1947:
1827:
3049:
2144:, 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
2191:
2077:
3221:
711:
1312:
1266:
2319:
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.
295:
2647:
2494:
2024:
603:
1709:
1220:
529:
133:
4081:
3864:
2776:
1763:
1057:
905:
246:
199:
5054:
4731:
4643:
4523:
4399:
4222:
2363:
365:
2945:
2430:
1140:; 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.
2317:
429:
2109:
1349:
1138:
979:
808:
775:
1605:
4859:
2528:
2326:
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
946:
1537:
2586:
4326:
2816:
4790:
2322:
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
3991:
1500:
1010:
835:
4819:
3765:
2282:
2138:
1819:
1636:
1568:
1106:
394:
3076:
2559:
2253:
1970:
1942:{\displaystyle \pi (x)=\operatorname {li} (x)-{\tfrac {1}{2}}\operatorname {li} ({\sqrt {x\,}})-\sum _{\rho }\operatorname {li} (x^{\rho })+{\text{smaller terms}}}
4735:
3249:
4034:
3817:
3144:
3124:
3100:
2856:
2836:
2226:
1659:
1170:
1077:
858:
630:
479:
449:
80:
4647:
2950:
26:
5141:
4085:
3079:
2208:, 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
2323:
2147:
2033:
5110:
4567:
717:
3152:
5415:
1777:
showed that this proportion is about 0.00000026, which is surprisingly large given how far one has to go to find the first example.
638:
4934:
1271:
1225:
5072:
1789:
251:
5058:(masters), Master's thesis, Manchester Institute for Mathematical Sciences, School of Mathematics, University of Manchester
2591:
2438:
1982:
5450:
5330:
5235:
5230:
5225:
5220:
5215:
5210:
5205:
5200:
537:
151:
5335:
5265:
5325:
1664:
1175:
484:
88:
4039:
3822:
2680:
1721:
1015:
863:
204:
157:
5134:
4971:
3798:
2141:
5280:
5012:
4689:
4601:
4481:
4357:
4180:
2329:
323:
5425:
4863:
4527:
4403:
3916:
3873:
2861:
2388:
5410:
5275:
5488:
4894:
2287:
399:
5519:
2140:. The other terms above are somewhat smaller, and moreover tend to have different, seemingly random complex
2082:
1321:
1111:
951:
320:
that there is such a number (and so, a first such number); and indeed found that the sign of the difference
305:
143:
5514:
5509:
5483:
5378:
5127:
780:
747:
5445:
5435:
5373:
1577:
737:
4824:
2499:
918:
732:
These upper bounds have since been reduced considerably by using large-scale computer calculations of
4340:
4005:
3783:
1509:
2564:
5320:
4296:
1711:
though computer calculations suggest some explicit numbers that are quite likely to satisfy this.
5368:
4994:
4951:
4917:
4422:
4330:
4156:
4094:
3995:
3935:
2785:
2027:
1973:
733:
458:
313:
5270:
4766:
5106:
4445:
4441:
3967:
1472:
988:
813:
4795:
3732:
2258:
2114:
1795:
1614:
1546:
1082:
370:
5290:
5240:
4986:
4943:
4909:
4872:
4744:
4672:
4664:
4656:
4576:
4536:
4457:
4412:
4280:
4266:
4255:
4237:
4228:
4164:
4140:
4120:
4104:
3951:
3925:
3911:
3900:
3882:
2650:
367:
changes infinitely many times. All numerical evidence then available seemed to suggest that
4963:
4886:
4756:
4588:
4550:
4471:
4434:
4251:
4152:
4116:
3947:
3896:
3054:
2537:
2231:
1955:
5430:
4959:
4932:(1941), "On the distribution function of the remainder term of the prime number theorem",
4882:
4752:
4676:
4668:
4584:
4546:
4467:
4430:
4351:
4284:
4259:
4247:
4168:
4148:
4124:
4112:
3955:
3943:
3904:
3892:
1770:
1715:
908:
4344:
4009:
3231:
5460:
5351:
5285:
5250:
4683:
4595:
4271:
4019:
3802:
3129:
3109:
3085:
2841:
2821:
2366:
2211:
2205:
2201:
1644:
1155:
1062:
843:
615:
464:
434:
309:
154:. Skewes's number is much larger, but it is now known that there is a crossing between
65:
60:
52:
4895:"On The Asymptotic Density Of Prime k-tuples and a Conjecture of Hardy and Littlewood"
4417:
5503:
5420:
5195:
5175:
5150:
4929:
4921:
2666:
2196:
The reason why the Skewes number is so large is that these smaller terms are quite a
721:
44:
36:
4998:
4877:
4160:
5455:
4990:
4580:
4558:
3044:{\displaystyle \operatorname {li_{P}} (x)=\int _{2}^{x}{\frac {dt}{(\ln t)^{k+1}}}}
2531:
48:
5083:
4541:
4131:
Kotnik, T. (2008), "The prime-counting function and its analytic approximations",
3887:
5465:
5405:
4562:
56:
4913:
720:: exhibiting some concrete upper bound for the first sign change. According to
5470:
5255:
4972:"The Skewes number for twin primes: counting sign changes of π2(x) − C2Li2(x)"
4660:
4144:
4108:
3727:
4748:
4462:
5361:
5356:
5190:
4242:
1144:
gave a small improvement and correction to the result of Bays and Hudson.
740:. The first estimate for the actual value of a crossover point was given by
5295:
3914:(1975), "Irregularities in the distribution of primes and twin primes",
1148:
found a smaller interval for a crossing, which was slightly improved by
5185:
4955:
4426:
3939:
1821:, whose leading terms are (ignoring some subtle convergence questions)
1785:
838:
15:
2228:
random complex numbers having roughly the same argument is about 1 in
5440:
5245:
5180:
4099:
5007:
4947:
4448:(1962), "Approximate formulas for some functions of prime numbers",
4016:
Chao, Kuok Fai; Plymen, Roger (2010), "A new bound for the smallest
3930:
431:
Littlewood's proof did not, however, exhibit a concrete such number
4000:
3254:
The table below shows the currently known Skewes numbers for prime
4478:
Saouter, Yannick; Demichel, Patrick (2010), "A sharp region where
4335:
2649:
is roughly analogous to a second-order correction accounting for
2186:{\displaystyle {\tfrac {1}{2}}\operatorname {li} ({\sqrt {x\,}})}
2072:{\displaystyle {\tfrac {1}{2}}\operatorname {li} ({\sqrt {x\,}})}
724:, this was at the time not considered obvious even in principle.
5119:
5123:
4176:
3216:{\displaystyle \pi _{P}(p)>C_{P}\operatorname {li} _{P}(p),}
5170:
2200:
smaller than the leading error term, mainly because the first
706:{\displaystyle e^{e^{e^{e^{7.705}}}}<10^{10^{10^{964}}}.}
4763:
Stoll, Douglas; Demichel, Patrick (2011), "The impact of
1307:{\displaystyle e^{727.9513386}<1.39717\times 10^{316}}
1261:{\displaystyle e^{727.9513468}<1.39718\times 10^{316}}
3102:
that violates the Hardy–Littlewood inequality for the (
1773:
of these positive integers does exist and is positive.
716:
Skewes's task was to make
Littlewood's existence proof
2665:
An equivalent definition of Skewes' number exists for
2596:
2443:
2152:
2038:
1865:
1059:. Bays and Hudson found a few much smaller values of
290:{\displaystyle e^{727.95133}<1.397\times 10^{316}.}
5015:
4827:
4798:
4769:
4692:
4604:
4484:
4360:
4299:
4183:
4042:
4022:
3970:
3825:
3805:
3735:
3234:
3155:
3132:
3112:
3088:
3057:
2953:
2864:
2844:
2824:
2788:
2683:
2642:{\displaystyle {\tfrac {1}{2}}\mathrm {li} (x^{1/2})}
2594:
2567:
2540:
2502:
2489:{\displaystyle {\tfrac {1}{2}}\mathrm {li} (x^{1/2})}
2441:
2391:
2332:
2290:
2261:
2234:
2214:
2150:
2117:
2085:
2036:
2019:{\displaystyle \pi (x)\approx \operatorname {li} (x)}
1985:
1958:
1830:
1798:
1724:
1667:
1647:
1617:
1580:
1549:
1512:
1475:
1324:
1274:
1228:
1178:
1158:
1114:
1085:
1065:
1018:
1012:
consecutive integers somewhere near this value where
991:
954:
921:
866:
846:
816:
783:
750:
641:
618:
540:
487:
467:
437:
402:
373:
326:
297:
It is not known whether it is the smallest crossing.
