1958:
1838:
3060:
2155:, so mostly cancel out. Occasionally however, several of the larger ones might happen to have roughly the same complex argument, in which case they will reinforce each other instead of cancelling and will overwhelm the term
2202:
2088:
3232:
722:
1323:
1277:
2330:
and also why it is rare for this to happen. It also shows why finding places where this happens depends on large scale calculations of millions of high precision zeros of the
Riemann zeta function.
306:
2658:
2505:
2035:
614:
1720:
1231:
540:
144:
4092:
3875:
2787:
1774:
1068:
916:
257:
210:
5065:
4742:
4654:
4534:
4410:
4233:
2374:
376:
2956:
2441:
1151:; the possibility that there are crossover points near these values does not seem to have been definitely ruled out yet, though computer calculations suggest they are unlikely to exist.
2328:
440:
2120:
1360:
1149:
990:
819:
786:
1616:
4870:
2539:
2337:
to show that sometimes many terms have about the same argument. In the event that the
Riemann hypothesis is false, the argument is much simpler, essentially because the terms
957:
1548:
2597:
4337:
2827:
4801:
2333:
The argument above is not a proof, as it assumes the zeros of the
Riemann zeta function are random, which is not true. Roughly speaking, Littlewood's proof consists of
4002:
1511:
1021:
846:
4830:
3776:
2293:
2149:
1830:
1647:
1579:
1117:
405:
3087:
2570:
2264:
1981:
1953:{\displaystyle \pi (x)=\operatorname {li} (x)-{\tfrac {1}{2}}\operatorname {li} ({\sqrt {x\,}})-\sum _{\rho }\operatorname {li} (x^{\rho })+{\text{smaller terms}}}
4746:
3260:
4045:
3828:
3155:
3135:
3111:
2867:
2847:
2237:
1670:
1181:
1088:
869:
641:
490:
460:
91:
4658:
2961:
37:
5152:
4096:
3090:
2219:, so a large number (several hundred) of them need to have roughly the same argument in order to overwhelm the dominant term. The chance of
2334:
2158:
2044:
5121:
4578:
728:
3163:
5426:
1788:
showed that this proportion is about 0.00000026, which is surprisingly large given how far one has to go to find the first example.
649:
4945:
1282:
1236:
5083:
1800:
262:
5069:(masters), Master's thesis, Manchester Institute for Mathematical Sciences, School of Mathematics, University of Manchester
2602:
2449:
1993:
5461:
5341:
5246:
5241:
5236:
5231:
5226:
5221:
5216:
5211:
548:
162:
5346:
5276:
5336:
1675:
1186:
495:
99:
4050:
3833:
2691:
1732:
1026:
874:
215:
168:
5145:
4982:
3809:
2152:
5291:
5023:
4700:
4612:
4492:
4368:
4191:
2340:
334:
5436:
4874:
4538:
4414:
3927:
3884:
2872:
2399:
5421:
5286:
5499:
4905:
2298:
410:
5530:
2151:. The other terms above are somewhat smaller, and moreover tend to have different, seemingly random complex
2093:
1332:
1122:
962:
331:
that there is such a number (and so, a first such number); and indeed found that the sign of the difference
316:
154:
5525:
5520:
5494:
5389:
5138:
791:
758:
5456:
5446:
5384:
1588:
748:
4835:
2510:
929:
743:
These upper bounds have since been reduced considerably by using large-scale computer calculations of
4351:
4016:
3794:
1520:
2575:
5331:
4307:
1722:
though computer calculations suggest some explicit numbers that are quite likely to satisfy this.
5379:
5005:
4962:
4928:
4433:
4341:
4167:
4105:
4006:
3946:
2796:
2038:
1984:
744:
469:
324:
5281:
4777:
5117:
4456:
4452:
3978:
1483:
999:
824:
4806:
3743:
2269:
2125:
1806:
1625:
1557:
1093:
381:
5301:
5251:
4997:
4954:
4920:
4883:
4755:
4683:
4675:
4667:
4587:
4547:
4468:
4423:
4291:
4277:
4266:
4248:
4239:
4175:
4151:
4131:
4115:
3962:
3936:
3922:
3911:
3893:
2661:
378:
changes infinitely many times. All numerical evidence then available seemed to suggest that
4974:
4897:
4767:
4599:
4561:
4482:
4445:
4262:
4163:
4127:
3958:
3907:
3065:
2548:
2242:
1966:
5441:
4970:
4943:(1941), "On the distribution function of the remainder term of the prime number theorem",
4893:
4763:
4687:
4679:
4595:
4557:
4478:
4441:
4362:
4295:
4270:
4258:
4179:
4159:
4135:
4123:
3966:
3954:
3915:
3903:
1781:
1726:
919:
4355:
4020:
3242:
5471:
5362:
5296:
5261:
4694:
4606:
4282:
4030:
3813:
3140:
3120:
3096:
2852:
2832:
2377:
2222:
2216:
2212:
1655:
1166:
1073:
854:
626:
475:
445:
320:
165:. Skewes's number is much larger, but it is now known that there is a crossing between
76:
71:
63:
4906:"On The Asymptotic Density Of Prime k-tuples and a Conjecture of Hardy and Littlewood"
4428:
5514:
5431:
5206:
5186:
5161:
4940:
4932:
2677:
2207:
The reason why the Skewes number is so large is that these smaller terms are quite a
732:
55:
47:
5009:
4888:
4171:
5466:
5001:
4591:
4569:
3055:{\displaystyle \operatorname {li_{P}} (x)=\int _{2}^{x}{\frac {dt}{(\ln t)^{k+1}}}}
2542:
59:
5094:
4552:
4142:
Kotnik, T. (2008), "The prime-counting function and its analytic approximations",
3898:
5476:
5416:
4573:
67:
4924:
731:: exhibiting some concrete upper bound for the first sign change. According to
5481:
5266:
4983:"The Skewes number for twin primes: counting sign changes of π2(x) − C2Li2(x)"
4671:
4155:
4119:
3738:
4759:
4473:
17:
5372:
5367:
5201:
4253:
1155:
gave a small improvement and correction to the result of Bays and Hudson.
751:. The first estimate for the actual value of a crossover point was given by
5306:
3925:(1975), "Irregularities in the distribution of primes and twin primes",
1159:
found a smaller interval for a crossing, which was slightly improved by
5196:
4966:
4437:
3950:
1832:, whose leading terms are (ignoring some subtle convergence questions)
1796:
849:
26:
2239:
random complex numbers having roughly the same argument is about 1 in
5451:
5256:
5191:
4110:
5018:
4958:
4459:(1962), "Approximate formulas for some functions of prime numbers",
4027:
Chao, Kuok Fai; Plymen, Roger (2010), "A new bound for the smallest
3941:
442:
Littlewood's proof did not, however, exhibit a concrete such number
4011:
3265:
The table below shows the currently known Skewes numbers for prime
4489:
Saouter, Yannick; Demichel, Patrick (2010), "A sharp region where
4346:
2660:
is roughly analogous to a second-order correction accounting for
2197:{\displaystyle {\tfrac {1}{2}}\operatorname {li} ({\sqrt {x\,}})}
2083:{\displaystyle {\tfrac {1}{2}}\operatorname {li} ({\sqrt {x\,}})}
735:, this was at the time not considered obvious even in principle.
5130:
5134:
4187:
3227:{\displaystyle \pi _{P}(p)>C_{P}\operatorname {li} _{P}(p),}
5181:
2211:
smaller than the leading error term, mainly because the first
717:{\displaystyle e^{e^{e^{e^{7.705}}}}<10^{10^{10^{964}}}.}
4774:
Stoll, Douglas; Demichel, Patrick (2011), "The impact of
1318:{\displaystyle e^{727.9513386}<1.39717\times 10^{316}}
1272:{\displaystyle e^{727.9513468}<1.39718\times 10^{316}}
3113:
that violates the Hardy–Littlewood inequality for the (
1784:
of these positive integers does exist and is positive.
