1646:
1346:
70:
Twin primes become increasingly rare as one examines larger ranges, in keeping with the general tendency of gaps between adjacent primes to become larger as the numbers themselves get larger. However, it is unknown whether there are infinitely many twin primes (the so-called
1505:
1212:
1098:
868:
596:
As of 14 April 2014, one year after Zhang's announcement, the bound has been reduced to 246. These improved bounds were discovered using a different approach that was simpler than Zhang's and was discovered independently by
2040:
number is divisible by 3, and therefore no three successive odd numbers can be prime unless one of them is 3. Therefore, 5 is the only prime that is part of two twin prime pairs. The lower member of a pair is by definition a
1358:
2150:
454:
873:
On the other hand, this result does not rule out that there may not be infinitely many intervals that contain two primes if we only allow the intervals to grow in size as, for example,
312:
1929:
1559:
1860:
1103:
is a major improvement on the
Goldston–Graham–Pintz–Yıldırım result. The Polymath Project optimization of Zhang's bound and the work of Maynard have reduced the bound: the
4593:
1603:
1176:
207:
2826:
4080:
712:
as long as we let these intervals grow slowly in size as we move to bigger and bigger primes. Here, "grow slowly" means that the length of these intervals can grow
481:
1341:{\displaystyle C_{2}=\prod _{\textstyle {p\;\mathrm {prime,} \atop p\geq 3}}\left(1-{\frac {1}{(p-1)^{2}}}\right)\approx 0.660161815846869573927812110014\ldots .}
984:
3312:
767:
3683:
4783:
1567:
of the prime distribution. This assumption, which is suggested by the prime number theorem, implies the twin prime conjecture, as shown in the formula for
216:; that is, the number between the two primes is a multiple of 6. As a result, the sum of any pair of twin primes (other than 3 and 5) is divisible by 12.
3070:
2993:
2284:
2188:
2016:
143:
563:
A stronger form of the twin prime conjecture, the Hardy–Littlewood conjecture (see below), postulates a distribution law for twin primes akin to the
4638:
3765:
123:
prime, this pair is the only pair of prime numbers that differ by one; thus twin primes are as closely spaced as possible for any other two primes.
104:
4621:
1619:
4586:
3688:
4616:
1121:
3602:
2403:
2490:(1915). "Über das Goldbachsche Gesetz und die Anzahl der Primzahlpaare" [On Goldbach's rule and the number of prime number pairs].
327:
4757:
3305:
4673:
889:
634:
4778:
4579:
3199:
2540:
1729:
2618:
3939:
4611:
3280:
3028:
2467:
2428:
2605:(1st Theorem. Every even number is equal to the difference of two consecutive prime numbers in an infinite number of ways ...)
4020:
1667:
2062:
4703:
3298:
3226:
3145:
598:
80:
4142:
3800:
1951:, have produced several record-largest twin primes. As of August 2022, the current largest twin prime pair known is
1710:
2704:
2382:
4643:
4167:
1682:
1500:{\displaystyle \pi _{2}(x)\sim 2C_{2}{\frac {x}{(\ln x)^{2}}}\sim 2C_{2}\int _{2}^{x}{\mathrm {d} t \over (\ln t)^{2}}}
3633:
3221:
1663:
4678:
2583:
557:
4752:
4075:
4628:
2721:
1689:
641:
4737:
4683:
4663:
264:
4742:
4710:
4698:
2603:
Tout nombre pair est égal à la différence de deux nombres premiers consécutifs d'une infinité de manières ..."
1742:
2304:
primes are isolated in the sense that the ratio of the number of isolated primes less than a given threshold
4727:
4668:
4648:
4633:
3708:
1696:
1656:
1629:
1129:
38:
that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair
4722:
4653:
4225:
3354:
1961:
An empirical analysis of all prime pairs up to 4.35 × 10 shows that if the number of such pairs less than
235:
1879:
1527:
4715:
4562:
4152:
3805:
3713:
2663:
1940:
661:
4688:
2821:
2753:
1678:
754:
732:
4732:
4693:
4132:
2037:
1813:
1516:
1137:
637:
and its generalized form, the
Polymath Project wiki states that the bound is 12 and 6, respectively.
564:
525:
254:. The modern version of Brun's argument can be used to show that the number of twin primes less than
120:
644:, if proved, would also prove there is an infinite number of twin primes, as would the existence of
4747:
4127:
3785:
1508:
1133:
4658:
4235:
4172:
4162:
4147:
3780:
3638:
3250:
3162:
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2914:
2888:
2861:
2835:
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2232:
is not part of a twin prime pair. For example, 23 is an isolated prime, since 21 and 25 are both
1572:
1145:
581:
that is less than 70 million, there are infinitely many pairs of primes that differ by
162:
88:
3559:
3243:
2025:
4204:
4179:
4157:
4137:
3760:
3732:
3425:
3258:
3195:
3122:
2757:
2636:
2536:
2499:
2297:
1944:
243:
239:
225:
1132:) is a generalization of the twin prime conjecture. It is concerned with the distribution of
1093:{\displaystyle \liminf _{n\to \infty }(p_{n+1}-p_{n})<N~\mathrm {with} ~N=7\times 10^{7},}
4114:
4104:
4099:
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3750:
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3154:
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2898:
2845:
2779:
2672:
2628:
2623:
2546:
2507:
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2233:
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590:
3216:
2969:
2910:
2857:
2791:
2686:
863:{\displaystyle \liminf _{n\to \infty }\left({\frac {p_{n+1}-p_{n}}{\log p_{n}}}\right)=0~.}
459:
3815:
3775:
3658:
3623:
3587:
3542:
3395:
3383:
2965:
2906:
2853:
2813:
2787:
2745:
2682:
2550:
2528:
2511:
1794:, but Zhang's result proves that it is true for at least one (currently unknown) value of
746:
728:
499:
The question of whether there exist infinitely many twin primes has been one of the great
2397:
1703:
700:. What this means is that we can find infinitely many intervals that contain two primes
155:
Five is the only prime that belongs to two pairs, as every twin prime pair greater than
4220:
4194:
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3959:
3810:
3770:
3755:
3627:
3518:
3483:
3438:
3363:
3345:
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2749:
2588:
1104:
750:
210:
4772:
4230:
3995:
3859:
3832:
3668:
3533:
3471:
3462:
3447:
3410:
3336:
3187:
2349:
2335:
2164:
1611:
657:
504:
17:
3285:
3042:
2918:
2865:
2799:
2374:
892:
or a slightly weaker version, they were able to show that there are infinitely many
4551:
4546:
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4526:
4521:
4516:
4511:
4506:
4501:
4496:
4491:
4486:
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4476:
4471:
4466:
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4456:
4451:
4446:
4441:
4436:
4431:
4426:
4421:
4416:
4411:
4406:
4401:
4396:
4391:
4386:
4381:
4184:
3907:
3790:
3673:
3663:
3648:
3643:
3607:
3321:
2654:
2417:
2325:
975:
717:
571:
500:
251:
76:
35:
3016:
2849:
1515:
approaches infinity. (The second ~ is not part of the conjecture and is proven by
91:
that there are infinitely many twin primes, but at present this remains unsolved.