254:
207:
160:
91:
68:
4269:(1914), "Sur la distribution des nombres premiers",
598:{\displaystyle e^{e^{e^{79}}}<10^{10^{10^{34}}}.}
5391:
5344:
5313:
5304:
5157:
1152:. The same source shows that there exists a number
5073:"The prime counting function and related subjects"
5048:
4853:
4813:
4784:
4725:
4637:
4517:
4393:
4320:
4216:
4075:
4028:
3985:
3858:
3811:
3759:
3243:
3215:
3138:
3118:
3094:
3070:
3043:
2939:
2850:
2830:
2810:
2770:
2641:
2580:
2553:
2522:
2488:
2424:
2357:
2311:
2276:
2247:
2220:
2185:
2132:
2103:
2071:
2018:
1964:
1941:
1813:
1757:
1704:{\displaystyle \pi (x)>\operatorname {li} (x),}
1703:
1653:
1630:
1599:
1562:
1531:
1494:
1343:
1306:
1260:
1215:{\displaystyle \pi (x)<\operatorname {li} (x),}
1214:
1164:
1132:
1100:
1071:
1051:
1004:
973:
940:
899:
852:
829:
802:
769:
705:
624:
597:
524:{\displaystyle \pi (x)<\operatorname {li} (x),}
523:
473:
443:
423:
388:
359:
289:
240:
193:
128:{\displaystyle \pi (x)>\operatorname {li} (x),}
127:
74:
4076:{\displaystyle \pi (x)>\operatorname {li} (x)}
3859:{\displaystyle \pi (x)>\operatorname {li} (x)}
2771:{\displaystyle P=(p,p+i_{1},p+i_{2},...,p+i_{k})}
2365:for zeros violating the Riemann hypothesis (with
1758:{\displaystyle \pi (x)>\operatorname {li} (x)}
1052:{\displaystyle \pi (x)>\operatorname {li} (x)}
900:{\displaystyle \pi (x)>\operatorname {li} (x)}
241:{\displaystyle \pi (x)>\operatorname {li} (x)}
194:{\displaystyle \pi (x)<\operatorname {li} (x)}
1774:
1469:proved that there are no crossover points below
1466:
4979:Computational Methods in Science and Technology
4902:Computational Methods in Science and Technology
1145:
5049:{\displaystyle \pi (x)-\operatorname {li} (x)}
4736:Proceedings of the London Mathematical Society
4726:{\displaystyle \pi (x)-\operatorname {li} (x)}
4638:{\displaystyle \pi (x)-\operatorname {li} (x)}
4518:{\displaystyle \pi (x)-\operatorname {li} (x)}
4394:{\displaystyle \pi (x)-\operatorname {li} (x)}
4217:{\displaystyle \pi (x)-\operatorname {li} (x)}
2358:{\displaystyle \operatorname {li} (x^{\rho })}
1974:non-trivial zeros of the Riemann zeta function
360:{\displaystyle \pi (x)-\operatorname {li} (x)}
5135:
2940:{\displaystyle p,p+i_{1},p+i_{2},...,p+i_{k}}
1571:
1315:
8:
3774:-tuples have a corresponding Skewes number.
2425:{\displaystyle \operatorname {li} (x^{1/2})}
2204:zero of the zeta function has quite a large
1979:The largest error term in the approximation
907:. Without assuming the Riemann hypothesis,
5310:
5142:
5128:
5120:
4648:Journal of the London Mathematical Society
3770:It is also unknown whether all admissible
3078:denote its Hardy–Littlewood constant (see
2534:, rather than the primes themselves, with
1141:
982:
317:
5014:
4876:
4843:
4838:
4826:
4797:
4768:
4691:
4603:
4540:
4483:
4461:
4416:
4359:
4334:
4298:
4241:
4182:
4098:
4041:
4021:
3999:
3969:
3929:
3886:
3824:
3804:
3734:
3233:
3192:
3182:
3160:
3154:
3131:
3111:
3087:
3062:
3056:
3026:
2999:
2993:
2988:
2962:
2954:
2952:
2931:
2900:
2881:
2863:
2843:
2823:
2793:
2787:
2759:
2728:
2709:
2682:
2626:
2622:
2607:
2595:
2593:
2568:
2566:
2545:
2539:
2503:
2501:
2473:
2469:
2454:
2442:
2440:
2409:
2405:
2390:
2346:
2331:
2289:
2260:
2239:
2233:
2213:
2177:
2172:
2151:
2149:
2116:
2084:
2063:
2058:
2037:
2035:
1984:
1957:
1934:
1922:
1903:
1890:
1885:
1864:
1829:
1797:
1723:
1666:
1646:
1622:
1616:
1591:
1579:
1554:
1548:
1523:
1511:
1486:
1474:
1335:
1323:
1298:
1279:
1273:
1252:
1233:
1227:
1177:
1157:
1113:
1084:
1064:
1017:
996:
990:
965:
953:
932:
920:
865:
845:
821:
815:
794:
782:
761:
749:
690:
685:
680:
661:
656:
651:
646:
640:
617:
608:Without assuming the Riemann hypothesis,
582:
577:
572:
555:
550:
545:
539:
486:
466:
436:
401:
372:
325:
278:
259:
253:
206:
159:
90:
67:
3784:Mertens' theorems § Changes in sign
3260:
1353:
1149:
912:
4354:(1987), "On the sign of the difference
2312:{\displaystyle \operatorname {li} (x),}
1766:
1661:known for certain to have the property
424:{\displaystyle \operatorname {li} (x).}
27:(more unsolved problems in mathematics)
4291:Platt, D. J.; Trudgian, T. S. (2014),
4086:International Journal of Number Theory
2104:{\displaystyle \operatorname {li} (x)}
1540:
1344:{\displaystyle 1.39716\times 10^{316}}
1133:{\displaystyle \operatorname {li} (x)}
974:{\displaystyle 1.39822\times 10^{316}}
741:
609:
454:
4133:Advances in Computational Mathematics
3726:The Skewes number (if it exists) for
1608:
1503:
23:What is the smallest Skewes's number?
7:
3314:
3293:
2674:
803:{\displaystyle 1.65\times 10^{1165}}
770:{\displaystyle 1.53\times 10^{1165}}
744:, who showed that somewhere between
612:proved that there exists a value of
1718:of the positive integers for which
2963:
2959:
2955:
2611:
2608:
2507:
2504:
2458:
2455:
1600:{\displaystyle 1.39\times 10^{17}}
32:Large number used in number theory
14:
5416:Indefinite and fictitious numbers
4854:{\displaystyle x<10^{10^{13}}}
4418:10.1090/s0025-5718-1987-0866118-6
3080:First Hardy–Littlewood conjecture
2324:Dirichlet's approximation theorem
1314:assuming the Riemann hypothesis.
5101:Asimov, I. (1976). "Skewered!".