727:
Skewes's task was to make
Littlewood's existence proof
2676:
An equivalent definition of Skewes' number exists for
2607:
2454:
2163:
2049:
1876:
1070:. Bays and Hudson found a few much smaller values of
301:{\displaystyle e^{727.95133}<1.397\times 10^{316}.}
5026:
4838:
4809:
4780:
4703:
4615:
4495:
4371:
4310:
4194:
4053:
4033:
3981:
3836:
3816:
3746:
3245:
3166:
3143:
3123:
3099:
3068:
2964:
2875:
2855:
2835:
2799:
2694:
2653:{\displaystyle {\tfrac {1}{2}}\mathrm {li} (x^{1/2})}
2605:
2578:
2551:
2513:
2500:{\displaystyle {\tfrac {1}{2}}\mathrm {li} (x^{1/2})}
2452:
2402:
2343:
2301:
2272:
2245:
2225:
2161:
2128:
2096:
2047:
2030:{\displaystyle \pi (x)\approx \operatorname {li} (x)}
1996:
1969:
1841:
1809:
1735:
1678:
1658:
1628:
1591:
1560:
1523:
1486:
1335:
1285:
1239:
1189:
1169:
1125:
1096:
1076:
1029:
1023:
consecutive integers somewhere near this value where
1002:
965:
932:
877:
857:
827:
794:
761:
652:
629:
551:
498:
478:
448:
413:
384:
337:
308:
It is not known whether it is the smallest crossing.
265:
218:
171:
102:
79:
4280:(1914), "Sur la distribution des nombres premiers",
609:{\displaystyle e^{e^{e^{79}}}<10^{10^{10^{34}}}.}
5402:
5355:
5324:
5315:
5168:
1163:. The same source shows that there exists a number
5084:"The prime counting function and related subjects"
5059:
4864:
4824:
4795:
4736:
4648:
4528:
4404:
4331:
4227:
4086:
4039:
3996:
3869:
3822:
3770:
3254:
3226:
3149:
3129:
3105:
3081:
3054:
2950:
2861:
2841:
2821:
2781:
2652:
2591:
2564:
2533:
2499:
2435:
2368:
2322:
2287:
2258:
2231:
2196:
2143:
2114:
2082:
2029:
1975:
1952:
1824:
1768:
1715:{\displaystyle \pi (x)>\operatorname {li} (x),}
1714:
1664:
1641:
1610:
1573:
1542:
1505:
1354:
1317:
1271:
1226:{\displaystyle \pi (x)<\operatorname {li} (x),}
1225:
1175:
1143:
1111:
1082:
1062:
1015:
984:
951:
910:
863:
840:
813:
780:
716:
635:
608:
535:{\displaystyle \pi (x)<\operatorname {li} (x),}
534:
484:
454:
434:
399:
370:
300:
251:
204:
139:{\displaystyle \pi (x)>\operatorname {li} (x),}
138:
85:
4087:{\displaystyle \pi (x)>\operatorname {li} (x)}
3870:{\displaystyle \pi (x)>\operatorname {li} (x)}
2782:{\displaystyle P=(p,p+i_{1},p+i_{2},...,p+i_{k})}
2376:for zeros violating the Riemann hypothesis (with
1769:{\displaystyle \pi (x)>\operatorname {li} (x)}
1063:{\displaystyle \pi (x)>\operatorname {li} (x)}
911:{\displaystyle \pi (x)>\operatorname {li} (x)}
252:{\displaystyle \pi (x)>\operatorname {li} (x)}
205:{\displaystyle \pi (x)<\operatorname {li} (x)}
1785:
1480:proved that there are no crossover points below
1477:
4990:Computational Methods in Science and Technology
4913:Computational Methods in Science and Technology
1156:
5060:{\displaystyle \pi (x)-\operatorname {li} (x)}
4747:Proceedings of the London Mathematical Society
4737:{\displaystyle \pi (x)-\operatorname {li} (x)}
4649:{\displaystyle \pi (x)-\operatorname {li} (x)}
4529:{\displaystyle \pi (x)-\operatorname {li} (x)}
4405:{\displaystyle \pi (x)-\operatorname {li} (x)}
4228:{\displaystyle \pi (x)-\operatorname {li} (x)}
2369:{\displaystyle \operatorname {li} (x^{\rho })}
1985:non-trivial zeros of the Riemann zeta function
371:{\displaystyle \pi (x)-\operatorname {li} (x)}
5146:
2951:{\displaystyle p,p+i_{1},p+i_{2},...,p+i_{k}}
1582:
1326:
8:
3785:-tuples have a corresponding Skewes number.
2436:{\displaystyle \operatorname {li} (x^{1/2})}
2215:zero of the zeta function has quite a large
1990:The largest error term in the approximation
918:. Without assuming the Riemann hypothesis,
5321:
5153:
5139:
5131:
4659:Journal of the London Mathematical Society
3781:It is also unknown whether all admissible
3089:denote its Hardy–Littlewood constant (see
2545:, rather than the primes themselves, with
1152:
993:
328:
5025:
4887:
4854:
4849:
4837:
4808:
4779:
4702:
4614:
4551:
4494:
4472:
4427:
4370:
4345:
4309:
4252:
4193:
4109:
4052:
4032:
4010:
3980:
3940:
3897:
3835:
3815:
3745:
3244:
3203:
3193:
3171:
3165:
3142:
3122:
3098:
3073:
3067:
3037:
3010:
3004:
2999:
2973:
2965:
2963:
2942:
2911:
2892:
2874:
2854:
2834:
2804:
2798:
2770:
2739:
2720:
2693:
2637:
2633:
2618:
2606:
2604:
2579:
2577:
2556:
2550:
2514:
2512:
2484:
2480:
2465:
2453:
2451:
2420:
2416:
2401:
2357:
2342:
2300:
2271:
2250:
2244:
2224:
2188:
2183:
2162:
2160:
2127:
2095:
2074:
2069:
2048:
2046:
1995:
1968:
1945:
1933:
1914:
1901:
1896:
1875:
1840:
1808:
1734:
1677:
1657:
1633:
1627:
1602:
1590:
1565:
1559:
1534:
1522:
1497:
1485:
1346:
1334:
1309:
1290:
1284:
1263:
1244:
1238:
1188:
1168:
1124:
1095:
1075:
1028:
1007:
1001:
976:
964:
943:
931:
876:
856:
832:
826:
805:
793:
772:
760:
701:
696:
691:
672:
667:
662:
657:
651:
628:
619:Without assuming the Riemann hypothesis,
593:
588:
583:
566:
561:
556:
550:
497:
477:
447:
412:
383:
336:
289:
270:
264:
217:
170:
101:
78:
3795:Mertens' theorems § Changes in sign
3271:
1364:
1160:
923:
4365:(1987), "On the sign of the difference
2323:{\displaystyle \operatorname {li} (x),}
1777:
1672:known for certain to have the property
435:{\displaystyle \operatorname {li} (x).}
38:(more unsolved problems in mathematics)
4302:Platt, D. J.; Trudgian, T. S. (2014),
4097:International Journal of Number Theory
2115:{\displaystyle \operatorname {li} (x)}
1551:
1355:{\displaystyle 1.39716\times 10^{316}}
1144:{\displaystyle \operatorname {li} (x)}
985:{\displaystyle 1.39822\times 10^{316}}
752:
620:
465:
4144:Advances in Computational Mathematics
3737:The Skewes number (if it exists) for
1619:
1514:
34:What is the smallest Skewes's number?
7:
3325:
3304:
2685:
814:{\displaystyle 1.65\times 10^{1165}}
781:{\displaystyle 1.53\times 10^{1165}}
755:, who showed that somewhere between
623:proved that there exists a value of
1729:of the positive integers for which
2974:
2970:
2966:
2622:
2619:
2518:
2515:
2469:
2466:
1611:{\displaystyle 1.39\times 10^{17}}
43:Large number used in number theory
25:
5427:Indefinite and fictitious numbers
4865:{\displaystyle x<10^{10^{13}}}
4429:10.1090/s0025-5718-1987-0866118-6
3091:First Hardy–Littlewood conjecture
2335:Dirichlet's approximation theorem
1325:assuming the Riemann hypothesis.
5112:Asimov, I. (1976). "Skewered!".