4376:
4371:
4366:
4361:
4356:
4351:
4346:
4341:
4336:
4331:
4326:
4321:
4316:
4311:
4306:
4301:
4296:
4291:
4286:
4281:
4276:
4122:
3795:
3703:
3698:
3678:
3592:
3495:
3371:
3275:
3261:
3237:
3183:
3060:
2983:
2902:
2700:
2677:
2658:
2279:
2275:
2271:
2267:
2263:
2259:
2255:
2251:
2247:
1790:. The conjecture has not yet been proven or disproven for any specific value of
1645:
1125:
645:
602:
586:
84:
4199:
4015:
3923:
3693:
3597:
2960:
2933:
2632:
2487:
2354:
2301:
2243:
2042:
247:
231:
2934:"Variants of the Selberg sieve, and bounded intervals containing many primes"
2640:
2503:
942:
are prime. Under a stronger hypothesis they showed that for infinitely many
4240:
4189:
4070:
3266:
3233:
2443:
2330:
1955:
with 388,342 decimal digits. It was discovered in
September 2016.
1948:
1772:
713:
52:
4571:
2988:"Sequence A005597 (Decimal expansion of the twin prime constant)"
2840:
2774:
1355:.) Then a special case of the first Hardy-Littlewood conjecture is that
3742:
3290:
3166:
3093:
2783:
1670: in this section. Unsourced material may be challenged and removed.
575:
93:
3158:
119:
is not considered to be a pair of twin primes. Since 2 is the only
59:
is used for a pair of twin primes; an alternative name for this is
3737:
3723:
3286:
Sudden
Progress on Prime Number Problem Has Mathematicians Buzzing
3065:"Sequence A007508 (Number of twin prime pairs below 10)"
2950:
2893:
1522:
The conjecture can be justified (but not proven) by assuming that
2418:"The first 100,000 twin primes (only first member of pair)"
2163: + 6 is also prime then the three primes are called a
4575:
3294:
528:
made the more general conjecture that for every natural number
2619:"First proof that infinitely many prime numbers come in pairs"
1958:
There are 808,675,888,577,436 twin prime pairs below 10.
1639:
618:
needed to guarantee that infinitely many intervals of width
390:
3064:
2987:
2288:
2192:
2020:
147:
633:
primes. Moreover (see also the next section) assuming the
75:) or if there is a largest pair. The breakthrough work of
4271:
4266:
4261:
4256:
1802:
did not exist, then for any positive even natural number
605:. This second approach also gave bounds for the smallest
3143:
P.A. Clement (1949). "Congruences for sets of primes".
2014:
is conjectured to equal twice the twin prime constant (
1610:
The fully general first Hardy–Littlewood conjecture on
2824:(2009). "Small gaps between primes or almost primes".
2523:
2521:
1745:
from 1849 states that for every positive even integer
1532:
1507:
in the sense that the quotient of the two expressions
1234:
735:
showed that the constant could be improved further to
488:
2145:{\displaystyle 4((m-1)!+1)\equiv -m{\pmod {m(m+2)}}.}
2065:
1882:
1816:
1575:
1530:
1361:
1215:
1148:
987:
770:
585:. Zhang's paper was accepted in early May 2013.
511:, which states that there are infinitely many primes
462:
449:{\displaystyle {\frac {8C_{2}N}{(\log N)^{2}}}\left,}
330:
267:
165:
2879:
Maynard, James (2015). "Small gaps between primes".
1749:, there are infinitely many consecutive prime pairs
652:
Other theorems weaker than the twin prime conjecture
4249:
4213:
4113:
4090:
4064:
3831:
3824:
3722:
3616:
3580:
3329:
51:In other words, a twin prime is a prime that has a
2716:
2714:
2144:
1923:
1854:
1597:
1553:
1499:
1340:
1170:
1092:
862:
475:
448:
306:
201:
87:and others, has made substantial progress towards
2827:Transactions of the American Mathematical Society
2577:
2575:
2375:"Yitang Zhang's spectacular mathematical journey"
2028:), according to the Hardy–Littlewood conjecture.
1348:(Here the product extends over all prime numbers
716:. This result was successively improved; in 1986
3953: = 0, 1, 2, 3, ...
2705:"Polymath proposal: Bounded gaps between primes"
2396:Tao, Terry, Ph.D. (presenter) (7 October 2014).
989:
772:
593:collaborative effort to optimize Zhang's bound.
3245:Introduction to Twin Primes and Brun's Constant
2584:"Recherches nouvelles sur les nombres premiers"
2056: + 2) is a twin prime if and only if
2564:Halberstam, Heini; Richert, Hans-Egon (2010).
2535:. World Scientific. pp. 313 and 334–335.
2406:Department of Mathematics – via YouTube.
250:and helped initiate the development of modern
4587:
3306:
8:
2444:"Are all primes (past 2 and 3) of the forms
761:can be chosen to be arbitrarily small, i.e.
2186:must end in the digit 0, 2, 3, 5, 7, or 8 (
1136:, including twin primes, in analogy to the
507:for many years. This is the content of the
4594:
4580:
4572:
3828:
3313:
3299:
3291:
1241:
307:{\displaystyle {\frac {CN}{(\log N)^{2}}}}
3071:On-Line Encyclopedia of Integer Sequences
2994:On-Line Encyclopedia of Integer Sequences
2959:
2949:
2892:
2839:
2773:
2676:
2586:[New research on prime numbers].
2228: + 2 is prime. In other words,
2108:
2064:
2003:tends to infinity. The limiting value of
1906:
1887:
1881:
1840:
1821:
1815:
1730:Learn how and when to remove this message
1580:
1574:
1531:
1529:
1488:
1462:
1459:
1453:
1448:
1438:
1419:
1397:
1391:
1366:
1360:
1312:
1290:
1242:
1235:
1233:
1220:
1214:
1153:
1147:
1081:
1048:
1030:
1011:
992:
986:
835:
817:
798:
791:
775:
769:
467:
461:
402:
389:
388:
368:
341:
331:
329:
295:
268:
266:
164:
27:Prime 2 more or 2 less than another prime
3092:Oliveira e Silva, Tomás (7 April 2008).
2492:Archiv for Mathematik og Naturvidenskab
2399:Small and large gaps between the primes
2365:
2308:and the number of all primes less than
1931:which would contradict Zhang's result.
126:The first several twin prime pairs are
105:(more unsolved problems in mathematics)
101:Are there infinitely many twin primes?
3281:Polymath: Bounded gaps between primes
3192:The Encyclopedia of Integer Sequences
2938:Research in the Mathematical Sciences
1628:This conjecture has been extended by
7:
2373:Thomas, Kelly Devine (Summer 2014).
2170:For a twin prime pair of the form (6
1668:adding citations to reliable sources
2764:. Series A. Mathematical Sciences.