3964:An analytic method for bounding
3797:Bays, C.; Hudson, R. H. (2000),
3226:(if such a prime exists) is the
2523:{\displaystyle \mathrm {li} (x)}
985:, who showed there are at least
941:{\displaystyle 7\times 10^{370}}
4935:American Journal of Mathematics
4878:10.1090/S0025-5718-2011-02477-4
4450:Illinois Journal of Mathematics
1532:{\displaystyle 8\times 10^{10}}
461:is true, there exists a number
457:proved that, assuming that the
18:Unsolved problem in mathematics
5043:
5037:
5025:
5019:
4991:10.12921/cmst.2011.17.01.87-92
4808:
4802:
4779:
4773:
4720:
4714:
4702:
4696:
4632:
4626:
4614:
4608:
4581:10.1080/10586458.1994.10504289
4512:
4506:
4494:
4488:
4388:
4382:
4370:
4364:
4309:
4303:
4211:
4205:
4193:
4187:
4070:
4064:
4052:
4046:
3980:
3974:
3853:
3847:
3835:
3829:
3799:"A new bound for the smallest
3754:
3736:
3207:
3201:
3172:
3166:
3023:
3010:
2978:
2972:
2805:
2799:
2765:
2690:
2636:
2615:
2581:{\displaystyle {\frac {1}{n}}}
2517:
2511:
2483:
2462:
2419:
2398:
2385:) are eventually larger than
2352:
2339:
2303:
2297:
2271:
2265:
2180:
2169:
2127:
2121:
2098:
2092:
2066:
2055:
2013:
2007:
1995:
1989:
1928:
1915:
1893:
1882:
1858:
1852:
1840:
1834:
1808:
1802:
1775:Rubinstein & Sarnak (1994)
1752:
1746:
1734:
1728:
1695:
1689:
1677:
1671:
1467:Rosser & Schoenfeld (1962)
1206:
1200:
1188:
1182:
1127:
1121:
1095:
1089:
1046:
1040:
1028:
1022:
894:
888:
876:
870:
515:
509:
497:
491:
415:
409:
383:
377:
354:
348:
336:
330:
235:
229:
217:
211:
188:
182:
170:
164:
119:
113:
101:
95:
1:
5331:Conway chained arrow notation
4542:10.1090/S0025-5718-10-02351-3
3888:10.1090/S0025-5718-99-01104-7
1146:Saouter & Demichel (2010)
152:logarithmic integral function
4321:{\displaystyle \theta (x)-x}
4293:On the first sign change of
5006:Zegowitz, Stefanie (2010),
4686:(1955), "On the difference
4598:(1933), "On the difference
4175:Lehman, R. Sherman (1966),
2811:{\displaystyle \pi _{P}(x)}
2496:is that, roughly speaking,
1641:There is no explicit value
1572:Platt & Trudgian (2014)
1316:Stoll & Demichel (2011)
915:) proved an upper bound of
5536:
5426:Largest known prime number
5103:Of Matters Great and Small
5009:On the positive region of
4914:10.12921/cmst.2019.0000033
4864:Mathematics of Computation
4528:Mathematics of Computation
4404:Mathematics of Computation
3917:Mathematics of Computation
3874:Mathematics of Computation
2530:actually counts powers of
1952:where the sum is over all
5479:
5411:Extended real number line
5326:Knuth's up-arrow notation
4785:{\displaystyle \zeta (s)}
4145:10.1007/s10444-007-9039-2
4109:10.1142/S1793042110003125
2284:is sometimes larger than
1268:. This can be reduced to
312:research supervisor, had
5336:Steinhaus–Moser notation
4568:Experimental Mathematics
3986:{\displaystyle \psi (x)}
3720:Pfoertner / Luhn (2021)
3677:Luhn / Pfoertner (2021)
3634:Pfoertner / Luhn (2021)
3126:, i.e., the first prime
3082:). Then the first prime
2782: + 1)-tuple,
2435:The reason for the term
1495:{\displaystyle x=10^{8}}
1142:Chao & Plymen (2010)
1005:{\displaystyle 10^{153}}
983:Bays & Hudson (2000)
948:. A better estimate was
830:{\displaystyle 10^{500}}
5105:. New York: Ace Books.
4814:{\displaystyle \pi (x)}
4661:10.1112/jlms/s1-8.4.277
4243:10.4064/aa-11-4-397-410
3760:{\displaystyle (p,p+6)}
3106: + 1)-tuple
2277:{\displaystyle \pi (x)}
2133:{\displaystyle \pi (x)}
2111:is usually larger than
1814:{\displaystyle \pi (x)}
1780:
1631:{\displaystyle 10^{19}}
1563:{\displaystyle 10^{14}}
1101:{\displaystyle \pi (x)}
389:{\displaystyle \pi (x)}
300:
144:prime-counting function
5379:Fast-growing hierarchy
5050:
4855:
4815:
4786:
4749:10.1112/plms/s3-5.1.48
4727:
4639:
4519:
4463:10.1215/ijm/1255631807
4395:
4322:
4218:
4077:
4030:
3987:
3860:
3813:
3761:
3245:
3217:
3140:
3120:
3096:
3072:
3045:
2941:
2852:
2832:
2812:
2772:
2643:
2582:
2555:
2524:
2490:
2426:
2359:
2313:
2278:
2249:
2222:
2187:
2134:
2105:
2073:
2020:
1966:
1943:
1815:
1759:
1705:
1655:
1632:
1601:
1564:
1533:
1496:
1345:
1308:
1262:
1216:
1166:
1134:
1102:
1073:
1053:
1006:
975:
942:
901:
854:
831:
804:
771:
707:
626:
599:
525:
475:
445:
425:
390:
361:
291:
242:
195:
129:
76:
5436:Long and short scales
5374:Grzegorczyk hierarchy
5051:
4893:Tóth, László (2019),
4856:
4816:
4787:
4728:
4640:
4520:
4396:
4323:
4219:
4078:
4031:
3988:
3861:
3814:
3762:
3246:
3218:
3141:
3121:
3097:
3073:
3071:{\displaystyle C_{P}}
3046:
2942:
2853:
2833:
2818:the number of primes
2813:
2773:
2657:Equivalent for prime
2644:
2583:
2556:
2554:{\displaystyle p^{n}}
2525:
2491:
2427:
2360:
2314:
2279:
2250:
2248:{\displaystyle 2^{N}}
2223:
2188:
2135:
2106:
2074:
2030:is true) is negative
2021:
1967:
1965:{\displaystyle \rho }
1944:
1816:
1760:
1706:
1656:
1633:
1602:
1565:
1534:
1497:
1437:Saouter and Demichel
1346:
1309:
1263:
1217:
1167:
1135:
1103:
1074:
1054:
1007:
976:
943:
909:H. J. J. te Riele
902:
855:
832:
805:
772:
738:Riemann zeta function
728:More recent estimates
708:
627:
600:
526:
476:
446:
426:
396:was always less than
391:
362:
292:
243:
196:
130:
77:
5071:Demichels, Patrick.
5013:
4970:Wolf, Marek (2011),
4825:
4796:
4767:
4690:
4602:
4482:
4358:
4297:
4181:
4040:
4020:
3968:
3823:
3803:
3733:
3232:
3153:
3130:
3110:
3086:
3055:
2951:
2862:
2842:
2822:
2786:
2681:
2592:
2565:
2538:
2500:
2439:
2389:
2330:
2288:
2259:
2255:. This explains why
2232:
2212:
2148:
2115:
2083:
2034:
1983:
1956:
1828:
1796:
1722:
1665:
1645:
1615:
1578:
1547:
1510:
1473:
1322:
1272:
1226:
1176:
1156:
1112:
1083:
1063:
1016:
989:
952:
919:
864:
844:
814:
810:there are more than
781:
748:
639:
616:
538:
485:
465:
435:
400:
371:
324:
252:
205:
158:
89:
66:
5451:Orders of magnitude
5321:Scientific notation
4345:2014arXiv1407.1914P
4177:"On the difference
4010:2015arXiv151102032B
3962:Büthe, Jan (2015),
3503: + 12,
3499: + 10,
3468: + 10,
2998:
2947:are all prime, let
1771:logarithmic density
1459:Stoll and Demichel
5369:Ackermann function
5046:
4871:(276): 2381–2394,
4851:
4811:
4782:
4723:
4635:
4563:"Chebyshev's bias"
4535:(272): 2395–2405,
4515:
4391:
4352:te Riele, H. J. J.
4318:
4214:
4073:
4026:
3983:
3881:(231): 1285–1296,
3856:
3809:
3767:is still unknown.