3975:An analytic method for bounding
3808:Bays, C.; Hudson, R. H. (2000),
3237:(if such a prime exists) is the
2534:{\displaystyle \mathrm {li} (x)}
996:, who showed there are at least
952:{\displaystyle 7\times 10^{370}}
4946:American Journal of Mathematics
4889:10.1090/S0025-5718-2011-02477-4
4461:Illinois Journal of Mathematics
1543:{\displaystyle 8\times 10^{10}}
472:is true, there exists a number
468:proved that, assuming that the
29:Unsolved problem in mathematics
5054:
5048:
5036:
5030:
5002:10.12921/cmst.2011.17.01.87-92
4819:
4813:
4790:
4784:
4731:
4725:
4713:
4707:
4643:
4637:
4625:
4619:
4592:10.1080/10586458.1994.10504289
4523:
4517:
4505:
4499:
4399:
4393:
4381:
4375:
4320:
4314:
4222:
4216:
4204:
4198:
4081:
4075:
4063:
4057:
3991:
3985:
3864:
3858:
3846:
3840:
3810:"A new bound for the smallest
3765:
3747:
3218:
3212:
3183:
3177:
3034:
3021:
2989:
2983:
2816:
2810:
2776:
2701:
2647:
2626:
2592:{\displaystyle {\frac {1}{n}}}
2528:
2522:
2494:
2473:
2430:
2409:
2396:) are eventually larger than
2363:
2350:
2314:
2308:
2282:
2276:
2191:
2180:
2138:
2132:
2109:
2103:
2077:
2066:
2024:
2018:
2006:
2000:
1939:
1926:
1904:
1893:
1869:
1863:
1851:
1845:
1819:
1813:
1786:Rubinstein & Sarnak (1994)
1763:
1757:
1745:
1739:
1706:
1700:
1688:
1682:
1478:Rosser & Schoenfeld (1962)
1217:
1211:
1199:
1193:
1138:
1132:
1106:
1100:
1057:
1051:
1039:
1033:
905:
899:
887:
881:
526:
520:
508:
502:
426:
420:
394:
388:
365:
359:
347:
341:
246:
240:
228:
222:
199:
193:
181:
175:
130:
124:
112:
106:
1:
5342:Conway chained arrow notation
4553:10.1090/S0025-5718-10-02351-3
3899:10.1090/S0025-5718-99-01104-7
1157:Saouter & Demichel (2010)
163:logarithmic integral function
4332:{\displaystyle \theta (x)-x}
4304:On the first sign change of
5017:Zegowitz, Stefanie (2010),
4697:(1955), "On the difference
4609:(1933), "On the difference
4186:Lehman, R. Sherman (1966),
2822:{\displaystyle \pi _{P}(x)}
2507:is that, roughly speaking,
1652:There is no explicit value
1583:Platt & Trudgian (2014)
1327:Stoll & Demichel (2011)
926:) proved an upper bound of
5547:
5437:Largest known prime number
5114:Of Matters Great and Small
5020:On the positive region of
4925:10.12921/cmst.2019.0000033
4875:Mathematics of Computation
4539:Mathematics of Computation
4415:Mathematics of Computation
3928:Mathematics of Computation
3885:Mathematics of Computation
2541:actually counts powers of
1963:where the sum is over all
5490:
5422:Extended real number line
5337:Knuth's up-arrow notation
4796:{\displaystyle \zeta (s)}
4156:10.1007/s10444-007-9039-2
4120:10.1142/S1793042110003125
2295:is sometimes larger than
1279:. This can be reduced to
323:research supervisor, had
5347:Steinhaus–Moser notation
4579:Experimental Mathematics
3997:{\displaystyle \psi (x)}
3731:Pfoertner / Luhn (2021)
3688:Luhn / Pfoertner (2021)
3645:Pfoertner / Luhn (2021)
3137:, i.e., the first prime
3093:). Then the first prime
2793: + 1)-tuple,
2446:The reason for the term
1506:{\displaystyle x=10^{8}}
1153:Chao & Plymen (2010)
1016:{\displaystyle 10^{153}}
994:Bays & Hudson (2000)
959:. A better estimate was
841:{\displaystyle 10^{500}}
5116:. New York: Ace Books.
4825:{\displaystyle \pi (x)}
4672:10.1112/jlms/s1-8.4.277
4254:10.4064/aa-11-4-397-410
3771:{\displaystyle (p,p+6)}
3117: + 1)-tuple
2288:{\displaystyle \pi (x)}
2144:{\displaystyle \pi (x)}
2122:is usually larger than
1825:{\displaystyle \pi (x)}
1791:
1642:{\displaystyle 10^{19}}
1574:{\displaystyle 10^{14}}
1112:{\displaystyle \pi (x)}
400:{\displaystyle \pi (x)}
311:
155:prime-counting function
5390:Fast-growing hierarchy
5061:
4866:
4826:
4797:
4760:10.1112/plms/s3-5.1.48
4738:
4650:
4530:
4474:10.1215/ijm/1255631807
4406:
4333:
4229:
4088:
4041:
3998:
3871:
3824:
3772:
3256:
3228:
3151:
3131:
3107:
3083:
3056:
2952:
2863:
2843:
2823:
2783:
2654:
2593:
2566:
2535:
2501:
2437:
2370:
2324:
2289:
2260:
2233:
2198:
2145:
2116:
2084:
2031:
1977:
1954:
1826:
1770:
1716:
1666:
1643:
1612:
1575:
1544:
1507:
1356:
1319:
1273:
1227:
1177:
1145:
1113:
1084:
1064:
1017:
986:
953:
912:
865:
842:
815:
782:
718:
637:
610:
536:
486:
456:
436:
401:
372:
302:
253:
206:
140:
87:
5447:Long and short scales
5385:Grzegorczyk hierarchy
5062:
4904:Tóth, László (2019),
4867:
4827:
4798:
4739:
4651:
4531:
4407:
4334:
4230:
4089:
4042:
3999:
3872:
3825:
3773:
3257:
3229:
3152:
3132:
3108:
3084:
3082:{\displaystyle C_{P}}
3057:
2953:
2864:
2844:
2829:the number of primes
2824:
2784:
2668:Equivalent for prime
2655:
2594:
2567:
2565:{\displaystyle p^{n}}
2536:
2502:
2438:
2371:
2325:
2290:
2261:
2259:{\displaystyle 2^{N}}
2234:
2199:
2146:
2117:
2085:
2041:is true) is negative
2032:
1978:
1976:{\displaystyle \rho }
1955:
1827:
1771:
1717:
1667:
1644:
1613:
1576:
1545:
1508:
1448:Saouter and Demichel
1357:
1320:
1274:
1228:
1178:
1146:
1114:
1085:
1065:
1018:
987:
954:
920:H. J. J. te Riele
913:
866:
843:
816:
783:
749:Riemann zeta function
739:More recent estimates
719:
638:
611:
537:
487:
457:
437:
407:was always less than
402:
373:
303:
254:
207:
141:
88:
5082:Demichels, Patrick.
5024:
4981:Wolf, Marek (2011),
4836:
4807:
4778:
4701:
4613:
4493:
4369:
4308:
4192:
4051:
4031:
3979:
3834:
3814:
3744:
3243:
3164:
3141:
3121:
3097:
3066:
2962:
2873:
2853:
2833:
2797:
2692:
2603:
2576:
2549:
2511:
2450:
2400:
2341:
2299:
2270:
2266:. This explains why
2243:
2223:
2159:
2126:
2094:
2045:
1994:
1967:
1839:
1807:
1733:
1676:
1656:
1626:
1589:
1558:
1521:
1484:
1333:
1283:
1237:
1187:
1167:
1123:
1094:
1074:
1027:
1000:
963:
930:
875:
855:
825:
821:there are more than
792:
759:
650:
627:
549:
496:
476:
446:
411:
382:
335:
263:
216:
169:
100:
77:
5462:Orders of magnitude
5332:Scientific notation
4356:2014arXiv1407.1914P
4188:"On the difference
4021:2015arXiv151102032B
3973:Büthe, Jan (2015),
3514: + 12,
3510: + 10,
3479: + 10,
3009:
2958:are all prime, let
1782:logarithmic density
1470:Stoll and Demichel
5380:Ackermann function
5057:
4882:(276): 2381–2394,
4862:
4822:
4793:
4734:
4646:
4574:"Chebyshev's bias"
4546:(272): 2395–2405,
4526:
4402:
4363:te Riele, H. J. J.