2116:
1999:and decreases towards about 1.3 as
1924:{\displaystyle p_{n+1}-p_{n}>N,}
1554:{\displaystyle {\tfrac {1}{\ln t}}}
532:, there are infinitely many primes
4784:Unsolved problems in number theory
2568:. Dover Publications. p. 117.
2239:The first few isolated primes are
2048:It has been proven that the pair (
1618:(not given here) implies that the
1463:
1255:
1252:
1249:
1246:
1243:
1236:
1058:
1055:
1052:
1049:
999:
782:
25:
3194:. San Diego, CA: Academic Press.
3043:"World record twin primes found!"
2758:"Small gaps between primes exist"
1122:first Hardy–Littlewood conjecture
1116:First Hardy–Littlewood conjecture
324:In fact, it is bounded above by
3689:Supersingular (moonshine theory)
3253:of 58711-digit twin prime record
2464:The Prime Pages (primes.utm.edu)
2425:The Prime Pages (primes.utm.edu)
1806:there are at most finitely many
1771:(i.e. there are infinitely many
1644:
1330:0.660161815846869573927812110014
574:announced a proof that for some
137:(59, 61), (71, 73), (101, 103),
2109:
1855:{\displaystyle p_{n+1}-p_{n}=m}
1655:needs additional citations for
96:Unsolved problem in mathematics
3684:Supersingular (elliptic curve)
3242:Xavier Gourdon, Pascal Sebah:
2135:
2132:
2120:
2110:
2096:
2084:
2072:
2069:
1592:
1586:
1485:
1472:
1416:
1403:
1378:
1372:
1309:
1296:
1165:
1159:
1036:
1004:
996:
779:
560:is the twin prime conjecture.
365:
352:
292:
279:
196:
166:
1:
3465:2 ± 2 ± 1
3146:American Mathematical Monthly
2850:10.1090/S0002-9947-09-04788-6
2726:Polymath (michaelnielsen.org)
2722:"Bounded gaps between primes"
2659:"Bounded gaps between primes"
2617:McKee, Maggie (14 May 2013).
2531:; Diamond, Harold G. (2004).
2178:+ 1) for some natural number
890:Elliott–Halberstam conjecture
696:denotes the next prime after
635:Elliott–Halberstam conjecture
242:. This famous result, called
134:(17, 19), (29, 31), (41, 43),
3276:The 20 000 first twin primes
2383:Institute for Advanced Study
1180:denote the number of primes
79:in 2013, as well as work by
3222:Encyclopedia of Mathematics
2903:10.4007/annals.2015.181.1.7
2678:10.4007/annals.2014.179.3.7
2032:Other elementary properties
2024:) (not to be confused with
1623:Hardy–Littlewood conjecture
1598:{\displaystyle \pi _{2}(x)}
1171:{\displaystyle \pi _{2}(x)}
670:and infinitely many primes
317:for some absolute constant
246:, was the first use of the
202:{\displaystyle (6n-1,6n+1)}
140:(107, 109), (137, 139), ...
131:(3, 5), (5, 7), (11, 13),
55:of two. Sometimes the term
4800:
3061:Sloane, N. J. A.
2984:Sloane, N. J. A.
2762:Japan Academy. Proceedings
1635:
1197:is also prime. Define the
896:such that at least two of
487:(slightly less than 2/3),
223:
4607:
4560:
2961:10.1186/s40687-014-0012-7
2932:Polymath, D.H.J. (2014).
2633:10.1038/nature.2013.12989
4779:Classes of prime numbers
4602:Prime number conjectures
4071:Mega (1,000,000+ digits)
3940:Arithmetic progression (
3251:"Official press release"
2582:de Polignac, A. (1849).
589:subsequently proposed a
558:de Polignac's conjecture
524:is also prime. In 1849,
4753:Schinzel's hypothesis H
1995:is about 1.7 for small
1953:2996863034895 × 2 ± 1 ,
1939:Beginning in 2007, two
720:showed that a constant
660:showed that there is a
570:On 17 April 2013,
238:of the twin primes was
234:showed that the sum of
219:
4226:Industrial-grade prime
3603:Newman–Shanks–Williams
3019:2996863034895 × 2 − 1
2533:Analytic Number Theory
2146:
1925:
1856:
1599:
1555:
1501:
1342:
1172:
1094:
864:
477:
450:
308:
203:
4758:Waring's prime number
4563:List of prime numbers
4021:Sophie Germain/Safe (
3094:"Tables of values of
2944:. artc. 12, 83.
2881:Annals of Mathematics
2748:; Motohashi, Yoichi;
2746:Goldston, Daniel Alan
2664:Annals of Mathematics
2147:
1941:distributed computing
1926:
1876:large enough we have
1857:
1788:twin prime conjecture
1743:Polignac's conjecture
1636:Polignac's conjecture
1600:
1556:
1502:
1343:
1173:
1095:
865:
727:can be used. In 2004
642:Goldbach’s conjecture
509:twin prime conjecture
495:Twin prime conjecture
478:
476:{\displaystyle C_{2}}
451:
309:
204:
73:twin prime conjecture
18:Twin Prime Conjecture
3745:(10 − 1)/9
3236:at Chris Caldwell's
3049:. 20 September 2016.
2754:Yıldırım, Cem Yalçın
2385:– via ias.edu.
2379:The Institute Letter
2216:) is a prime number
2063:
1880:
1814:
1798:. Indeed, if such a
1664:improve this article
1630:Dickson's conjecture
1573:
1528:
1517:integration by parts
1359:
1213:
1146:
1138:prime number theorem
1134:prime constellations
985:
768:
565:prime number theorem
460:
328:
265:
163:
4723:Legendre's constant
4054: ± 7, ...
3581:By integer sequence
3366:(2 + 1)/3
3015:Caldwell, Chris K.
2442:Caldwell, Chris K.
2316:tends to infinity.
2224: − 2 nor
1458:
1199:twin prime constant
640:A strengthening of
548:is also prime. The
485:twin prime constant
4674:Elliott–Halberstam
4659:Chinese hypothesis
4236:Formula for primes
3869: + 2 or
3801:Smarandache–Wellin
3259:Weisstein, Eric W.
3234:Top-20 Twin Primes
3025:The Prime Database
2784:10.3792/pjaa.82.61
2220:such that neither
2159: − 4 or
2142:
1921:
1852:
1595:
1551:
1549:
1497:
1444:
1338:
1278:
1276:
1168:
1090:
1003:
946:, at least two of
860:
786:
473:
446:
304:
199:
4766:
4765:
4694:Landau's problems
4569:
4568:
4180:Carmichael number
4115:Composite numbers
4050: ± 3, 8
4046: ± 1, 4
4009: ± 1, …
4005: ± 1, 4
4001: ± 1, 2
3991:
3990:
3536:3·2 − 1
3441:2·3 + 1
3355:Double Mersenne (
3123:Aveiro University
3074:. OEIS Foundation
2997:. OEIS Foundation
2883:. Second Series.
2834:(10): 5285–5330.
