50031 57685 V 50253 57685 V 3304 59124 V 3526 59124 V 4101 58693 a Fh(z)p Fi(924)p 6989 59124 V 1969 w(1686994940955803)p 18655 59124 V 18877 59124 V 1329 w(1006)p 22408 59124 V 1363 w(37343192296558573)p 34073 59124 V 34295 59124 V 1329 w Fh(z)p Fi(1184)p 38365 59124 V 1363 w(43841547845541059)p 50031 59124 V 50253 59124 V 3304 60563 V 3526 60563 V 4640 60132 a(926)p 6989 60563 V 1969 w(6381944136489827)p 18655 60563 V 18877 60563 V 1329 w(1008)p 22408 60563 V 1969 w(5356763933625179)p 34073 60563 V 34295 60563 V 38365 60563 V 50031 60563 V 50253 60563 V 3326 60608 46949 45 v 1848 62842 a Ft(T)-125 b(able)500 b(1.)664 b(First)500 b(o)42 b(ccurrence)499 b(prime)g(gaps)h(b)42 b(et)-42 b(w)g(een)499 b(10)33889 62360 y Fg(15)35527 62842 y Ft(and)g(5)296 b Fq(\243)f Ft(10)42277 62360 y Fg(16)43418 62842 y Ft(.)664 b Fq(z)498 b Ft(denotes)h(a)22618 64392 y(maximal)g(gap)p Black 800 68086 a Fs(b)36 b(ounding)391 b(in)-36 b(tegers)392 b(ha)-36 b(v)g(e)393 b(only)g(b)36 b(een)392 b(pro)-36 b(v)g(ed)391 b(strong)i(probable)e(primes)h (\(based)g(on)g(m)-36 b(ultiple)392 b(Miller's)800 69635 y(tests\).)p Black Black eop %%Page: 5 5 5 4 bop Black 0 TeXcolorgray Black 52288 -3114 a Fj(5)p Black 2128 -458 a Fs(The)305 b(largest)g(gap)f(herein)g(established)g (as)g(a)h(\257rst)e(o)36 b(ccurrence)304 b(is)h(the)e(maximal)j(gap)e (of)h(1184)h(follo)-36 b(wing)800 1092 y(the)593 b(prime)f (43841547845541059,)639 b(disco)-36 b(v)g(ered)593 b(31)h(August)e (2002)j(b)-36 b(y)593 b(Nyman.)1057 b(The)593 b(smallest)h(gap)800 2642 y(whose)434 b(\257rst)f(o)36 b(ccurrence)433 b(remains)h (uncertain)e(is)i(the)f(gap)h(of)g(1048.)2128 4192 y(The)482 b(maximal)g(gap)f(of)h(1132)h(follo)-36 b(wing)483 b(the)e(prime)f (1693182318746371,)499 b(disco)-36 b(v)g(ered)481 b(24)h(Jan)-36 b(uary)800 5741 y(1999)514 b(b)-36 b(y)513 b(Nyman,)533 b(is)513 b(the)f(\257rst)g(o)36 b(ccurrence)512 b(of)h(an)-36 b(y)513 b(\\kilogap",)535 b(i.e.,)f(an)-36 b(y)513 b(gap)g(of)g (measure)g(1000)h(or)800 7291 y(greater.)567 b(Its)399 b(maximalit)-36 b(y)400 b(p)36 b(ersists)398 b(throughout)g(an)g (extraordinarily)i(large)f(in)-36 b(terv)-72 b(al;)411 b(the)398 b(succeeding)800 8841 y(maximal)512 b(gap)f(is)g(the)f(gap)g (of)h(1184)h(follo)-36 b(wing)513 b(the)d(prime)g(43841547845541059.) 815 b(The)510 b(ratio)h(of)h(the)800 10390 y(initial)f(primes)g(of)g (these)f(t)-36 b(w)g(o)511 b(successiv)-36 b(e)511 b(maximal)h(gaps)f (is)g Fq(\274)500 b Fs(25)p Fn(:)p Fs(89,)532 b(far)511 b(exceeding)g(the)f(previous)800 11940 y(extreme)389 b(ratio)g(of)g Fq(\274)369 b Fs(7)p Fn(:)p Fs(20)390 b(for)f(the)f(maximal)h(gaps)g(of)g(34)g(\(follo)-36 b(wing)391 b(1327\))e(and)f(36)h(\(follo)-36 b(wing)391 b(9551\),)800 13490 y(eac)-36 b(h)389 b(disco)-36 b(v)g(ered)390 b(b)-36 b(y)389 b(Glaisher)h([)p 0 1 0 0 TeXcolorcmyk(7)p (#cite.Gl) [[226 596 232 608] [1 1 1 [3 3]] [0 0 1]] pdfm Black(])g(in)f(1877.)565 b(F)-108 b(urthermore,)397 b(the)389 b(gap)g(of)i(1132)f(has)g(the)e(greatest)i(merit)800 15040 y(\()p Fq(\274)400 b Fs(32)p Fn(:)p Fs(28\))453 b(of)f(an)-36 b(y)452 b(kno)-36 b(wn)451 b(gap;)462 b(the)451 b(maximal)i(gap)e(of)i(1184)g(is)f(the)f(only)h(other)f(one)h(b)36 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b(to)d(Nicely)g(\(1995{2003\).)p Black 1354 3527 a(7.)p Black 554 w(J.)512 b(W.)f(L.)g(Glaisher,)548 b(On)511 b(long)i(successions)d(of)i(comp)31 b(osite)513 b(n)-31 b(um)g(b)31 b(ers,)547 b Fd(Messenger)527 b(of)g(Mathematics)601 b Fe(7)512 b Fk(\(1877\),)2768 4856 y(102{106,)373 b(171{176.)p Black 1354 6184 a(8.)p Black 554 w(Andrew)359 b(Gran)-31 b(ville,)363 b(Unexp)31 b(ected)359 b(irregularities)i(in)e(the)h (distribution)h(of)e(prime)h(n)-31 b(um)g(b)31 b(ers,)361 b(in)e Fd(Pr)-57 b(o)g(c)g(e)g(e)g(dings)389 b(of)e(the)2768 7512 y(International)c(Congr)-57 b(ess)383 b(of)h(Mathematicians,)h(V) -85 b(ol.)384 b(I)f(\(Z)14 b(\304)-580 b(urich,)386 b(1994\))p Fk(,)357 b(Birkh\304)-553 b(auser,)358 b(Basel,)h(1995,)h(pp.)355 b(388{399.)2768 8841 y(MR)369 b Fe(97d:)p Fk(11139.)p Black 1354 10169 a(9.)p Black 554 w(L.)i(J.)g(Lander)g(and)h(Thomas)g (R.)f(P)-31 b(arkin,)373 b(On)e(\257rst)g(app)31 b(earance)371 b(of)h(prime)f(di\256erences,)g Fd(Math.)398 b(Comp.)371 b Fe(21)h 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