4329:
4225:
4084:
4037:
3994:
3892:(231): 1285–1296,
3867:
3820:
3778:is still unknown.
3768:
3728:750247439134737983
3685:523250002674163757
3518: + 16)
3506: + 6,
3502: + 4,
3483: + 12)
3475: +6 ,
3471: +4 ,
3452: + 12)
3448: + 8,
3444: + 6,
3440: + 2,
3421: + 10)
3417: +6 ,
3413: + 4,
3390: + 6,
3386: + 2,
3363: + 4,
3340: + 2,
3255:{\displaystyle P.}
3252:
3239:Skewes number for
3224:
3147:
3127:
3103:
3079:
3052:
2995:
2948:
2859:
2839:
2819:
2779:
2650:
2616:
2589:
2562:
2531:
2497:
2463:
2433:
2366:
2320:
2285:
2256:
2229:
2194:
2172:
2141:
2112:
2080:
2058:
2039:Riemann hypothesis
2027:
1973:
1950:
1919:
1885:
1822:
1766:
1712:
1662:
1639:
1608:
1571:
1540:
1503:
1352:
1315:
1269:
1223:
1173:
1141:
1109:
1080:
1060:
1013:
982:
949:
908:
861:
838:
811:
778:
714:
633:
606:
532:
482:
470:Riemann hypothesis
452:
432:
397:
368:
298:
249:
202:
136:
83:
54:is any of several
5508:
5507:
5398:
5397:
4803:complex zeros on
4278:Littlewood, J. E.
4040:{\displaystyle x}
3823:{\displaystyle x}
3735:
3734:
3602:Pfoertner (2020)
3563:Pfoertner (2020)
3394: + 8)
3367: + 6)
3344: + 6)
3319: + 4)
3298: + 2)
3150:{\displaystyle p}
3130:{\displaystyle P}
3106:{\displaystyle p}
3050:
2862:{\displaystyle x}
2842:{\displaystyle p}
2615:
2587:
2462:
2232:{\displaystyle N}
2189:
2171:
2075:
2057:
1948:
1910:
1902:
1884:
1792:Riemann's formula
1665:{\displaystyle x}
1474:
1473:
1176:{\displaystyle x}
1083:{\displaystyle x}
864:{\displaystyle x}
636:{\displaystyle x}
485:{\displaystyle x}
455:{\displaystyle x}
329:Littlewood (1914)
86:{\displaystyle x}
70:for the smallest
16:(Redirected from
5538:
5322:
5252:Eddington number
5197:Hundred thousand
5155:
5148:
5141:
5132:
5127:
5108:
5106:
5105:
5099:
5093:. Archived from
5088:
5070:
5066:
5064:
5063:
5058:
5012:
4987:
4977:
4935:
4910:
4900:
4891:
4871:
4869:
4868:
4863:
4861:
4860:
4859:
4858:
4831:
4829:
4828:
4823:
4802:
4800:
4799:
4794:
4770:
4743:
4741:
4740:
4735:
4690:
4655:
4653:
4652:
4647:
4602:
4568:Rubinstein, M.;
4564:
4555:
4535:
4533:
4532:
4527:
4485:
4476:
4448:
4431:
4422:(177): 323–328,
4411:
4409:
4408:
4403:
4358:
4349:
4338:
4336:
4335:
4330:
4298:
4273:
4256:
4240:Acta Arithmetica
4234:
4232:
4231:
4226:
4182:
4138:
4113:
4093:
4091:
4090:
4085:
4046:
4044:
4043:
4038:
4023:
4014:
4003:
4001:
4000:
3995:
3969:
3944:
3918:
3901:
3881:
3876:
3874:
3873:
3868:
3829:
3827:
3826:
3821:
3777:
3775:
3774:
3769:
3642:1203255673037261
3272:
3261:
3259:
3258:
3253:
3233:
3231:
3230:
3225:
3208:
3207:
3198:
3197:
3176:
3175:
3156:
3154:
3153:
3148:
3136:
3134:
3133:
3128:
3112:
3110:
3109:
3104:
3088:
3086:
3085:
3080:
3078:
3077:
3061:
3059:
3058:
3053:
3051:
3049:
3048:
3047:
3019:
3011:
3008:
3003:
2979:
2978:
2977:
2957:
2955:
2954:
2949:
2947:
2946:
2916:
2915:
2897:
2896:
2868:
2866:
2865:
2860:
2848:
2846:
2845:
2840:
2828:
2826:
2825:
2820:
2809:
2808:
2789:denote a prime (
2788:
2786:
2785:
2780:
2775:
2774:
2744:
2743:
2725:
2724:
2659:
2657:
2656:
2651:
2646:
2645:
2641:
2625:
2617:
2608:
2598:
2596:
2595:
2590:
2588:
2580:
2571:
2569:
2568:
2563:
2561:
2560:
2540:
2538:
2537:
2532:
2521:
2506:
2504:
2503:
2498:
2493:
2492:
2488:
2472:
2464:
2455:
2442:
2440:
2439:
2434:
2429:
2428:
2424:
2395:
2393:
2392:
2389:
2386:
2375:
2373:
2372:
2367:
2362:
2361:
2329:
2327:
2326:
2321:
2294:
2292:
2291:
2286:
2265:
2263:
2262:
2257:
2255:
2254:
2238:
2236:
2235:
2230:
2203:
2201:
2200:
2195:
2190:
2184:
2173:
2164:
2150:
2148:
2147:
2142:
2121:
2119:
2118:
2113:
2089:
2087:
2086:
2081:
2076:
2070:
2059:
2050:
2036:
2034:
2033:
2028:
1982:
1980:
1979:
1974:
1959:
1957:
1956:
1951:
1949:
1946:
1938:
1937:
1918:
1903:
1897:
1886:
1877:
1831:
1829:
1828:
1823:
1801:explicit formula
1780:showed that the
1776:does not exist,
1775:
1773:
1772:
1767:
1725:Even though the
1721:
1719:
1718:
1713:
1671:
1669:
1668:
1663:
1648:
1646:
1645:
1640:
1638:
1637:
1617:
1615:
1614:
1609:
1607:
1606:
1580:
1578:
1577:
1572:
1570:
1569:
1549:
1547:
1546:
1541:
1539:
1538:
1512:
1510:
1509:
1504:
1502:
1501:
1466:
1459:
1444:
1437:
1426:Chao and Plymen
1422:
1415:
1404:Bays and Hudson
1400:
1393:
1365:
1361:
1359:
1358:
1353:
1351:
1350:
1324:
1322:
1321:
1316:
1314:
1313:
1295:
1294:
1278:
1276:
1275:
1270:
1268:
1267:
1249:
1248:
1232:
1230:
1229:
1224:
1182:
1180:
1179:
1174:
1150:
1148:
1147:
1142:
1118:
1116:
1115:
1110:
1089:
1087:
1086:
1081:
1069:
1067:
1066:
1061:
1022:
1020:
1019:
1014:
1012:
1011:
991:
989:
988:
983:
981:
980:
958:
956:
955:
950:
948:
947:
917:
915:
914:
909:
870:
868:
867:
862:
847:
845:
844:
839:
837:
836:
820:
818:
817:
812:
810:
809:
787:
785:
784:
779:
777:
776:
723:
721:
720:
715:
710:
709:
708:
707:
706:
705:
683:
682:
681:
680:
679:
678:
677:
676:
642:
640:
639:
634:
615:
613:
612:
607:
602:
601:
600:
599:
598:
597:
575:
574:
573:
572:
571:
570:
541:
539:
538:
533:
491:
489:
488:
483:
461:
459:
458:
453:
441:
439:
438:
433:
406:
404:
403:
398:
377:
375:
374:
369:
312:Skewes's numbers
307:
305:
304:
299:
294:
293:
275:
274:
258:
256:
255:
250:
211:
209:
208:
203:
160:
152:
145:
143:
142:
137:
92:
90:
89:
84:
30:
21:
5546:
5545:
5541:
5540:
5539:
5537:
5536:
5535:
5511:
5510:
5509:
5504:
5486:
5442:List of numbers
5410:
5408:
5406:
5404:
5394:
5351:
5317:
5311:
5282:Graham's number
5272:Skewes's number
5174:
5172:
5170:
5164:
5159:
5124:
5111:
5103:
5101:
5097:
5086:
5081:
5078:
5073:
5022:
5021:
5016:
4985:
4980:
4959:10.2307/2371519
4939:
4908:
4903:
4850:
4845:
4834:
4833:
4805:
4804:
4776:
4775:
4773:
4699:
4698:
4693:
4611:
4610:
4605:
4567:
4491:
4490:
4488:
4451:
4367:
4366:
4361:
4306:
4305:
4301:
4276:
4190:
4189:
4185:
4141:
4049:
4048:
4029:
4028:
4026:
3977:
3976:
3972:
3942:10.2307/2005460
3921:
3879:
3832:
3831:
3812:
3811:
3807:
3803:
3791:
3742:
3741:
3599:214159878489239
3241:
3240:
3199:
3189:
3167:
3162:
3161:
3139:
3138:
3119:
3118:
3095:
3094:
3069:
3064:
3063:
3033:
3020:
3012:
2969:
2960:
2959:
2938:
2907:
2888:
2871:
2870:
2851:
2850:
2831:
2830:
2800:
2795:
2794:
2766:
2735:
2716:
2690:
2689:
2674:
2629:
2601:
2600:
2574:
2573:
2552:
2547:
2546:
2509:
2508:
2476:
2448:
2447:
2412:
2398:
2397:
2390:
2387:
2384:
2383:
2381:
2353:
2339:
2338:
2297:
2296:
2268:
2267:
2246:
2241:
2240:
2221:
2220:
2157:
2156:
2124:
2123:
2092:
2091:
2090:, showing that
2043:
2042:
1992:
1991:
1965:
1964:
1929:
1837:
1836:
1805:
1804:
1794:
1731:
1730:
1727:natural density
1674:
1673:
1654:
1653:
1629:
1624:
1623:
1598:
1587:
1586:
1561:
1556:
1555:
1530:
1519:
1518:
1493:
1482:
1481:
1464:
1457:
1442:
1435:
1420:
1413:
1398:
1391:
1378:
1342:
1331:
1330:
1305:
1286:
1281:
1280:
1259:
1240:
1235:
1234:
1185:
1184:
1165:
1164:
1161:Zegowitz (2010)
1121:
1120:
1092:
1091:
1072:
1071:
1025:
1024:
1003:
998:
997:
972:
961:
960:
939:
928:
927:
873:
872:
853:
852:
828:
823:
822:
801:
790:
789:
768:
757:
756:
741:
697:
692:
687:
668:
663:
658:
653:
648:
647:
625:
624:
589:
584:
579:
562:
557:
552:
547:
546:
494:
493:
474:
473:
444:
443:
409:
408:
380:
379:
333:
332:
317:J.E. Littlewood
314:
285:
266:
261:
260:
214:
213:
167:
166:
158:
150:
98:
97:
75:
74:
52:Skewes's number
44:
41:
40:
35:
32:
28:
23:
22:
15:
12:
11:
5:
5544:
5542:
5534:
5533:
5531:Large integers
5528:
5523:
5513:
5512:
5506:
5505:
5503:
5502:
5497:
5491:
5488:
5487:
5485:
5484:
5479:
5474:
5472:Power of three
5469:
5464:
5459:
5454:
5452:Number systems
5449:
5444:
5439:
5434:
5429:
5424:
5419:
5413:
5411:
5407:(alphabetical
5400:
5399:
5396:
5395:
5393:
5392:
5387:
5382:
5377:
5376:
5375:
5370:
5363:Hyperoperation
5359:
5357:
5353:
5352:
5350:
5349:
5344:
5339:
5334:
5328:
5326:
5319:
5313:
5312:
5310:
5309:
5304:
5299:
5294:
5289:
5284:
5279:
5277:Moser's number
5274:
5269:
5264:
5262:Shannon number
5259:
5254:
5249:
5244:
5239:
5234:
5229:
5224:
5219:
5214:
5209:
5204:
5199:
5194:
5189:
5184:
5178:
5176:
5166:
5165:
5160:
5158:
5157:
5150:
5143:
5135:
5129:
5128:
5123:978-0441610723
5122:
5109:
5100:on Sep 8, 2006
5077:
5076:External links
5074:
5072:
5071:
5056:
5053:
5050:
5047:
5044:
5041:
5038:
5035:
5032:
5029:
5014:
4978:
4953:(2): 233–248,
4937:
4901:
4857:
4853:
4848:
4844:
4841:
4821:
4818:
4815:
4812:
4792:
4789:
4786:
4783:
4771:
4733:
4730:
4727:
4724:
4721:
4718:
4715:
4712:
4709:
4706:
4691:
4645:
4642:
4639:
4636:
4633:
4630:
4627:
4624:
4621:
4618:
4603:
4586:(3): 173–197,
4565:
4536:is positive",
4525:
4522:
4519:
4516:
4513:
4510:
4507:
4504:
4501:
4498:
4486:
4457:Schoenfeld, L.
4449:
4401:
4398:
4395:
4392:
4389:
4386:
4383:
4380:
4377:
4374:
4359:
4328:
4325:
4322:
4319:
4316:
4313:
4299:
4283:Comptes Rendus
4274:
4224:
4221:
4218:
4215:
4212:
4209:
4206:
4203:
4200:
4197:
4183:
4139:
4104:(3): 681–690,
4083:
4080:
4077:
4074:
4071:
4068:
4065:
4062:
4059:
4056:
4036:
4024:
3993:
3990:
3987:
3984:
3970:
3935:(129): 43–56,
3919:
3866:
3863:
3860:
3857:
3854:
3851:
3848:
3845:
3842:
3839:
3819:
3804:
3802:
3799:
3798:
3797:
3790:
3787:
3767:
3764:
3761:
3758:
3755:
3752:
3749:
3733:
3732:
3729:
3726:
3690:
3689:
3686:
3683:
3647:
3646:
3643:
3640:
3604:
3603:
3600:
3597:
3565:
3564:
3561:
3558:
3526:
3525:
3522:
3519:
3491:
3490:
3487:
3484:
3460:
3459:
3456:
3453:
3429:
3428:
3425:
3422:
3402:
3401:
3398:
3395:
3375:
3374:
3371:
3368:
3352:
3351:
3348:
3345:
3329:
3328:
3323:
3320:
3308:
3307:
3302:
3299:
3287:
3286:
3283:
3280:
3251:
3248:
3235:
3234:
3223:
3220:
3217:
3214:
3211:
3206:
3202:
3196:
3192:
3188:
3185:
3182:
3179:
3174:
3170:
3146:
3126:
3102:
3076:
3072:
3046:
3043:
3040:
3036:
3032:
3029:
3026:
3023:
3018:
3015:
3007:
3002:
2998:
2994:
2991:
2988:
2985:
2982:
2976:
2972:
2968:
2945:
2941:
2937:
2934:
2931:
2928:
2925:
2922:
2919:
2914:
2910:
2906:
2903:
2900:
2895:
2891:
2887:
2884:
2881:
2878:
2858:
2838:
2818:
2815:
2812:
2807:
2803:
2778:
2773:
2769:
2765:
2762:
2759:
2756:
2753:
2750:
2747:
2742:
2738:
2734:
2731:
2728:
2723:
2719:
2715:
2712:
2709:
2706:
2703:
2700:
2697:
2673:
2666:
2649:
2644:
2640:
2636:
2632:
2628:
2624:
2621:
2614:
2611:
2586:
2583:
2559:
2555:
2530:
2527:
2524:
2520:
2517:
2496:
2491:
2487:
2483:
2479:
2475:
2471:
2468:
2461:
2458:
2432:
2427:
2423:
2419:
2415:
2411:
2408:
2405:
2365:
2360:
2356:
2352:
2349:
2346:
2319:
2316:
2313:
2310:
2307:
2304:
2284:
2281:
2278:
2275:
2253:
2249:
2228:
2217:imaginary part
2193:
2187:
2182:
2179:
2176:
2170:
2167:
2140:
2137:
2134:
2131:
2111:
2108:
2105:
2102:
2099:
2079:
2073:
2068:
2065:
2062:
2056:
2053:
2026:
2023:
2020:
2017:
2014:
2011:
2008:
2005:
2002:
1999:
1983:in the set of
1972:
1961:
1960:
1944:
1941:
1936:
1932:
1928:
1925:
1922:
1917:
1913:
1909:
1906:
1900:
1895:
1892:
1889:
1883:
1880:
1874:
1871:
1868:
1865:
1862:
1859:
1856:
1853:
1850:
1847:
1844:
1821:
1818:
1815:
1812:
1793:
1790:
1778:Wintner (1941)
1765:
1762:
1759:
1756:
1753:
1750:
1747:
1744:
1741:
1738:
1711:
1708:
1705:
1702:
1699:
1696:
1693:
1690:
1687:
1684:
1681:
1661:
1636:
1632:
1605:
1601:
1597:
1594:
1568:
1564:
1537:
1533:
1529:
1526:
1513:, improved by
1500:
1496:
1492:
1489:
1472:
1471:
1468:
1461:
1454:
1450:
1449:
1446:
1439:
1432:
1428:
1427:
1424:
1417:
1410:
1406:
1405:
1402:
1395:
1388:
1384:
1383:
1380:
1375:
1369:
1349:
1345:
1341:
1338:
1312:
1308:
1304:
1301:
1298:
1293:
1289:
1266:
1262:
1258:
1255:
1252:
1247:
1243:
1222:
1219:
1216:
1213:
1210:
1207:
1204:
1201:
1198:
1195:
1192:
1172:
1140:
1137:
1134:
1131:
1128:
1119:gets close to
1108:
1105:
1102:
1099:
1079:
1059:
1056:
1053:
1050:
1047:
1044:
1041:
1038:
1035:
1032:
1010:
1006:
992:discovered by
979:
975:
971:
968:
946:
942:
938:
935:
907:
904:
901:
898:
895:
892:
889:
886:
883:
880:
860:
835:
831:
808:
804:
800:
797:
775:
771:
767:
764:
740:
737:
725:
724:
713:
704:
700:
695:
690:
686:
675:
671:
666:
661:
656:
632:
617:
616:
605:
596:
592:
587:
582:
578:
569:
565:
560:
555:
531:
528:
525:
522:
519:
516:
513:
510:
507:
504:
501:
481:
451:
431:
428:
425:
422:
419:
416:
396:
393:
390:
387:
367:
364:
361:
358:
355:
352:
349:
346:
343:
340:
313:
310:
297:
292:
288:
284:
281:
278:
273:
269:
248:
245:
242:
239:
236:
233:
230:
227:
224:
221:
201:
198:
195:
192:
189:
186:
183:
180:
177:
174:
147:
146:
135:
132:
129:
126:
123:
120:
117:
114:
111:
108:
105:
82:
72:natural number
64:Stanley Skewes
62:mathematician
42:
36:
33:
27:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5543:
5532:
5529:
5527:
5526:Number theory
5524:
5522:
5521:Large numbers
5519:
5518:
5516:
5501:
5498:
5496:
5493:
5492:
5489:
5483:
5480:
5478:
5475:
5473:
5470:
5468:
5465:
5463:
5460:
5458:
5455:
5453:
5450:
5448:
5445:
5443:
5440:
5438:
5435:
5433:
5432:Infinitesimal
5430:
5428:
5425:
5423:
5420:
5418:
5415:
5414:
5412:
5401:
5391:
5388:
5386:
5383:
5381:
5378:
5374:
5371:
5369:
5366:
5365:
5364:
5361:
5360:
5358:
5354:
5348:
5345:
5343:
5340:
5338:
5335:
5333:
5330:
5329:
5327:
5323:
5320:
5314:
5308:
5305:
5303:
5302:Rayo's number
5300:
5298:
5295:
5293:
5290:
5288:
5285:
5283:
5280:
5278:
5275:
5273:
5270:
5268:
5265:
5263:
5260:
5258:
5255:
5253:
5250:
5248:
5245:
5243:
5240:
5238:
5235:
5233:
5230:
5228:
5225:
5223:
5220:
5218:
5215:
5213:
5210:
5208:
5205:
5203:
5200:
5198:
5195:
5193:
5190:
5188:
5185:
5183:
5180:
5179:
5177:
5167:
5163:
5162:Large numbers
5156:
5151:
5149:
5144:
5142:
5137:
5136:
5133:
5125:
5119:
5115:
5110:
5096:
5092:
5085:
5080:
5079:
5075:
5068:
5067:
5051:
5045:
5042:
5039:
5033:
5027:
5015:
5011:
5007:
5003:
4999:
4995:
4991:
4984:
4979:
4976:
4972:
4968:
4964:
4960:
4956:
4952:
4948:
4947:
4942:
4938:
4934:
4930:
4926:
4922:
4918:
4914:
4907:
4902:
4899:
4895:
4890:
4885:
4881:
4877:
4876:
4855:
4851:
4846:
4842:
4839:
4816:
4810:
4787:
4781:
4772:
4769:
4765:
4761:
4757:
4753:
4749:
4748:
4728:
4722:
4719:
4716:
4710:
4704:
4696:
4692:
4689:
4685:
4681:
4677:
4673:
4669:
4665:
4661:
4660:
4640:
4634:
4631:
4628:
4622:
4616:
4608:
4604:
4601:
4597:
4593:
4589:
4585:
4581:
4580:
4575:
4571:
4566:
4563:
4559:
4554:
4549:
4545:
4541:
4540:
4520:
4514:
4511:
4508:
4502:
4496:
4487:
4484:
4480:
4475:
4470:
4466:
4462:
4458:
4454:
4453:Rosser, J. B.
4450:
4447:
4443:
4439:
4435:
4430:
4425:
4421:
4417:
4416:
4396:
4390:
4387:
4384:
4378:
4372:
4364:
4360:
4357:
4353:
4348:
4343:
4339:
4326:
4323:
4317:
4311:
4300:
4297:
4293:
4290:: 1869–1872,
4289:
4285:
4284:
4279:
4275:
4272:
4268:
4264:
4260:
4255:
4250:
4246:
4242:
4241:
4236:
4219:
4213:
4210:
4207:
4201:
4195:
4184:
4181:
4177:
4173:
4169:
4165:
4161:
4157:
4153:
4149:
4145:
4140:
4137:
4133:
4129:
4125:
4121:
4117:
4112:
4107:
4103:
4099:
4098:
4078:
4072:
4069:
4066:
4060:
4054:
4034:
4025:
4022:
4018:
4013:
4008:
4004:
3988:
3982:
3971:
3968:
3964:
3960:
3956:
3952:
3948:
3943:
3938:
3934:
3930:
3929:
3924:
3920:
3917:
3913:
3909:
3905:
3900:
3895:
3891:
3887:
3886:
3878:
3861:
3855:
3852:
3849:
3843:
3837:
3817:
3806:
3805:
3800:
3796:
3793:
3792:
3788:
3786:
3784:
3779:
3762:
3759:
3756:
3753:
3750:
3740:
3730:
3727:
3724:
3720:
3716:
3712:
3708:
3704:
3700:
3696:
3692:
3691:
3687:
3684:
3681:
3677:
3673:
3669:
3665:
3661:
3657:
3653:
3649:
3648:
3644:
3641:
3638:
3634:
3630:
3626:
3622:
3618:
3614:
3610:
3606:
3605:
3601:
3598:
3595:
3591:
3587:
3583:
3579:
3575:
3571:
3567:
3566:
3562:
3560:7572964186421
3559:
3556:
3552:
3548:
3544:
3540:
3536:
3532:
3528:
3527:
3523:
3520:
3517:
3513:
3509:
3505:
3501:
3497:
3493:
3492:
3488:
3485:
3482:
3478:
3474:
3470:
3466:
3462:
3461:
3457:
3454:
3451:
3447:
3443:
3439:
3435:
3431:
3430:
3426:
3423:
3420:
3416:
3412:
3408:
3404:
3403:
3399:
3396:
3393:
3389:
3385:
3381:
3377:
3376:
3372:
3369:
3366:
3362:
3358:
3354:
3353:
3349:
3346:
3343:
3339:
3335:
3331:
3330:
3327:
3324:
3321:
3318:
3314:
3310:
3309:
3306:
3303:
3300:
3297:
3293:
3289:
3288:
3284:
3282:Skewes number
3281:
3278:
3274:
3273:
3270:
3268:
3263:
3262:
3249:
3246:
3221:
3215:
3209:
3204:
3200:
3194:
3190:
3186:
3180:
3172:
3168:
3160:
3159:
3158:
3144:
3124:
3116:
3100:
3092:
3074:
3070:
3044:
3041:
3038:
3030:
3027:
3024:
3016:
3013:
3005:
3000:
2996:
2992:
2986:
2980:
2943:
2939:
2935:
2932:
2929:
2926:
2923:
2920:
2917:
2912:
2908:
2904:
2901:
2898:
2893:
2889:
2885:
2882:
2879:
2876:
2856:
2836:
2813:
2805:
2801:
2792:
2771:
2767:
2763:
2760:
2757:
2754:
2751:
2748:
2745:
2740:
2736:
2732:
2729:
2726:
2721:
2717:
2713:
2710:
2707:
2704:
2698:
2695:
2687:
2683:
2681:
2671:
2667:
2665:
2663:
2642:
2638:
2634:
2630:
2612:
2609:
2584:
2581:
2557:
2553:
2544:
2525:
2489:
2485:
2481:
2477:
2459:
2456:
2444:
2425:
2421:
2417:
2413:
2406:
2403:
2380:greater than
2379:
2358:
2354:
2347:
2344:
2336:
2331:
2317:
2311:
2305:
2302:
2279:
2273:
2251:
2247:
2226:
2218:
2214:
2210:
2205:
2185:
2177:
2174:
2168:
2165:
2154:
2135:
2129:
2106:
2100:
2097:
2071:
2063:
2060:
2054:
2051:
2040:
2021:
2015:
2012:
2009:
2003:
1997:
1988:
1986:
1970:
1947:smaller terms
1942:
1934:
1930:
1923:
1920:
1915:
1911:
1907:
1898:
1890:
1887:
1881:
1878:
1872:
1866:
1860:
1857:
1854:
1848:
1842:
1835:
1834:
1833:
1816:
1810:
1802:
1798:
1789:
1787:
1783:
1779:
1760:
1754:
1751:
1748:
1742:
1736:
1728:
1723:
1709:
1703:
1697:
1694:
1691:
1685:
1679:
1659:
1650:
1634:
1630:
1621:
1603:
1599:
1595:
1592:
1584:
1566:
1562:
1553:
1552:Kotnik (2008)
1535:
1531:
1527:
1524:
1516:
1498:
1494:
1490:
1487:
1479:
1469:
1462:
1455:
1452:
1451:
1447:
1440:
1433:
1430:
1429:
1425:
1418:
1411:
1408:
1407:
1403:
1396:
1389:
1386:
1385:
1381:
1376:
1374:
1370:
1367:
1366:
1363:
1347:
1343:
1339:
1336:
1328:
1310:
1306:
1302:
1299:
1296:
1291:
1287:
1264:
1260:
1256:
1253:
1250:
1245:
1241:
1220:
1214:
1208:
1205:
1202:
1196:
1190:
1170:
1162:
1158:
1154:
1135:
1129:
1126:
1103:
1097:
1077:
1054:
1048:
1045:
1042:
1036:
1030:
1008:
1004:
995:
977:
973:
969:
966:
944:
940:
936:
933:
925:
921:
902:
896:
893:
890:
884:
878:
858:
851:
833:
829:
806:
802:
798:
795:
773:
769:
765:
762:
754:
753:Lehman (1966)
750:
746:
738:
736:
734:
733:Georg Kreisel
730:
711:
702:
698:
693:
688:
684:
673:
669:
664:
659:
654:
646:
645:
644:
630:
622:
621:Skewes (1955)
603:
594:
590:
585:
580:
576:
567:
563:
558:
553:
545:
544:
543:
529:
523:
517:
514:
511:
505:
499:
479:
471:
467:
466:Skewes (1933)
463:
449:
429:
423:
417:
414:
391:
385:
362:
356:
353:
350:
344:
338:
330:
326:
322:
318:
309:
295:
290:
286:
282:
279:
276:
271:
267:
243:
237:
234:
231:
225:
219:
196:
190:
187:
184:
178:
172:
164:
156:
133:
127:
121:
118:
115:
109:
103:
96:
95:
94:
80:
73:
69:
65:
61:
60:South African
57:
56:large numbers
53:
49:
48:number theory
39:
19:
18:Skewes number
5467:Power of two
5457:Number names
5271:
5192:Ten thousand
5113:
5102:. Retrieved
5095:the original
5090:
5019:
4993:
4989:
4950:
4944:
4916:
4912:
4879:
4873:
4751:
4745:
4663:
4657:
4583:
4577:
4543:
4537:
4464:
4460:
4419:
4413:
4303:
4287:
4281:
4244:
4238:
4150:(1): 55–70,
4147:
4143:
4111:math/0509312
4101:
4095:
3974:
3932:
3926:
3923:Brent, R. P.
3889:
3883:
3782:
3780:
3736:
3722:
3718:
3714:
3710:
3706:
3702:
3698:
3694:
3679:
3675:
3671:
3667:
3663:
3659:
3655:
3651:
3636:
3632:
3628:
3624:
3620:
3616:
3612:
3608:
3593:
3589:
3585:
3581:
3577:
3573:
3569:
3554:
3550:
3546:
3542:
3538:
3534:
3530:
3524:Tóth (2019)
3521:251331775687
3515:
3511:
3507:
3503:
3499:
3495:
3489:Tóth (2019)
3480:
3476:
3472:
3468:
3464:
3458:Tóth (2019)
3449:
3445:
3441:
3437:
3433:
3427:Tóth (2019)
3418:
3414:
3410:
3406:
3400:Tóth (2019)
3391:
3387:
3383:
3379:
3373:Tóth (2019)
3364:
3360:
3356:
3350:Tóth (2019)
3341:
3337:
3333:
3316:
3312:
3295:
3291:
3276:
3266:
3264:
3238:
3236:
3114:
2790:
2679:
2675:
2669:
2572:weighted by
2445:
2332:
2208:
2206:
1989:
1962:
1795:
1724:
1651:
1620:Büthe (2015)
1515:Brent (1975)
1476:Rigorously,
1475:
1377:# of complex
1372:
848:consecutive
742:
726:
618:
464:
315:
148:
68:upper bounds
58:used by the
51:
45:
5477:Power of 10
5417:Busy beaver
5222:Quintillion
5217:Quadrillion
4941:Wintner, A.
4666:: 277–283,
4247:: 397–410,
3739:sexy primes
3326:Tóth (2019)
3305:Wolf (2011)
2686:Tóth (2019)
2664:of primes.
2599:. The term
1292:727.9513386
1246:727.9513468
5515:Categories
5482:Sagan Unit
5316:Expression
5267:Googolplex
5232:Septillion
5227:Sextillion
5173:numerical
5104:2009-09-29
4695:Skewes, S.
4688:0007.34003
4680:59.0370.02
4607:Skewes, S.
4570:Sarnak, P.