2402:(video lecture).
2381:. Princeton, NJ:
1945:Twin Prime Search
1935:Large twin primes
1740:
1739:
1732:
1714:
1548:
1495:
1426:
1319:
1274:
1229:
1064:
1047:
988:
856:
842:
771:
757:established that
629:contain at least
432:
375:
302:
115:Usually the pair
16:(Redirected from
4791:
4612:Hardy–Littlewood
4596:
4589:
4582:
4573:
4100:Eisenstein prime
4055:
4031:
4010:
3982:
3954:
3934:
3918:
3902:
3897: + 6,
3893: + 2,
3878:
3873: + 4,
3854:
3829:
3746:
3709:Highly cototient
3571:
3570:
3564:
3554:
3537:
3528:
3513:
3490:
3489:·2 − 1
3478:
3477:·2 + 1
3466:
3457:
3442:
3433:
3420:
3405:
3390:
3378:
3377:·2 + 1
3367:
3358:
3349:
3340:
3315:
3308:
3301:
3292:
3272:
3271:
3230:
3205:
3171:
3170:
3140:
3134:
3133:
3131:
3129:
3118:
3104:
3089:
3083:
3082:
3080:
3079:
3057:
3051:
3050:
3039:
3033:
3032:
3020:
3012:
3006:
3005:
3003:
3002:
2980:
2974:
2973:
2963:
2953:
2929:
2923:
2922:
2896:
2876:
2870:
2869:
2843:
2816:; Graham, S.W.;
2810:
2804:
2803:
2777:
2742:
2736:
2735:
2733:
2732:
2718:
2709:
2708:
2697:
2691:
2690:
2680:
2671:(3): 1121–1174.
2651:
2645:
2644:
2614:
2608:
2607:
2579:
2570:
2569:
2561:
2555:
2554:
2529:Bateman, Paul T.
2525:
2516:
2515:
2484:
2478:
2477:
2475:
2474:
2459:
2451:
2439:
2433:
2432:
2422:
2414:
2408:
2407:
2393:
2387:
2386:
2370:
2345:Prime quadruplet
2296:It follows from
2291:
2195:
2151:
2149:
2148:
2143:
2138:
2023:
2013:
2002:
1998:
1994:
1983:
1964:
1954:
1930:
1928:
1927:
1922:
1911:
1910:
1898:
1897:
1875:
1871:
1861:
1859:
1858:
1853:
1845:
1844:
1832:
1831:
1809:
1805:
1801:
1797:
1793:
1785:
1778:
1770:
1756:
1752:
1748:
1735:
1728:
1724:
1721:
1715:
1713:
1672:
1648:
1640:
1615:
1606:
1604:
1602:
1601:
1596:
1585:
1584:
1565:density function
1562:
1560:
1558:
1557:
1552:
1550:
1547:
1533:
1514:
1506:
1504:
1503:
1498:
1496:
1494:
1493:
1492:
1470:
1466:
1460:
1457:
1452:
1443:
1442:
1427:
1425:
1424:
1423:
1398:
1396:
1395:
1371:
1370:
1354:
1347:
1345:
1344:
1339:
1325:
1321:
1320:
1318:
1317:
1316:
1291:
1277:
1275:
1273:
1262:
1261:
1225:
1224:
1208:
1196:
1189:
1179:
1177:
1175:
1174:
1169:
1158:
1157:
1107:is at most 246.
1099:
1097:
1096:
1091:
1086:
1085:
1062:
1061:
1045:
1035:
1034:
1022:
1021:
1002:
970:
963:
956:
949:
945:
941:
934:
927:
920:
913:
906:
899:
895:
888:By assuming the
885:
883:
869:
867:
866:
861:
854:
847:
843:
841:
840:
839:
823:
822:
821:
809:
808:
792:
785:
760:
744:
742:
726:
711:
699:
695:
691:
673:
669:
632:
628:
617:
616:
591:Polymath Project
584:
580:
555:
553:
547:
546:
535:
531:
523:
522:
514:
482:
480:
479:
474:
472:
471:
455:
453:
452:
447:
442:
438:
437:
433:
431:
420:
403:
394:
393:
376:
374:
373:
372:
350:
346:
345:
332:
323:
321:
313:
311:
310:
305:
303:
301:
300:
299:
277:
269:
258:does not exceed
257:
215:
208:
206:
205:
200:
158:
150:
141:
138:
135:
132:
118:
97:
50:
48:
43:
42:
21:
4799:
4798:
4794:
4793:
4792:
4790:
4789:
4788:
4769:
4768:
4767:
4762:
4603:
4600:
4570:
4565:
4556:
4250:First 60 primes
4245:
4209:
4109:
4092:Complex numbers
4086:
4060:
4038:
4022:
3997:
3996:Bi-twin chain (
3987:
3961:
3941:
3925:
3909:
3885:
3861:
3845:
3820:
3806:Strobogrammatic
3744:
3718:
3612:
3576:
3568:
3562:
3561:
3544:
3535:
3520:
3497:
3485:
3473:
3464:
3449:
3440:
3427:
3419:# + 1
3417:
3412:
3404:# ± 1
3402:
3397:
3389:! ± 1
3385:
3373:
3365:
3357:2 − 1
3356:
3348:2 − 1
3347:
3339:2 + 1
3338:
3325:
3319:
3257:
3256:
3215:
3212:
3202:
3182:
3179:
3177:Further reading
3174:
3159:10.2307/2305816
3142:
3141:
3137:
3127:
3125:
3112:
3106:
3095:
3091:
3090:
3086:
3077:
3075:
3059:
3058:
3054:
3041:
3040:
3036:
3018:
3014:
3013:
3009:
3000:
2998:
2982:
2981:
2977:
2931:
2930:
2926:
2878:
2877:
2873:
2841:math.NT/0506067
2812:
2811:
2807:
2775:math.NT/0505300
2744:
2743:
2739:
2730:
2728:
2720:
2719:
2712:
2703:(4 June 2013).