3757:
3717:750247439134737983
3674:523250002674163757
3507: + 16)
3495: + 6,
3491: + 4,
3472: + 12)
3464: +6 ,
3460: +4 ,
3441: + 12)
3437: + 8,
3433: + 6,
3429: + 2,
3410: + 10)
3406: +6 ,
3402: + 4,
3379: + 6,
3375: + 2,
3352: + 4,
3329: + 2,
3244:{\displaystyle P.}
3241:
3228:Skewes number for
3213:
3136:
3116:
3092:
3068:
3041:
2984:
2937:
2848:
2828:
2808:
2768:
2639:
2605:
2578:
2551:
2520:
2486:
2452:
2422:
2355:
2309:
2274:
2245:
2218:
2183:
2161:
2130:
2101:
2069:
2047:
2028:Riemann hypothesis
2016:
1962:
1939:
1908:
1874:
1811:
1755:
1701:
1651:
1628:
1597:
1560:
1529:
1492:
1341:
1304:
1258:
1212:
1162:
1130:
1098:
1069:
1049:
1002:
971:
938:
897:
850:
827:
800:
767:
703:
622:
595:
521:
471:
459:Riemann hypothesis
441:
421:
386:
357:
287:
238:
191:
125:
72:
43:is any of several
5497:
5496:
5387:
5386:
4792:complex zeros on
4267:Littlewood, J. E.
4029:{\displaystyle x}
3812:{\displaystyle x}
3724:
3723:
3591:Pfoertner (2020)
3552:Pfoertner (2020)
3383: + 8)
3356: + 6)
3333: + 6)
3308: + 4)
3287: + 2)
3139:{\displaystyle p}
3119:{\displaystyle P}
3095:{\displaystyle p}
3039:
2851:{\displaystyle x}
2831:{\displaystyle p}
2604:
2576:
2451:
2221:{\displaystyle N}
2178:
2160:
2064:
2046:
1937:
1899:
1891:
1873:
1781:Riemann's formula
1654:{\displaystyle x}
1463:
1462:
1165:{\displaystyle x}
1072:{\displaystyle x}
853:{\displaystyle x}
625:{\displaystyle x}
474:{\displaystyle x}
444:{\displaystyle x}
318:Littlewood (1914)
75:{\displaystyle x}
59:for the smallest
5527:
5311:
5241:Eddington number
5186:Hundred thousand
5144:
5137:
5130:
5121:
5116:
5097:
5095:
5094:
5088:
5082:. Archived from
5077:
5059:
5055:
5053:
5052:
5047:
5001:
4976:
4966:
4924:
4899:
4889:
4880:
4860:
4858:
4857:
4852:
4850:
4849:
4848:
4847:
4820:
4818:
4817:
4812:
4791:
4789:
4788:
4783:
4759:
4732:
4730:
4729:
4724:
4679:
4644:
4642:
4641:
4636:
4591:
4557:Rubinstein, M.;
4553:
4544:
4524:
4522:
4521:
4516:
4474:
4465:
4437:
4420:
4411:(177): 323–328,
4400:
4398:
4397:
4392:
4347:
4338:
4327:
4325:
4324:
4319:
4287:
4262:
4245:
4229:Acta Arithmetica
4223:
4221:
4220:
4215:
4171:
4127:
4102:
4082:
4080:
4079:
4074:
4035:
4033:
4032:
4027:
4012:
4003:
3992:
3990:
3989:
3984:
3958:
3933:
3907:
3890:
3870:
3865:
3863:
3862:
3857:
3818:
3816:
3815:
3810:
3766:
3764:
3763:
3758:
3631:1203255673037261
3261:
3250:
3248:
3247:
3242:
3222:
3220:
3219:
3214:
3197:
3196:
3187:
3186:
3165:
3164:
3145:
3143:
3142:
3137:
3125:
3123:
3122:
3117:
3101:
3099:
3098:
3093:
3077:
3075:
3074:
3069:
3067:
3066:
3050:
3048:
3047:
3042:
3040:
3038:
3037:
3036:
3008:
3000:
2997:
2992:
2968:
2967:
2966:
2946:
2944:
2943:
2938:
2936:
2935:
2905:
2904:
2886:
2885:
2857:
2855:
2854:
2849:
2837:
2835:
2834:
2829:
2817:
2815:
2814:
2809:
2798:
2797:
2778:denote a prime (
2777:
2775:
2774:
2769:
2764:
2763:
2733:
2732:
2714:
2713:
2648:
2646:
2645:
2640:
2635:
2634:
2630:
2614:
2606:
2597:
2587:
2585:
2584:
2579:
2577:
2569:
2560:
2558:
2557:
2552:
2550:
2549:
2529:
2527:
2526:
2521:
2510:
2495:
2493:
2492:
2487:
2482:
2481:
2477:
2461:
2453:
2444:
2431:
2429:
2428:
2423:
2418:
2417:
2413:
2384:
2382:
2381:
2378:
2375:
2364:
2362:
2361:
2356:
2351:
2350:
2318:
2316:
2315:
2310:
2283:
2281:
2280:
2275:
2254:
2252:
2251:
2246:
2244:
2243:
2227:
2225:
2224:
2219:
2192:
2190:
2189:
2184:
2179:
2173:
2162:
2153:
2139:
2137:
2136:
2131:
2110:
2108:
2107:
2102:
2078:
2076:
2075:
2070:
2065:
2059:
2048:
2039:
2025:
2023:
2022:
2017:
1971:
1969:
1968:
1963:
1948:
1946:
1945:
1940:
1938:
1935:
1927:
1926:
1907:
1892:
1886:
1875:
1866:
1820:
1818:
1817:
1812:
1790:explicit formula
1769:showed that the
1765:does not exist,
1764:
1762:
1761:
1756:
1714:Even though the
1710:
1708:
1707:
1702:
1660:
1658:
1657:
1652:
1637:
1635:
1634:
1629:
1627:
1626:
1606:
1604:
1603:
1598:
1596:
1595:
1569:
1567:
1566:
1561:
1559:
1558:
1538:
1536:
1535:
1530:
1528:
1527:
1501:
1499:
1498:
1493:
1491:
1490:
1455:
1448:
1433:
1426:
1415:Chao and Plymen
1411:
1404:
1393:Bays and Hudson
1389:
1382:
1354:
1350:
1348:
1347:
1342:
1340:
1339:
1313:
1311:
1310:
1305:
1303:
1302:
1284:
1283:
1267:
1265:
1264:
1259:
1257:
1256:
1238:
1237:
1221:
1219:
1218:
1213:
1171:
1169:
1168:
1163:
1139:
1137:
1136:
1131:
1107:
1105:
1104:
1099:
1078:
1076:
1075:
1070:
1058:
1056:
1055:
1050:
1011:
1009:
1008:
1003:
1001:
1000:
980:
978:
977:
972:
970:
969:
947:
945:
944:
939:
937:
936:
906:
904:
903:
898:
859:
857:
856:
851:
836:
834:
833:
828:
826:
825:
809:
807:
806:
801:
799:
798:
776:
774:
773:
768:
766:
765:
712:
710:
709:
704:
699:
698:
697:
696:
695:
694:
672:
671:
670:
669:
668:
667:
666:
665:
631:
629:
628:
623:
604:
602:
601:
596:
591:
590:
589:
588:
587:
586:
564:
563:
562:
561:
560:
559:
530:
528:
527:
522:
480:
478:
477:
472:
450:
448:
447:
442:
430:
428:
427:
422:
395:
393:
392:
387:
366:
364:
363:
358:
301:Skewes's numbers
296:
294:
293:
288:
283:
282:
264:
263:
247:
245:
244:
239:
200:
198:
197:
192:
149:
141:
134:
132:
131:
126:
81:
79:
78:
73:
19:
5535:
5534:
5530:
5529:
5528:
5526:
5525:
5524:
5500:
5499:
5498:
5493:
5475:
5431:List of numbers
5399:
5397:
5395:
5393:
5383:
5340:
5306:
5300:
5271:Graham's number
5261:Skewes's number
5163:
5161:
5159:
5153:
5148:
5113:
5100:
5092:
5090:
5086:
5075:
5070:
5067:
5062:
5011:
5010:
5005:
4974:
4969:
4948:10.2307/2371519
4928:
4897:
4892:
4839:
4834:
4823:
4822:
4794:
4793:
4765:
4764:
4762:
4688:
4687:
4682:
4600:
4599:
4594:
4556:
4480:
4479:
4477:
4440:
4356:
4355:
4350:
4295:
4294:
4290:
4265:
4179:
4178:
4174:
4130:
4038:
4037:
4018:
4017:
4015:
3966:
3965:
3961:
3931:10.2307/2005460
3910:
3868:
3821:
3820:
3801:
3800:
3796:
3792:
3780:
3731:
3730:
3588:214159878489239
3230:
3229:
3188:
3178:
3156:
3151:
3150:
3128:
3127:
3108:
3107:
3084:
3083:
3058:
3053:
3052:
3022:
3009:
3001:
2958:
2949:
2948:
2927:
2896:
2877:
2860:
2859:
2840:
2839:
2820:
2819:
2789:
2784:
2783:
2755:
2724:
2705:
2679:
2678:
2663:
2618:
2590:
2589:
2563:
2562:
2541:
2536:
2535:
2498:
2497:
2465:
2437:
2436:
2401:
2387:
2386:
2379:
2376:
2373:
2372:
2370:
2342:
2328:
2327:
2286:
2285:
2257:
2256:
2235:
2230:
2229:
2210:
2209:
2146:
2145:
2113:
2112:
2081:
2080:
2079:, showing that
2032:
2031:
1981:
1980:
1954:
1953:
1918:
1826:
1825:
1794:
1793:
1783:
1720:
1719:
1716:natural density
1663:
1662:
1643:
1642:
1618:
1613:
1612:
1587:
1576:
1575:
1550:
1545:
1544:
1519:
1508:
1507:
1482:
1471:
1470:
1453:
1446:
1431:
1424:
1409:
1402:
1387:
1380:
1367:
1331:
1320:
1319:
1294:
1275:
1270:
1269:
1248:
1229:
1224:
1223:
1174:
1173:
1154:
1153:
1150:Zegowitz (2010)
1110:
1109:
1081:
1080:
1061:
1060:
1014:
1013:
992:
987:
986:
961:
950:
949:
928:
917:
916:
862:
861:
842:
841:
817:
812:
811:
790:
779:
778:
757:
746:
745:
730:
686:
681:
676:
657:
652:
647:
642:
637:
636:
614:
613:
578:
573:
568:
551:
546:
541:
536:
535:
483:
482:
463:
462:
433:
432:
398:
397:
369:
368:
322:
321:
306:J.E. Littlewood
303:
274:
255:
250:
249:
203:
202:
156:
155:
147:
139:
87:
86:
64:
63:
41:Skewes's number
33:
30:
29:
24:
21:
17:
12:
11:
5:
5533:
5531:
5523:
5522:
5520:Large integers
5517:
5512:
5502:
5501:
5495:
5494:
5492:
5491:
5486:
5480:
5477:
5476:
5474:
5473:
5468:
5463:
5461:Power of three
5458:
5453:
5448:
5443:
5441:Number systems
5438:
5433:
5428:
5423:
5418:
5413:
5408:
5402:
5400:
5396:(alphabetical
5389:
5388:
5385:
5384:
5382:
5381:
5376:
5371:
5366:
5365:
5364:
5359:
5352:Hyperoperation
5348:
5346:
5342:
5341:
5339:
5338:
5333:
5328:
5323:
5317:
5315:
5308:
5302:
5301:
5299:
5298:
5293:
5288:
5283:
5278:
5273:
5268:
5266:Moser's number
5263:
5258:
5253:
5251:Shannon number
5248:
5243:
5238:
5233:
5228:
5223:
5218:
5213:
5208:
5203:
5198:
5193:
5188:
5183:
5178:
5173:
5167:
5165:
5155:
5154:
5149:
5147:
5146:
5139:
5132:
5124:
5118:
5117:
5112:978-0441610723
5111:
5098:
5089:on Sep 8, 2006
5066:
5065:External links
5063:
5061:
5060:
5045:
5042:
5039:
5036:
5033:
5030:
5027:
5024:
5021:
5018:
5003:
4967:
4942:(2): 233–248,
4926:
4890:
4846:
4842:
4837:
4833:
4830:
4810:
4807:
4804:
4801:
4781:
4778:
4775:
4772:
4760:
4722:
4719:
4716:
4713:
4710:
4707:
4704:
4701:
4698:
4695:
4680:
4634:
4631:
4628:
4625:
4622:
4619:
4616:
4613:
4610:
4607:
4592:
4575:(3): 173–197,
4554:
4525:is positive",
4514:
4511:
4508:
4505:
4502:
4499:
4496:
4493:
4490:
4487:
4475:
4446:Schoenfeld, L.
4438:
4390:
4387:
4384:
4381:
4378:
4375:
4372:
4369:
4366:
4363:
4348:
4317:
4314:
4311:
4308:
4305:
4302:
4288:
4272:Comptes Rendus
4263:
4213:
4210:
4207:
4204:
4201:
4198:
4195:
4192:
4189:
4186:
4172:
4128:
4093:(3): 681–690,
4072:
4069:
4066:
4063:
4060:
4057:
4054:
4051:
4048:
4045:
4025:
4013:
3982:
3979:
3976:
3973:
3959:
3924:(129): 43–56,
3908:
3855:
3852:
3849:
3846:
3843:
3840:
3837:
3834:
3831:
3828:
3808:
3793:
3791:
3788:
3787:
3786:
3779:
3776:
3756:
3753:
3750:
3747:
3744:
3741:
3738:
3722:
3721:
3718:
3715:
3679:
3678:
3675:
3672:
3636:
3635:
3632:
3629:
3593:
3592:
3589:
3586:
3554:
3553:
3550:
3547:
3515:
3514:
3511:
3508:
3480:
3479:
3476:
3473:
3449:
3448:
3445:
3442:
3418:
3417:
3414:
3411:
3391:
3390:
3387:
3384:
3364:
3363:
3360:
3357:
3341:
3340:
3337:
3334:
3318:
3317:
3312:
3309:
3297:
3296:
3291:
3288:
3276:
3275:
3272:
3269:
3240:
3237:
3224:
3223:
3212:
3209:
3206:
3203:
3200:
3195:
3191:
3185:
3181:
3177:
3174:
3171:
3168:
3163:
3159:
3135:
3115:
3091:
3065:
3061:
3035:
3032:
3029:
3025:
3021:
3018:
3015:
3012:
3007:
3004:
2996:
2991:
2987:
2983:
2980:
2977:
2974:
2971:
2965:
2961:
2957:
2934:
2930:
2926:
2923:
2920:
2917:
2914:
2911:
2908:
2903:
2899:
2895:
2892:
2889:
2884:
2880:
2876:
2873:
2870:
2867:
2847:
2827:
2807:
2804:
2801:
2796:
2792:
2767:
2762:
2758:
2754:
2751:
2748:
2745:
2742:
2739:
2736:
2731:
2727:
2723:
2720:
2717:
2712:
2708:
2704:
2701:
2698:
2695:
2692:
2689:
2686:
2662:
2655:
2638:
2633:
2629:
2625:
2621:
2617:
2613:
2610:
2603:
2600:
2575:
2572:
2548:
2544:
2519:
2516:
2513:
2509:
2506:
2485:
2480:
2476:
2472:
2468:
2464:
2460:
2457:
2450:
2447:
2421:
2416:
2412:
2408:
2404:
2400:
2397:
2394:
2354:
2349:
2345:
2341:
2338:
2335:
2308:
2305:
2302:
2299:
2296:
2293:
2273:
2270:
2267:
2264:
2242:
2238:
2217:
2206:imaginary part
2182:
2176:
2171:
2168:
2165:
2159:
2156:
2129:
2126:
2123:
2120:
2100:
2097:
2094:
2091:
2088:
2068:
2062:
2057:
2054:
2051:
2045:
2042:
2015:
2012:
2009:
2006:
2003:
2000:
1997:
1994:
1991:
1988:
1972:in the set of
1961:
1950:
1949:
1933:
1930:
1925:
1921:
1917:
1914:
1911:
1906:
1902:
1898:
1895:
1889:
1884:
1881:
1878:
1872:
1869:
1863:
1860:
1857:
1854:
1851:
1848:
1845:
1842:
1839:
1836:
1833:
1810:
1807:
1804:
1801:
1782:
1779:
1767:Wintner (1941)
1754:
1751:
1748:
1745:
1742:
1739:
1736:
1733:
1730:
1727:
1700:
1697:
1694:
1691:
1688:
1685:
1682:
1679:
1676:
1673:
1670:
1650:
1625:
1621:
1594:
1590:
1586:
1583:
1557:
1553:
1526:
1522:
1518:
1515:
1502:, improved by
1489:
1485:
1481:
1478:
1461:
1460:
1457:
1450:
1443:
1439:
1438:
1435:
1428:
1421:
1417:
1416:
1413:
1406:
1399:
1395:
1394:
1391:
1384:
1377:
1373:
1372:
1369:
1364:
1358:
1338:
1334:
1330:
1327:
1301:
1297:
1293:
1290:
1287:
1282:
1278:
1255:
1251:
1247:
1244:
1241:
1236:
1232:
1211:
1208:
1205:
1202:
1199:
1196:
1193:
1190:
1187:
1184:
1181:
1161:
1129:
1126:
1123:
1120:
1117:
1108:gets close to
1097:
1094:
1091:
1088:
1068:
1048:
1045:
1042:
1039:
1036:
1033:
1030:
1027:
1024:
1021:
999:
995:
981:discovered by
968:
964:
960:
957:
935:
931:
927:
924:
896:
893:
890:
887:
884:
881:
878:
875:
872:
869:
849:
824:
820:
797:
793:
789:
786:
764:
760:
756:
753:
729:
726:
714:
713:
702:
693:
689:
684:
679:
675:
664:
660:
655:
650:
645:
621:
606:
605:
594:
585:
581:
576:
571:
567:
558:
554:
549:
544:
520:
517:
514:
511:
508:
505:
502:
499:
496:
493:
490:
470:
440:
420:
417:
414:
411:
408:
405:
385:
382:
379:
376:
356:
353:
350:
347:
344:
341:
338:
335:
332:
329:
302:
299:
286:
281:
277:
273:
270:
267:
262:
258:
237:
234:
231:
228:
225:
222:
219:
216:
213:
210:
190:
187:
184:
181:
178:
175:
172:
169:
166:
163:
136:
135:
124:
121:
118:
115:
112:
109:
106:
103:
100:
97:
94:
71:
61:natural number
53:Stanley Skewes
51:mathematician
31:
25:
22:
16:
13:
10:
9:
6:
4:
3:
2:
5532:
5521:
5518:
5516:
5515:Number theory
5513:
5511:
5510:Large numbers
5508:
5507:
5505:
5490:
5487:
5485:
5482:
5481:
5478:
5472:
5469:
5467:
5464:
5462:
5459:
5457:
5454:
5452:
5449:
5447:
5444:
5442:
5439:
5437:
5434:
5432:
5429:
5427:
5424:
5422:
5421:Infinitesimal
5419:
5417:
5414:
5412:
5409:
5407:
5404:
5403:
5401:
5390:
5380:
5377:
5375:
5372:
5370:
5367:
5363:
5360:
5358:
5355:
5354:
5353:
5350:
5349:
5347:
5343:
5337:
5334:
5332:
5329:
5327:
5324:
5322:
5319:
5318:
5316:
5312:
5309:
5303:
5297:
5294:
5292:
5291:Rayo's number
5289:
5287:
5284:
5282:
5279:
5277:
5274:
5272:
5269:
5267:
5264:
5262:
5259:
5257:
5254:
5252:
5249:
5247:
5244:
5242:
5239:
5237:
5234:
5232:
5229:
5227:
5224:
5222:
5219:
5217:
5214:
5212:
5209:
5207:
5204:
5202:
5199:
5197:
5194:
5192:
5189:
5187:
5184:
5182:
5179:
5177:
5174:
5172:
5169:
5168:
5166:
5156:
5152:
5151:Large numbers
5145:
5140:
5138:
5133:
5131:
5126:
5125:
5122:
5114:
5108:
5104:
5099:
5085:
5081:
5074:
5069:
5068:
5064:
5057:
5056:
5040:
5034:
5031:
5028:
5022:
5016:
5004:
5000:
4996:
4992:
4988:
4984:
4980:
4973:
4968:
4965:
4961:
4957:
4953:
4949:
4945:
4941:
4937:
4936:
4931:
4927:
4923:
4919:
4915:
4911:
4907:
4903:
4896:
4891:
4888:
4884:
4879:
4874:
4870:
4866:
4865:
4844:
4840:
4835:
4831:
4828:
4805:
4799:
4776:
4770:
4761:
4758:
4754:
4750:
4746:
4742:
4738:
4737:
4717:
4711:
4708:
4705:
4699:
4693:
4685:
4681:
4678:
4674:
4670:
4666:
4662:
4658:
4654:
4650:
4649:
4629:
4623:
4620:
4617:
4611:
4605:
4597:
4593:
4590:
4586:
4582:
4578:
4574:
4570:
4569:
4564:
4560:
4555:
4552:
4548:
4543:
4538:
4534:
4530:
4529:
4509:
4503:
4500:
4497:
4491:
4485:
4476:
4473:
4469:
4464:
4459:
4455:
4451:
4447:
4443:
4442:Rosser, J. B.
4439:
4436:
4432:
4428:
4424:
4419:
4414:
4410:
4406:
4405:
4385:
4379:
4376:
4373:
4367:
4361:
4353:
4349:
4346:
4342:
4337:
4332:
4328:
4315:
4312:
4306:
4300:
4289:
4286:
4282:
4279:: 1869–1872,
4278:
4274:
4273:
4268:
4264:
4261:
4257:
4253:
4249:
4244:
4239:
4235:
4231:
4230:
4225:
4208:
4202:
4199:
4196:
4190:
4184:
4173:
4170:
4166:
4162:
4158:
4154:
4150:
4146:
4142:
4138:
4134:
4129:
4126:
4122:
4118:
4114:
4110:
4106:
4101:
4096:
4092:
4088:
4087:
4067:
4061:
4058:
4055:
4049:
4043:
4023:
4014:
4011:
4007:
4002:
3997:
3993:
3977:
3971:
3960:
3957:
3953:
3949:
3945:
3941:
3937:
3932:
3927:
3923:
3919:
3918:
3913:
3909:
3906:
3902:
3898:
3894:
3889:
3884:
3880:
3876:
3875:
3867:
3850:
3844:
3841:
3838:
3832:
3826:
3806:
3795:
3794:
3789:
3785:
3782:
3781:
3777:
3775:
3773:
3768:
3751:
3748:
3745:
3742:
3739:
3729:
3719:
3716:
3713:
3709:
3705:
3701:
3697:
3693:
3689:
3685:
3681:
3680:
3676:
3673:
3670:
3666:
3662:
3658:
3654:
3650:
3646:
3642:
3638:
3637:
3633:
3630:
3627:
3623:
3619:
3615:
3611:
3607:
3603:
3599:
3595:
3594:
3590:
3587:
3584:
3580:
3576:
3572:
3568:
3564:
3560:
3556:
3555:
3551:
3549:7572964186421
3548:
3545:
3541:
3537:
3533:
3529:
3525:
3521:
3517:
3516:
3512:
3509:
3506:
3502:
3498:
3494:
3490:
3486:
3482:
3481:
3477:
3474:
3471:
3467:
3463:
3459:
3455:
3451:
3450:
3446:
3443:
3440:
3436:
3432:
3428:
3424:
3420:
3419:
3415:
3412:
3409:
3405:
3401:
3397:
3393:
3392:
3388:
3385:
3382:
3378:
3374:
3370:
3366:
3365:
3361:
3358:
3355:
3351:
3347:
3343:
3342:
3338:
3335:
3332:
3328:
3324:
3320:
3319:
3316:
3313:
3310:
3307:
3303:
3299:
3298:
3295:
3292:
3289:
3286:
3282:
3278:
3277:
3273:
3271:Skewes number
3270:
3267:
3263:
3262:
3259:
3257:
3252:
3251:
3238:
3235:
3210:
3204:
3198:
3193:
3189:
3183:
3179:
3175:
3169:
3161:
3157:
3149:
3148:
3147:
3133:
3113:
3105:
3089:
3081:
3063:
3059:
3033:
3030:
3027:
3019:
3016:
3013:
3005:
3002:
2994:
2989:
2985:
2981:
2975:
2969:
2932:
2928:
2924:
2921:
2918:
2915:
2912:
2909:
2906:
2901:
2897:
2893:
2890:
2887:
2882:
2878:
2874:
2871:
2868:
2865:
2845:
2825:
2802:
2794:
2790:
2781:
2760:
2756:
2752:
2749:
2746:
2743:
2740:
2737:
2734:
2729:
2725:
2721:
2718:
2715:
2710:
2706:
2702:
2699:
2696:
2693:
2687:
2684:
2676:
2672:
2670:
2660:
2656:
2654:
2652:
2631:
2627:
2623:
2619:
2601:
2598:
2573:
2570:
2546:
2542:
2533:
2514:
2478:
2474:
2470:
2466:
2448:
2445:
2433:
2414:
2410:
2406:
2402:
2395:
2392:
2369:greater than
2368:
2347:
2343:
2336:
2333:
2325:
2320:
2306:
2300:
2294:
2291:
2268:
2262:
2240:
2236:
2215:
2207:
2203:
2199:
2194:
2174:
2166:
2163:
2157:
2154:
2143:
2124:
2118:
2095:
2089:
2086:
2060:
2052:
2049:
2043:
2040:
2029:
2010:
2004:
2001:
1998:
1992:
1986:
1977:
1975:
1959:
1936:smaller terms
1931:
1923:
1919:
1912:
1909:
1904:
1900:
1896:
1887:
1879:
1876:
1870:
1867:
1861:
1855:
1849:
1846:
1843:
1837:
1831:
1824:
1823:
1822:
1805:
1799:
1791:
1787:
1778:
1776:
1772:
1768:
1749:
1743:
1740:
1737:
1731:
1725:
1717:
1712:
1698:
1692:
1686:
1683:
1680:
1674:
1668:
1648:
1639:
1623:
1619:
1610:
1592:
1588:
1584:
1581:
1573:
1555:
1551:
1542:
1541:Kotnik (2008)
1524:
1520:
1516:
1513:
1505:
1487:
1483:
1479:
1476:
1468:
1458:
1451:
1444:
1441:
1440:
1436:
1429:
1422:
1419:
1418:
1414:
1407:
1400:
1397:
1396:
1392:
1385:
1378:
1375:
1374:
1370:
1365:
1363:
1359:
1356:
1355:
1352:
1336:
1332:
1328:
1325:
1317:
1299:
1295:
1291:
1288:
1285:
1280:
1276:
1253:
1249:
1245:
1242:
1239:
1234:
1230:
1209:
1203:
1197:
1194:
1191:
1185:
1179:
1159:
1151:
1147:
1143:
1124:
1118:
1115:
1092:
1086:
1066:
1043:
1037:
1034:
1031:
1025:
1019:
997:
993:
984:
966:
962:
958:
955:
933:
929:
925:
922:
914:
910:
891:
885:
882:
879:
873:
867:
847:
840:
822:
818:
795:
791:
787:
784:
762:
758:
754:
751:
743:
742:Lehman (1966)
739:
735:
727:
725:
723:
722:Georg Kreisel
719:
700:
691:
687:
682:
677:
673:
662:
658:
653:
648:
643:
635:
634:
633:
619:
611:
610:Skewes (1955)
592:
583:
579:
574:
569:
565:
556:
552:
547:
542:
534:
533:
532:
518:
512:
506:
503:
500:
494:
488:
468:
460:
456:
455:Skewes (1933)
452:
438:
418:
412:
406:
403:
380:
374:
351:
345:
342:
339:
333:
327:
319:
315:
311:
307:
298:
284:
279:
275:
271:
268:
265:
260:
256:
232:
226:
223:
220:
214:
208:
185:
179:
176:
173:
167:
161:
153:
145:
122:
116:
110:
107:
104:
98:
92:
85:
84:
83:
69:
62:
58:
54:
50:
49:South African
46:
45:large numbers
42:
38:
37:number theory
28:
5456:Power of two
5446:Number names
5260:
5181:Ten thousand
5102:
5091:. Retrieved
5084:the original
5079:
5008:
4982:
4978:
4939:
4933:
4905:
4901:
4868:
4862:
4740:
4734:
4652:
4646:
4572:
4566:
4532:
4526:
4453:
4449:
4408:
4402:
4292:
4276:
4270:
4233:
4227:
4139:(1): 55–70,
4136:
4132:
4100:math/0509312
4090:
4084:
3963:
3921:
3915:
3912:Brent, R. P.
3878:
3872:
3771:
3769:
3725:
3711:
3707:
3703:
3699:
3695:
3691:
3687:
3683:
3668:
3664:
3660:
3656:
3652:
3648:
3644:
3640:
3625:
3621:
3617:
3613:
3609:
3605:
3601:
3597:
3582:
3578:
3574:
3570:
3566:
3562:
3558:
3543:
3539:
3535:
3531:
3527:
3523:
3519:
3513:Tóth (2019)
3510:251331775687
3504:
3500:
3496:
3492:
3488:
3484:
3478:Tóth (2019)
3469:
3465:
3461:
3457:
3453:
3447:Tóth (2019)
3438:
3434:
3430:
3426:
3422:
3416:Tóth (2019)
3407:
3403:
3399:
3395:
3389:Tóth (2019)
3380:
3376:
3372:
3368:
3362:Tóth (2019)
3353:
3349:
3345:
3339:Tóth (2019)
3330:
3326:
3322:
3305:
3301:
3284:
3280:
3265:
3255:
3253:
3227:
3225:
3103:
2779:
2668:
2664:
2658:
2561:weighted by
2434:
2321:
2197:
2195:
1978:
1951:
1784:
1713:
1640:
1609:Büthe (2015)
1504:Brent (1975)
1465:Rigorously,
1464:
1366:# of complex
1361:
837:consecutive
731:
715:
607:
453:
304:
137:
57:upper bounds
47:used by the
40:
34:
5466:Power of 10
5406:Busy beaver
5211:Quintillion
5206:Quadrillion
4930:Wintner, A.
4655:: 277–283,
4236:: 397–410,
3728:sexy primes
3315:Tóth (2019)
3294:Wolf (2011)
2675:Tóth (2019)
2653:of primes.
2588:. The term
1281:727.9513386
1235:727.9513468
5504:Categories
5471:Sagan Unit
5305:Expression
5256:Googolplex
5221:Septillion
5216:Sextillion
5162:numerical
5093:2009-09-29
4684:Skewes, S.
4677:0007.34003
4669:59.0370.02
4596:Skewes, S.
4559:Sarnak, P.