4296:45.0305.01
4271:0151.04101
4180:1149.11004
4136:1215.11084
4012:1511.02032
3967:0295.10002
3916:1042.11001
3801:References
3157:such that
2869:such that
1379:zeros used
1183:violating
492:violating
319:, who was
93:for which
5373:Pentation
5368:Tetration
5356:Operators
5325:Notations
5247:Decillion
5242:Nonillion
5237:Octillion
5169:Examples
5046:
5040:−
5028:π
4996:: 87–92,
4933:203836016
4811:π
4782:ζ
4754:: 48–70,
4723:
4717:−
4705:π
4635:
4629:−
4617:π
4515:
4509:−
4497:π
4467:: 64–94,
4391:
4385:−
4373:π
4347:1407.1914
4324:−
4312:θ
4214:
4208:−
4196:π
4073:
4055:π
3983:ψ
3856:
3838:π
3486:216646267
3424:827929093
3285:Found by
3269:-tuples:
3210:
3169:π
3028:
2997:∫
2981:
2802:π
2407:
2378:real part
2359:ρ
2348:
2306:
2274:π
2178:
2153:arguments
2130:π
2101:
2064:
2016:
2010:≈
1998:π
1971:ρ
1935:ρ
1924:
1916:ρ
1912:∑
1908:−
1891:
1873:−
1861:
1843:π
1811:π
1755:
1737:π
1698:
1680:π
1618:, and by
1596:×
1528:×
1340:×
1303:×
1257:×
1209:
1191:π
1130:
1098:π
1049:
1031:π
970:×
937:×
897:
879:π
799:×
766:×
729:effective
518:
500:π
418:
386:π
357:
351:−
339:π
283:×
272:727.95133
238:
220:π
191:
173:π
122:
104:π
5405:articles
5403:Related
5307:Infinity
5212:Trillion
5187:Thousand
5091:Demichel
5010:59578795
4572:(1994),
4172:18991347
3789:See also
3455:21432401
3347:87613571
3062:and let
2037:(if the
1799:gave an
1456:1.397162
1434:1.397166
850:integers
321:Skewes's
5500:History
5318:methods
5292:SSCG(3)
5287:TREE(3)
5207:Billion
5202:Million
5182:Hundred
4975:0004255
4967:2371519
4898:2813366
4768:0067145
4744:(II)",
4600:1329368
4562:2684372
4483:0137689
4446:0866118
4438:2007893
4352:Bibcode
4263:0202686
4164:2420864
4128:2652902
4017:Bibcode
3959:0369287
3951:2005460
3908:1752093
3397:1172531
3322:5206837
3301:1369391
2688:). Let
2682:-tuples
2672:-tuples
2662:squares
2394:
2382:
2213:complex
1797:Riemann
1412:1.39801
1390:1.39822
1337:1.39716
1300:1.39717
1254:1.39718
967:1.39822
922: (
747:of the
161:is the
153:is the
5409:order)
5257:Googol
5120:
5008:
4973:
4965:
4931:
4896:
4766:
4686:
4678:
4598:
4560:
4481:
4444:
4436:
4294:
4269:
4261:
4178:
4170:
4162:
4134:
4126:
3965:
3957:
3949:
3914:
3906:
3370:337867
3279:-tuple
3275:Prime
2849:below
2678:prime
2543:primes
1233:below
1090:where
643:below
542:below
325:proved
151:π
149:where
5495:Names
5297:BH(3)
5175:order
5098:(PDF)
5087:(PDF)
5006:S2CID
4986:(PDF)
4963:JSTOR
4929:S2CID
4919:(3),
4909:(PDF)
4434:JSTOR
4342:arXiv
4168:S2CID
4106:arXiv
4047:with
4007:arXiv
3947:JSTOR
3880:(PDF)
3830:with
3721:+24,
3717:+20,
3713:+18,
3709:+14,
3678:+24,
3674:+20,
3670:+14,
3666:+12,
3635:+20,
3631:+18,
3627:+12,
3592:+18,
3588:+14,
3584:+12,
3553:+18,
3549:+12,
1581:, by
1550:, by
1371:near
1329:gave
871:with
745:zeros
674:7.705
280:1.397
259:near
5118:ISBN
4843:<
4832:for
4067:>
3850:>
3725:+26)
3705:+8,
3701:+6,
3682:+26)
3662:+6,
3658:+2,
3639:+26)
3623:+8,
3619:+6,
3615:+2,
3596:+20)
3580:+8,
3576:+2,
3557:+20)
3545:+8,
3541:+6,
3537:+2,
3187:>
1803:for
1749:>
1692:>
1593:1.39
1453:2011
1431:2010
1409:2010
1387:2000
1368:Year
1297:<
1251:<
1203:<
1043:>
924:1987
891:>
807:1165
796:1.65
788:and
774:1165
763:1.53
685:<
577:<
512:<
277:<
232:>
212:and
185:<
157:and
116:>
4998:doi
4955:doi
4921:doi
4884:doi
4872:",
4756:doi
4684:Zbl
4676:JFM
4668:doi
4656:",
4588:doi
4548:doi
4469:doi
4424:doi
4412:",
4292:JFM
4288:158
4267:Zbl
4249:doi
4176:Zbl
4152:doi
4132:Zbl
4116:doi
4094:",
3963:Zbl
3937:doi
3912:Zbl
3894:doi
2209:lot
1622:to
1585:to
1554:to
1517:to
1463:2.0
1441:2.2
1382:by
1348:316
1311:316
1265:316
1009:153
978:316
945:370
834:500
703:964
327:in
291:316
66:as
46:In
5517::
5171:in
5089:.
5043:li
5004:,
4994:17
4992:,
4988:,
4971:MR
4969:,
4961:,
4951:63
4949:,
4927:,
4917:25
4915:,
4911:,
4894:MR
4892:,
4880:80
4878:,
4856:13
4852:10
4847:10
4764:MR
4762:,
4750:,
4720:li
4682:,
4674:,
4662:,
4632:li
4596:MR
4594:,
4582:,
4576:,
4558:MR
4556:,
4544:79
4542:,
4512:li
4479:MR
4477:,
4463:,
4455:;
4442:MR
4440:,
4432:,
4420:48
4418:,
4388:li
4350:,
4340:,
4286:,
4265:,
4259:MR
4257:,
4245:11
4243:,
4237:,
4211:li
4174:,
4166:,
4160:MR
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4148:29
4146:,
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4124:MR
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4070:li
4015:,
4005:,
3961:,
3955:MR
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3945:,
3933:29
3931:,
3910:,
3904:MR
3902:,
3890:69
3888:,
3882:,
3853:li
3697:,
3654:,
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3572:,
3533:,
3498:,
3467:,
3436:,
3409:,
3382:,
3359:,
3336:,
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3294:,
3201:li
3025:ln
2443:.
2404:li
2345:li
2303:li
2204:.
2175:li
2098:li
2061:li
2013:li
1987:.
1921:li
1888:li
1858:li
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1649:.
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1600:10
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1460:10
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1362:.
1344:10
1307:10
1261:10
1206:li
1127:li
1046:li
1005:10
974:10
941:10
894:li
830:10
803:10
770:10
699:10
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591:10
586:10
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568:79
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462:.
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5000::
4957::
4936:.
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4886::
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4814:(
4791:)
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4220:x
4217:(
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4058:(
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3865:)
3862:x
3859:(
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3760:+
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3754:,
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3748:(
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3711:p
3707:p
3703:p
3699:p
3695:p
3693:(
3680:p
3676:p
3672:p
3668:p
3664:p
3660:p
3656:p
3652:p
3650:(
3637:p
3633:p
3629:p
3625:p
3621:p
3617:p
3613:p
3609:p
3607:(
3594:p
3590:p
3586:p
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3570:p
3568:(
3555:p
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3539:p
3535:p
3531:p
3529:(
3516:p
3512:p
3508:p
3504:p
3500:p
3496:p
3494:(
3481:p
3477:p
3473:p
3469:p
3465:p
3463:(
3450:p
3446:p
3442:p
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3432:(
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3415:p
3411:p
3407:p
3405:(
3392:p
3388:p
3384:p
3380:p
3378:(
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3357:p
3355:(
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3338:p
3334:p
3332:(
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3313:p
3311:(
3296:p
3292:p
3290:(
3277:k
3267:k
3250:.
3247:P
3222:,
3219:)
3216:p
3213:(
3205:P
3195:P
3191:C
3184:)
3181:p
3178:(
3173:P
3145:p
3125:P
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3101:p
3075:P
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3045:1
3042:+
3039:k
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3031:t
3022:(
3017:t
3014:d
3006:x
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2993:=
2990:)
2987:x
2984:(
2975:P
2971:i
2967:l
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2940:i
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2909:i
2905:+
2902:p
2899:,
2894:1
2890:i
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2883:p
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2814:x
2811:(
2806:P
2791:k
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2772:k
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2696:P
2684:(
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2670:k
2648:)
2643:2
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2631:x
2627:(
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2486:/
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2410:(
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2351:(
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2309:(
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2248:2
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2078:)
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2067:(
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2022:x
2019:(
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2004:x
2001:(
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1931:x
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1488:x
1465:×
1458:×
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1436:×
1421:×
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1399:×
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