2699:
2698:
2694:
2653:
2652:
2648:
2616:
2615:
2611:
2581:
2580:
2573:
2563:
2562:
2558:
2543:
2527:
2526:
2519:
2486:
2485:
2481:
2472:
2470:
2453:
2445:
2441:
2440:
2436:
2420:
2416:
2415:
2411:
2395:
2394:
2390:
2372:
2371:
2367:
2363:
2322:
2283:
2208:(also known as
2202:
2187:
2061:
2060:
2034:
2026:Brun's constant
2015:
2004:
2000:
1996:
1985:
1966:
1962:
1952:
1937:
1902:
1883:
1878:
1877:
1873:
1863:
1836:
1817:
1812:
1811:
1807:
1803:
1799:
1795:
1791:
1780:
1776:
1758:
1754:
1750:
1746:
1736:
1725:
1719:
1716:
1673:
1671:
1661:
1649:
1638:
1613:
1576:
1571:
1570:
1568:
1537:
1526:
1525:
1523:
1512:
1484:
1471:
1461:
1434:
1415:
1402:
1387:
1362:
1357:
1356:
1349:
1308:
1295:
1283:
1279:
1263:
1237:
1216:
1211:
1210:
1207:
1201:
1191:
1181:
1149:
1144:
1143:
1141:
1130:John Littlewood
1118:
1113:
1077:
1026:
1007:
983:
982:
965:
958:
951:
947:
943:
936:
929:
922:
915:
908:
901:
897:
893:
875:
874:
831:
824:
813:
794:
793:
787:
766:
765:
758:
737:
736:
729:Daniel Goldston
721:
714:logarithmically
701:
697:
693:
675:
671:
664:
654:
630:
619:
607:
606:
582:
578:
551:
549:
538:
537:
533:
529:
517:
516:
512:
497:
463:
458:
457:
421:
404:
398:
381:
377:
364:
351:
337:
333:
326:
325:
319:
318:
291:
278:
270:
263:
262:
255:
228:
222:
213:
161:
160:
159:is of the form
156:
142:
139:
136:
133:
130:
116:
113:
108:
107:
102:
99:
95:
46:
45:
40:
39:
28:
23:
22:
15:
12:
11:
5:
4797:
4795:
4787:
4786:
4781:
4771:
4770:
4764:
4763:
4761:
4760:
4755:
4750:
4745:
4740:
4735:
4730:
4725:
4720:
4719:
4718:
4713:
4708:
4707:
4706:
4691:
4686:
4681:
4676:
4671:
4666:
4661:
4656:
4651:
4646:
4641:
4636:
4631:
4626:
4625:
4624:
4619:
4608:
4605:
4604:
4601:
4599:
4598:
4591:
4584:
4576:
4567:
4566:
4561:
4558:
4557:
4555:
4554:
4549:
4544:
4539:
4534:
4529:
4524:
4519:
4514:
4509:
4504:
4499:
4494:
4489:
4484:
4479:
4474:
4469:
4464:
4459:
4454:
4449:
4444:
4439:
4434:
4429:
4424:
4419:
4414:
4409:
4404:
4399:
4394:
4389:
4384:
4379:
4374:
4369:
4364:
4359:
4354:
4349:
4344:
4339:
4334:
4329:
4324:
4319:
4314:
4309:
4304:
4299:
4294:
4289:
4284:
4279:
4274:
4269:
4264:
4259:
4253:
4251:
4247:
4246:
4244:
4243:
4238:
4233:
4228:
4223:
4221:Probable prime
4217:
4215:
4214:Related topics
4211:
4210:
4208:
4207:
4202:
4197:
4195:Sphenic number
4192:
4187:
4182:
4177:
4176:
4175:
4170:
4165:
4160:
4155:
4150:
4145:
4140:
4135:
4130:
4119:
4117:
4111:
4110:
4108:
4107:
4105:Gaussian prime
4102:
4096:
4094:
4088:
4087:
4085:
4084:
4083:
4073:
4068:
4066:
4062:
4061:
4059:
4058:
4034:
4030: + 1
4018:
4013:
3992:
3989:
3988:
3986:
3985:
3957:
3937:
3933: + 6
3921:
3917: + 4
3905:
3901: + 8
3881:
3877: + 6
3857:
3853: + 2
3840:
3838:
3826:
3822:
3821:
3819:
3818:
3813:
3808:
3803:
3798:
3793:
3788:
3783:
3778:
3773:
3768:
3763:
3758:
3753:
3748:
3740:
3735:
3729:
3727:
3720:
3719:
3717:
3716:
3711:
3706:
3701:
3696:
3691:
3686:
3681:
3676:
3671:
3666:
3661:
3656:
3651:
3646:
3641:
3636:
3631:
3620:
3618:
3614:
3613:
3611:
3610:
3605:
3600:
3595:
3590:
3584:
3582:
3578:
3577:
3575:
3574:
3557:
3553: − 1
3540:
3531:
3516:
3493:
3481:
3469:
3460:
3445:
3436:
3432: + 1
3423:
3415:
3408:
3400:
3393:
3381:
3369:
3361:
3352:
3343:
3333:
3331:
3327:
3326:
3320:
3318:
3317:
3310:
3303:
3295:
3289:
3288:
3283:
3278:
3273:
3254:
3248:
3240:
3231:
3211:
3210:External links
3208:
3207:
3206:
3200:
3188:Plouffe, Simon
3178:
3175:
3173:
3172:
3135:
3110:
3084:
3052:
3034:
3027:. Martin, TN:
3007:
2975:
2924:
2887:(1): 383–413.
2871:
2822:Yıldırım, C.Y.
2814:Goldston, D.A.
2805:
2737:
2710:
2692:
2646:
2609:
2589:Comptes rendus
2571:
2556:
2541:
2517:
2479:
2466:. Martin, TN:
2434:
2427:. Martin, TN:
2409:
2388:
2364:
2362:
2359:
2358:
2357:
2352:
2347:
2342:
2333:
2328:
2321:
2318:
2312:tends to 1 as
2298:Brun's theorem
2294:
2293:
2214:non-twin prime
2206:isolated prime
2201:
2200:Isolated prime
2198:
2153:
2152:
2141:
2137:
2134:
2131:
2128:
2125:
2122:
2119:
2115:
2112:
2107:
2104:
2101:
2098:
2095:
2092:
2089:
2086:
2083:
2080:
2077:
2074:
2071:
2068:
2033:
2030:
1936:
1933:
1920:
1917:
1914:
1909:
1905:
1901:
1896:
1893:
1890:
1886:
1851:
1848:
1843:
1839:
1835:
1830:
1827:
1824:
1820:
1738:
1737:
1652:
1650:
1643:
1637:
1634:
1594:
1591:
1588:
1583:
1579:
1563:describes the
1546:
1543:
1540:
1536:
1491:
1487:
1483:
1480:
1477:
1474:
1469:
1465:
1456:
1451:
1447:
1441:
1437:
1433:
1430:
1422:
1418:
1414:
1411:
1408:
1405:
1401:
1394:
1390:
1386:
1383:
1380:
1377:
1374:
1369:
1365:
1337:
1334:
1331:
1328:
1324:
1315:
1311:
1307:
1304:
1301:
1298:
1294:
1289:
1286:
1282:
1272:
1269:
1266:
1260:
1257:
1254:
1251:
1248:
1245:
1240:
1232:
1228:
1223:
1219:
1205:
1167:
1164:
1161:
1156:
1152:
1117:
1114:
1112:
1109:
1105:limit inferior
1101:
1100:
1089:
1084:
1080:
1076:
1073:
1070:
1067:
1060:
1057:
1054:
1051:
1044:
1041:
1038:
1033:
1029:
1025:
1020:
1017:
1014:
1010:
1006:
1001:
998:
995:
991:
990:lim inf
974:The result of
871:
870:
859:
853:
850:
846:
838:
834:
830:
827:
820:
816:
812:
807:
804:
801:
797:
790:
784:
781:
778:
774:
773:lim inf
741:= 0.085786...