4285:45.0305.01
4260:0151.04101
4169:1149.11004
4125:1215.11084
4001:1511.02032
3956:0295.10002
3905:1042.11001
3790:References
3146:such that
2858:such that
1368:zeros used
1172:violating
481:violating
308:, who was
82:for which
5362:Pentation
5357:Tetration
5345:Operators
5314:Notations
5236:Decillion
5231:Nonillion
5226:Octillion
5158:Examples
5035:
5029:−
5017:π
4985:: 87–92,
4922:203836016
4800:π
4771:ζ
4743:: 48–70,
4712:
4706:−
4694:π
4624:
4618:−
4606:π
4504:
4498:−
4486:π
4456:: 64–94,
4380:
4374:−
4362:π
4336:1407.1914
4313:−
4301:θ
4203:
4197:−
4185:π
4062:
4044:π
3972:ψ
3845:
3827:π
3475:216646267
3413:827929093
3274:Found by
3258:-tuples:
3199:
3158:π
3017:
2986:∫
2970:
2791:π
2396:
2367:real part
2348:ρ
2337:
2295:
2263:π
2167:
2142:arguments
2119:π
2090:
2053:
2005:
1999:≈
1987:π
1960:ρ
1924:ρ
1913:
1905:ρ
1901:∑
1897:−
1880:
1862:−
1850:
1832:π
1800:π
1744:
1726:π
1687:
1669:π
1607:, and by
1585:×
1517:×
1329:×
1292:×
1246:×
1198:
1180:π
1119:
1087:π
1038:
1020:π
959:×
926:×
886:
868:π
788:×
755:×
718:effective
507:
489:π
407:
375:π
346:
340:−
328:π
272:×
261:727.95133
227:
209:π
180:
162:π
111:
93:π
5394:articles
5392:Related
5296:Infinity
5201:Trillion
5176:Thousand
5080:Demichel
4999:59578795
4561:(1994),
4161:18991347
3778:See also
3444:21432401
3336:87613571
3051:and let
2026:(if the
1788:gave an
1445:1.397162
1423:1.397166
839:integers
310:Skewes's
5489:History
5307:methods
5281:SSCG(3)
5276:TREE(3)
5196:Billion
5191:Million
5171:Hundred
4964:0004255
4956:2371519
4887:2813366
4757:0067145
4733:(II)",
4589:1329368
4551:2684372
4472:0137689
4435:0866118
4427:2007893
4341:Bibcode
4252:0202686
4153:2420864
4117:2652902
4006:Bibcode
3948:0369287
3940:2005460
3897:1752093
3386:1172531
3311:5206837
3290:1369391
2677:). Let
2671:-tuples
2661:-tuples
2651:squares
2383:
2371:
2202:complex
1786:Riemann
1401:1.39801
1379:1.39822
1326:1.39716
1289:1.39717
1243:1.39718
956:1.39822
911: (
736:of the
150:is the
142:is the
5398:order)
5246:Googol
5109:
4997:
4962:
4954:
4920:
4885:
4755:
4675:
4667:
4587:
4549:
4470:
4433:
4425:
4283:
4258:
4250:
4167:
4159:
4151:
4123:
4115:
3954:
3946:
3938:
3903:
3895:
3359:337867
3268:-tuple
3264:Prime
2838:below
2667:prime
2532:primes
1222:below
1079:where
632:below
531:below
314:proved
140:π
138:where
5484:Names
5286:BH(3)
5164:order
5087:(PDF)
5076:(PDF)
4995:S2CID
4975:(PDF)
4952:JSTOR
4918:S2CID
4908:(3),
4898:(PDF)
4423:JSTOR
4331:arXiv
4157:S2CID
4095:arXiv
4036:with
3996:arXiv
3936:JSTOR
3869:(PDF)
3819:with
3710:+24,
3706:+20,
3702:+18,
3698:+14,
3667:+24,
3663:+20,
3659:+14,
3655:+12,
3624:+20,
3620:+18,
3616:+12,
3581:+18,
3577:+14,
3573:+12,
3542:+18,
3538:+12,
1570:, by
1539:, by
1360:near
1318:gave
860:with
734:zeros
663:7.705
269:1.397
248:near
5107:ISBN
4832:<
4821:for
4056:>
3839:>
3714:+26)
3694:+8,
3690:+6,
3671:+26)
3651:+6,
3647:+2,
3628:+26)
3612:+8,
3608:+6,
3604:+2,
3585:+20)
3569:+8,
3565:+2,
3546:+20)
3534:+8,
3530:+6,
3526:+2,
3176:>
1792:for
1738:>
1681:>
1582:1.39
1442:2011
1420:2010
1398:2010
1376:2000
1357:Year
1286:<
1240:<
1192:<
1032:>
913:1987
880:>
796:1165
785:1.65
777:and
763:1165
752:1.53
674:<
566:<
501:<
266:<
221:>
201:and
174:<
146:and
105:>
4987:doi
4944:doi
4910:doi
4873:doi
4861:",
4745:doi
4673:Zbl
4665:JFM
4657:doi
4645:",
4577:doi
4537:doi
4458:doi
4413:doi
4401:",
4281:JFM
4277:158
4256:Zbl
4238:doi
4165:Zbl
4141:doi
4121:Zbl
4105:doi
4083:",
3952:Zbl
3926:doi
3901:Zbl
3883:doi
2198:lot
1611:to
1574:to
1543:to
1506:to
1452:2.0
1430:2.2
1371:by
1337:316
1300:316
1254:316
998:153
967:316
934:370
823:500
692:964
316:in
280:316
55:as
35:In
5506::
5160:in
5078:.
5032:li
4993:,
4983:17
4981:,
4977:,
4960:MR
4958:,
4950:,
4940:63
4938:,
4916:,
4906:25
4904:,
4900:,
4883:MR
4881:,
4869:80
4867:,
4845:13
4841:10
4836:10
4753:MR
4751:,
4739:,
4709:li
4671:,
4663:,
4651:,
4621:li
4585:MR
4583:,
4571:,
4565:,
4547:MR
4545:,
4533:79
4531:,
4501:li
4468:MR
4466:,
4452:,
4444:;
4431:MR
4429:,
4421:,
4409:48
4407:,
4377:li
4339:,
4329:,
4275:,
4254:,
4248:MR
4246:,
4234:11
4232:,
4226:,
4200:li
4163:,
4155:,
4149:MR
4147:,
4137:29
4135:,
4119:,
4113:MR
4111:,
4103:,
4089:,
4059:li
4004:,
3994:,
3950:,
3944:MR
3942:,
3934:,
3922:29
3920:,
3899:,
3893:MR
3891:,
3879:69
3877:,
3871:,
3842:li
3686:,
3643:,
3600:,
3561:,
3522:,
3487:,
3456:,
3425:,
3398:,
3371:,
3348:,
3325:,
3304:,
3283:,
3190:li
3014:ln
2432:.
2393:li
2334:li
2292:li
2193:.
2164:li
2087:li
2050:li
2002:li
1976:.
1910:li
1877:li
1847:li
1741:li
1684:li
1638:.
1624:19
1620:10
1593:17
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1449:10
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1412:10
1405:10
1390:10
1383:10
1351:.
1333:10
1296:10
1250:10
1195:li
1116:li
1035:li
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963:10
930:10
883:li
819:10
792:10
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688:10
683:10
678:10
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580:10
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570:10
557:79
504:li
451:.
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343:li
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224:li
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5143:e
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5115:.
5096:.
5044:)
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5038:(
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5023:x
5020:(
5002:.
4989::
4946::
4925:.
4912::
4875::
4829:x
4809:)
4806:x
4803:(
4780:)
4777:s
4774:(
4747::
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4718:x
4715:(
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4697:(
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4627:(
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4612:x
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4510:x
4507:(
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4415::
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4365:(
4343::
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4316:x
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4307:x
4304:(
4240::
4224:"
4212:)
4209:x
4206:(
4194:)
4191:x
4188:(
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4068:x
4065:(
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4050:x
4047:(
4024:x
4008::
3998::
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3978:x
3975:(
3928::
3885::
3866:"
3854:)
3851:x
3848:(
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3833:x
3830:(
3807:x
3772:k
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3752:6
3749:+
3746:p
3743:,
3740:p
3737:(
3712:p
3708:p
3704:p
3700:p
3696:p
3692:p
3688:p
3684:p
3682:(
3669:p
3665:p
3661:p
3657:p
3653:p
3649:p
3645:p
3641:p
3639:(
3626:p
3622:p
3618:p
3614:p
3610:p
3606:p
3602:p
3598:p
3596:(
3583:p
3579:p
3575:p
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3557:(
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3536:p
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3528:p
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3518:(
3505:p
3501:p
3497:p
3493:p
3489:p
3485:p
3483:(
3470:p
3466:p
3462:p
3458:p
3454:p
3452:(
3439:p
3435:p
3431:p
3427:p
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3421:(
3408:p
3404:p
3400:p
3396:p
3394:(
3381:p
3377:p
3373:p
3369:p
3367:(
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3344:(
3331:p
3327:p
3323:p
3321:(
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3300:(
3285:p
3281:p
3279:(
3266:k
3256:k
3239:.
3236:P
3211:,
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3205:p
3202:(
3194:P
3184:P
3180:C
3173:)
3170:p
3167:(
3162:P
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3114:P
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3090:p
3064:P
3060:C
3034:1
3031:+
3028:k
3024:)
3020:t
3011:(
3006:t
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2995:x
2990:2
2982:=
2979:)
2976:x
2973:(
2964:P
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2925:+
2922:p
2919:,
2916:.
2913:.
2910:.
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2883:1
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2875:+
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2806:)
2803:x
2800:(
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2780:k
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2744:.
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2703:+
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2697:,
2694:p
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2685:P
2673:(
2669:k
2659:k
2637:)
2632:2
2628:/
2624:1
2620:x
2616:(
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2609:l
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2543:p
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2475:/
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2399:(
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2340:(
2307:,
2304:)
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2298:(
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2122:(
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2093:(
2067:)
2061:x
2056:(
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2041:1
2014:)
2011:x
2008:(
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1993:x
1990:(
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1477:x
1454:×
1447:×
1432:×
1425:×
1410:×
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