653:
650:
501:open questions
496:
493:
470:
466:
445:
441:
436:
430:
427:
424:
419:
416:
413:
410:
407:
401:
397:
392:
387:
384:
380:
371:
367:
363:
360:
357:
354:
349:
344:
340:
336:
315:
314:
298:
294:
290:
287:
284:
281:
276:
273:
244:Brun's theorem
226:Brun's theorem
224:Main article:
221:
220:Brun's theorem
218:
211:natural number
198:
195:
192:
189:
186:
183:
180:
177:
174:
171:
168:
153:
152:
112:
109:
103:
100:
94:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4796:
4785:
4782:
4780:
4777:
4776:
4774:
4759:
4756:
4754:
4751:
4749:
4746:
4744:
4741:
4739:
4736:
4734:
4731:
4729:
4726:
4724:
4721:
4717:
4714:
4712:
4709:
4705:
4702:
4701:
4700:
4697:
4696:
4695:
4692:
4690:
4687:
4685:
4682:
4680:
4679:Firoozbakht's
4677:
4675:
4672:
4670:
4667:
4665:
4662:
4660:
4657:
4655:
4652:
4650:
4647:
4645:
4642:
4640:
4637:
4635:
4632:
4630:
4627:
4623:
4620:
4618:
4615:
4614:
4613:
4610:
4609:
4606:
4597:
4592:
4590:
4585:
4583:
4578:
4577:
4574:
4564:
4559:
4553:
4550:
4548:
4545:
4543:
4540:
4538:
4535:
4533:
4530:
4528:
4525:
4523:
4520:
4518:
4515:
4513:
4510:
4508:
4505:
4503:
4500:
4498:
4495:
4493:
4490:
4488:
4485:
4483:
4480:
4478:
4475:
4473:
4470:
4468:
4465:
4463:
4460:
4458:
4455:
4453:
4450:
4448:
4445:
4443:
4440:
4438:
4435:
4433:
4430:
4428:
4425:
4423:
4420:
4418:
4415:
4413:
4410:
4408:
4405:
4403:
4400:
4398:
4395:
4393:
4390:
4388:
4385:
4383:
4380:
4378:
4375:
4373:
4370:
4368:
4365:
4363:
4360:
4358:
4355:
4353:
4350:
4348:
4345:
4343:
4340:
4338:
4335:
4333:
4330:
4328:
4325:
4323:
4320:
4318:
4315:
4313:
4310:
4308:
4305:
4303:
4300:
4298:
4295:
4293:
4290:
4288:
4285:
4283:
4280:
4278:
4275:
4273:
4270:
4268:
4265:
4263:
4260:
4258:
4255:
4254:
4252:
4248:
4242:
4239:
4237:
4234:
4232:
4231:Illegal prime
4229:
4227:
4224:
4222:
4219:
4218:
4216:
4212:
4206:
4203:
4201:
4198:
4196:
4193:
4191:
4188:
4186:
4183:
4181:
4178:
4174:
4171:
4169:
4166:
4164:
4161:
4159:
4156:
4154:
4151:
4149:
4146:
4144:
4141:
4139:
4136:
4134:
4131:
4129:
4126:
4125:
4124:
4121:
4120:
4118:
4116:
4112:
4106:
4103:
4101:
4098:
4097:
4095:
4093:
4089:
4082:
4079:
4078:
4077:
4076:Largest known
4074:
4072:
4069:
4067:
4063:
4057:
4053:
4049:
4045:
4041:
4035:
4033:
4029:
4025:
4019:
4017:
4014:
4012:
4008:
4004:
4000:
3994:
3993:
3984:
3981:
3978: +
3977:
3973:
3969:
3966: −
3965:
3958:
3956:
3952:
3948:
3945: +
3944:
3938:
3936:
3932:
3928:
3922:
3920:
3916:
3912:
3906:
3904:
3900:
3896:
3892:
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3882:
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3876:
3872:
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3839:
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3809:
3807:
3804:
3802:
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3794:
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3772:
3769:
3767:
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3731:
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3728:
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3715:
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3710:
3707:
3705:
3702:
3700:
3697:
3695:
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3690:
3687:
3685:
3682:
3680:
3677:
3675:
3672:
3670:
3667:
3665:
3662:
3660:
3657:
3655:
3652:
3650:
3647:
3645:
3642:
3640:
3637:
3635:
3632:
3629:
3625:
3622:
3621:
3619:
3615:
3609:
3606:
3604:
3601:
3599:
3596:
3594:
3591:
3589:
3586:
3585:
3583:
3579:
3573:
3567:
3558:
3556:
3552:
3548:
3541:
3539:
3532:
3530:
3527:
3524: +
3523:
3517:
3515:
3512:
3509: −
3508:
3504:
3501: −
3500:
3494:
3492:
3488:
3482:
3480:
3476:
3470:
3468:
3461:
3459:
3456:
3453: +
3452:
3446:
3444:
3437:
3435:
3431:
3426:Pythagorean (
3424:
3422:
3418:
3409:
3407:
3403:
3394:
3392:
3388:
3382:
3380:
3376:
3370:
3368:
3362:
3360:
3353:
3351:
3344:
3342:
3335:
3334:
3332:
3328:
3323:
3316:
3311:
3309:
3304:
3302:
3297:
3296:
3293:
3287:
3284:
3282:
3279:
3277:
3274:
3269:
3268:
3263:
3262:"Twin Primes"
3260:
3255:
3252:
3249:
3247:
3246:
3241:
3239:
3235:
3232:
3228:
3224:
3223:
3218:
3214:
3213:
3209:
3203:
3201:0-12-558630-2
3197:
3193:
3189:
3185:
3181:
3180:
3176:
3168:
3164:
3160:
3156:
3152:
3148:
3147:
3139:
3136:
3124:
3120:
3116:
3109:
3102:
3098:
3088:
3085:
3073:
3072:
3066:
3062:
3056:
3053:
3048:
3047:primegrid.com
3044:
3038:
3035:
3030:
3026:
3022:
3011:
3008:
2996:
2995:
2989:
2985:
2979:
2976:
2971:
2967:
2962:
2957:
2952:
2947:
2943:
2939:
2935:
2928:
2925:
2920:
2916:
2912:
2908:
2904:
2900:
2895:
2890:
2886:
2882:
2875:
2872:
2867:
2863:
2859:
2855:
2851:
2847:
2842:
2837:
2833:
2829:
2828:
2823:
2819:
2815:
2809:
2806:
2801:
2797:
2793:
2789:
2785:
2781:
2776:
2771:
2767:
2763:
2759:
2755:
2751:
2747:
2741:
2738:
2727:
2723:
2717:
2715:
2711:
2706:
2702:
2696:
2693:
2688:
2684:
2679:
2674:
2670:
2666:
2665:
2660:
2656:
2655:Zhang, Yitang
2650:
2647:
2642:
2638:
2634:
2630:
2626:
2625:
2620:
2613:
2610:
2606:
2604:
2600:
2595:
2592:(in French).
2591:
2590:
2585:
2578:
2576:
2572:
2567:
2566:Sieve Methods
2560:
2557:
2552:
2548:
2544:
2542:981-256-080-7
2538:
2534:
2530:
2524:
2522:
2518:
2513:
2509:
2505:
2501:
2497:
2494:(in German).
2493:
2489:
2483:
2480:
2469:
2465:
2461:
2457:
2449:
2438:
2435:
2430:
2426:
2419:
2413:
2410:
2405:
2401:
2400:
2392:
2389:
2384:
2380:
2376:
2369:
2366:
2360:
2356:
2353:
2351:
2350:Prime triplet
2348:
2346:
2343:
2341:
2339:
2334:
2332:
2329:
2327:
2324:
2323:
2319:
2317:
2315:
2311:
2307:
2303:
2299:
2290:
2286:
2281:
2277:
2273:
2269:
2265:
2261:
2257:
2253:
2249:
2245:
2242:
2241:
2240:
2237:
2235:
2231:
2227:
2223:
2219:
2215:
2211:
2207:
2199:
2197:
2194:
2190:
2185:
2181:
2177:
2173:
2168:
2166:
2165:prime triplet
2162:
2158:
2139:
2129:
2126:
2123:
2117:
2113:
2105:
2102:
2099:
2093:
2090:
2087:
2081:
2078:
2075:
2066:
2059:
2058:
2057:
2055:
2051:
2046:
2044:
2039:
2031:
2029:
2027:
2022:
2018:
2011:
2007:
1992:
1988:
1981:
1978: /(log
1977:
1973:
1969:
1959:
1956:
1950:
1946:
1942:
1934:
1932:
1918:
1915:
1912:
1907:
1903:
1899:
1894:
1891:
1888:
1884:
1870:
1866:
1849:
1846:
1841:
1837:
1833:
1828:
1825:
1822:
1818:
1789:
1783:
1774:
1769:
1765:
1761:
1744:
1734:
1731:
1723:
1712:
1709:
1705:
1702:
1698:
1695:
1691:
1688:
1684:
1681: –
1680:
1676:
1675:Find sources:
1669:
1665:
1659:
1658:
1653:This section
1651:
1647:
1642:
1641:
1633:
1631:
1626:
1624:
1622:
1617:
1608:
1589:
1581:
1577:
1566:
1544:
1541:
1538:
1534:
1520:
1518:
1510:
1489:
1481:
1478:
1475:
1467:
1454:
1449:
1445:
1439:
1435:
1431:
1428:
1420:
1412:
1409:
1406:
1399:
1392:
1388:
1384:
1381:
1375:
1367:
1363:
1352:
1335:
1332:
1329:
1326:
1322:
1313:
1305:
1302:
1299:
1292:
1287:
1284:
1280:
1270:
1267:
1264:
1258:
1238:
1230:
1226:
1221:
1217:
1204:
1200:
1194:
1188:
1184:
1162:
1154:
1150:
1139:
1135:
1131:
1127:
1124:(named after
1123:
1115:
1110:
1108:
1106:
1087:
1082:
1078:
1074:
1071:
1068:
1065:
1042:
1039:
1031:
1027:
1023:
1018:
1015:
1012:
1008:
993:
981:
980:
979:
977:
972:
968:
961:
954:
939:
932:
925:
918:
911:
904:
891:
886:
882:
878:
857:
851:
848:
844:
836:
832:
828:
825:
818:
814:
810:
805:
802:
799:
795:
788:
776:
764:
763:
762:
756:
752:
748:
740:
734:
730:
724:
719:
715:
709:
705:
690:
686:
682:
678:
667:
663:
659:
651:
649:
647:
646:Siegel zeroes
643:
638:
636:
626:
622:
614:
610:
604:
600:
599:James Maynard
594:
592:
588:
577:
573:
568:
566:
561:
559:
545:
541:
527:
520:
510:
506:
505:number theory
502:
494:
492:
490:
486:
468:
464:
443:
439:
434:
428:
425:
422:
417:
414:
411:
408:
405:
399:
395:
385:
382:
378:
369:
361:
358:
355:
347:
342:
338:
334:
296:
288:
285:
282:
274:
271:
261:
260:
259:
253:
249:
245:
241:
237:
233:
227:
217:
212:
193:
190:
187:
184:
181:
178:
175:
172:
169:
149:
145:
129:
128:
127:
124:
122:
110:
106:
92:
90:
86:
82:
81:James Maynard
78:
74:
68:
66:
62:
58:
54:
37:
33:
19:
4644:Bateman–Horn
4185:Almost prime
4143:Euler–Jacobi
4051:
4047:
4043:
4039:
4037:Cunningham (
4027:
4023:
4006:
4002:
3998:
3979:
3975:
3971:
3967:
3963:
3962:consecutive
3950:
3946:
3942:
3930:
3926:
3914:
3910:
3898:
3894:
3890:
3886:
3884:Quadruplet (
3874:
3870:
3866:
3862:
3850:
3846:
3843:
3833:
3781:Full reptend
3639:Wolstenholme
3634:Wall–Sun–Sun
3565:
3550:
3546:
3525:
3521:
3510:
3506:
3502:
3498:
3486:
3474:
3454:
3450:
3429:
3413:
3398:
3386:
3374:
3322:Prime number
3265:
3244:
3220:
3191:
3184:Sloane, Neil
3150:
3144:
3138:
3126:. Retrieved
3114:
3107:
3100:
3096:
3087:
3076:. Retrieved
3068:
3055:
3046:
3037:
3024:
3010:
2999:. Retrieved
2991:
2978:
2941:
2937:
2927:
2884:
2880:
2874:
2831:
2825:
2808:
2768:(4): 61–65.
2765:
2761:
2750:Pintz, János
2740:
2729:. Retrieved
2725:
2701:Tao, Terence
2695:
2668:
2662:
2649:
2622:
2612:
2602:
2598:
2597:
2593:
2587:
2565:
2559:
2532:
2495:
2491:
2482:
2471:. Retrieved
2463:
2455:
2447:
2437:
2424:
2421:(plain text)
2412:
2398:
2391:
2378:
2368:
2337:
2326:Cousin prime
2313:
2309:
2305:
2295:
2238:
2229:
2225:
2221:
2217:
2213:
2210:single prime
2209:
2205:
2203:
2183:
2179:
2175:
2171:
2169:
2160:
2156:
2154:
2053:
2049:
2047:
2036:Every third
2035:
2009:
2005:
1990:
1986:
1979:
1975:
1971:
1967:
1960:
1957:
1938:
1868:
1864:
1787:
1781:
1779:). The case
1767:
1763:
1759:
1741:
1726:
1717:
1707:
1700:
1693:
1686:
1679:"Twin prime"
1674:
1662:Please help
1657:verification
1654:
1627:
1620:
1609:
1521:
1350:
1202:
1198:
1192:
1186:
1182:
1119:
1102:
976:Yitang Zhang
973:
966:
959:
952:
937:
930:
923:
916:
909:
902:
887:
880:
876:
872:
738:
733:Cem Yıldırım
722:
718:Helmut Maier
707:
703:
688:
684:
680:
676:
665:
655:
639:
624:
620:
612:
608:
595:
572:Yitang Zhang
569:
562:
543:
539:
518:
508:
498:
484:
316:
252:sieve theory
229:
154:
125:
114:
77:Yitang Zhang
72:
69:
64:
60:
56:
36:prime number
31:
29:
4738:Oppermann's
4684:Gilbreath's
4654:Bunyakovsky
4168:Somer–Lucas
4123:Pseudoprime
3761:Truncatable
3733:Palindromic
3617:By property
3396:Primorial (
3384:Factorial (
3238:Prime Pages
2596:: 397–401.
2498:(8): 3–19.
2468:U.T. Martin
2429:U.T. Martin
1872:and so for
1720:August 2020
1126:G. H. Hardy
1111:Conjectures
971:are prime.
603:Terence Tao
587:Terence Tao
526:de Polignac
489:given below
236:reciprocals
85:Terence Tao
4773:Categories
4743:Polignac's
4716:Twin prime
4711:Legendre's
4699:Goldbach's
4629:Agoh–Giuga
4205:Pernicious
4200:Interprime
3960:Balanced (
3751:Permutable
3726:-dependent
3543:Williams (
3439:Pierpont (
3364:Wagstaff
3346:Mersenne (
3330:By formula
3078:2019-11-01
3001:2019-11-01
2731:2014-03-27
2551:1074.11001
2512:45.0330.16
2473:2018-09-27
2361:References
2355:Sexy prime
2302:almost all
2043:Chen prime
1943:projects,
1810:such that
1773:prime gaps
1757:such that
1690:newspapers
1625:is false.
1190:such that
674:such that
658:Paul Erdős
536:such that
515:such that
248:Brun sieve
240:convergent
232:Viggo Brun
111:Properties
65:prime pair
61:prime twin
57:twin prime
32:twin prime
4728:Lemoine's
4669:Dickson's
4649:Brocard's
4634:Andrica's
4241:Prime gap
4190:Semiprime
4153:Frobenius
3860:Triplet (
3659:Ramanujan
3654:Fortunate
3624:Wieferich
3588:Fibonacci
3519:Leyland (
3484:Woodall (
3463:Solinas (
3448:Quartan (
3267:MathWorld
3227:EMS Press
3153:: 23–25.
3128:7 January
3029:UT Martin
2951:1407.4897
2894:1311.4600
2818:Pintz, J.
2641:0028-0836
2601:Théorème.
2504:0365-4524
2423:. Lists.
2331:Prime gap
2234:composite
2103:−
2100:≡
2079:−
1974:) ·
1949:PrimeGrid
1900:−
1834:−
1578:π
1542:
1479:
1446:∫
1429:∼
1410:
1382:∼
1364:π
1333:…
1327:≈
1303:−
1288:−
1268:≥
1231:∏
1151:π
1075:×
1024:−
1000:∞
997:→
829:
811:−
783:∞
780:→
745:In 2005,
725:< 0.25
656:In 1940,
426:
415:
409:
396:
359:
286:
230:In 1915,
209:for some
176:−
53:prime gap
4733:Mersenne
4664:Cramér's
4133:Elliptic
3908:Cousin (
3825:Patterns
3816:Tetradic
3811:Dihedral
3776:Primeval
3771:Delicate
3756:Circular
3743:Repunit
3534:Thabit (
3472:Cullen (
3411:Euclid (
3337:Fermat (
3190:(1995).
2919:55175056
2866:12127823
2800:18847478
2756:(2006).
2657:(2014).
2488:Brun, V.
2320:See also
2182:> 1,
2008: (
1989: (
1970: (
1862:for all
1775:of size
1509:tends to
755:Yıldırım
747:Goldston
662:constant
47:(41, 43)
41:(17, 19)
4689:Grimm's
4639:Artin's
4128:Catalan
4065:By size
3836:-tuples
3766:Minimal
3669:Regular
3560:Mills (
3496:Cuban (
3372:Proth (
3324:classes
3229:, 2001
3217:"Twins"
3167:2305816
3105:and of
3063:(ed.).
2986:(ed.).
2970:3373710
2911:3272929
2858:2515812
2792:2222213
2687:3171761
2289:A007510
2287::
2193:A002822
2191::
2052:,
2021:A114907
2019::
1786:is the
1704:scholar
1616:-tuples
1607:above.
1605:
1569:
1561:
1524:
1178:
1142:
576:integer
483:is the
322:> 0.
157:(3, 5)
148:A077800
146::
89:proving
4173:Strong
4163:Perrin
4148:Fermat
3924:Sexy (
3844:Twin (
3786:Unique
3714:Unique
3674:Strong
3664:Pillai
3644:Wilson
3608:Perrin
3198:
3165:
3108:π
3097:π
2968:
2917:
2909:
2864:
2856:
2798:
2790:
2685:
2639:
2624:Nature
2549:
2539:
2510:
2502:
2340:-tuple
2336:Prime
2282:, ...
2174:− 1, 6
1706:
1699:
1692:
1685:
1677:
1621:second
1612:prime
1140:. Let
1063:
1046:
964:, and
879:ln ln
855:
753:, and
692:where
668:< 1
456:where
117:(2, 3)
4748:Pólya
4158:Lucas
4138:Euler
3791:Happy
3738:Emirp
3704:Higgs
3699:Super
3679:Stern
3649:Lucky
3593:Lucas
3163:JSTOR
2946:arXiv
2915:S2CID
2889:arXiv
2862:S2CID
2836:arXiv
2796:S2CID
2770:arXiv
2300:that
1984:then
1867:<
1711:JSTOR
1697:books
1511:1 as
935:, or
751:Pintz
683:<
550:case
34:is a
4704:weak
4081:list
4016:Chen
3796:Self
3724:Base
3694:Good
3628:pair
3598:Pell
3549:−1)·
3196:ISBN
3130:2011
3069:The
2992:The
2637:ISSN
2537:ISBN
2500:ISSN
2452:and
2404:UCLA
2285:OEIS
2189:OEIS
2017:OEIS
1947:and
1913:>
1762:′ −
1753:and
1683:news
1128:and
1120:The
1040:<
940:+ 20
933:+ 18
926:+ 12
731:and
679:′ −
601:and
144:OEIS
121:even
4622:2nd
4617:1st
4552:281
4547:277
4542:271
4537:269
4532:263
4527:257
4522:251
4517:241
4512:239
4507:233
4502:229
4497:227
4492:223
4487:211
4482:199
4477:197
4472:193
4467:191
4462:181
4457:179
4452:173
4447:167
4442:163
4437:157
4432:151
4427:149
4422:139
4417:137
4412:131
4407:127
4402:113
4397:109
4392:107
4387:103
4382:101
4042:, 2
4026:, 2
3947:a·n
3505:)/(
3155:doi
2956:doi
2899:doi
2885:181
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