1273:
811:
297:
and sometimes this formulation is called Cramér's conjecture. However, this stronger version is not supported by more accurate heuristic models, which nevertheless support the first version of Cramér's conjecture. Neither form has yet been proven or disproven.
614:
292:
1015:
1258:
In fact, due to the work done by
Granville, it is now widely believed that Cramér's conjecture is false. Indeed, there some theorems concerning short intervals between primes, such as Maier's theorem, which contradict Cramér's
599:
1580:
1802:
1032:
shows that the Cramér random model does not adequately describe the distribution of primes on short intervals, and a refinement of Cramér's model taking into account divisibility by small primes suggests that
1891:
404:
1079:
1128:
As a result of the work of H. Maier on gaps between consecutive primes, the exact formulation of Cramér's conjecture has been called into question It is still probably true that for every constant
1412:
145:
487:
1634:
2295:
and Kevin McCurley, Open
Problems in Number Theoretic Complexity, II. Algorithmic number theory (Ithaca, NY, 1994), 291–322, Lecture Notes in Comput. Sci., 877, Springer, Berlin, 1994.
1332:
882:
1667:
806:{\displaystyle \limsup _{n\to \infty }{\frac {p_{n+1}-p_{n}}{\log p_{n}}}\cdot {\frac {\left(\log \log \log p_{n}\right)^{2}}{\log \log p_{n}\log \log \log \log p_{n}}}>0.}
2604:
Equidistribution in number theory, an introduction. Proceedings of the NATO Advanced Study
Institute on equidistribution in number theory, Montréal, Canada, July 11--22, 2005
1246:
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189:
1914:
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921:
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509:
1685:
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1083:
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2744:
2727:
2692:
2722:
2615:
2523:
1484:
1699:
2863:
2779:
2685:
1817:
1116:
320:
2717:
2884:
1972:
1036:
1339:
1112:
2809:
836:
57:
2305:
2749:
893:
311:
2784:
1940:
419:
2858:
1811:
has calculated many large prime gaps. He measures the quality of fit to Cramér's conjecture by measuring the ratio
820:
2734:
2253:
2843:
2789:
824:
1588:
503:
In the other direction, E. Westzynthius proved in 1931 that prime gaps grow more than logarithmically. That is,
2848:
2816:
2804:
1930:
2833:
2774:
2754:
2739:
2176:
Ford, Kevin; Green, Ben; Konyagin, Sergei; Maynard, James; Tao, Terence (2018). "Long gaps between primes".
1934:
1447:
1021:
2046:
Westzynthius, E. (1931), "Über die
Verteilung der Zahlen die zu den n ersten Primzahlen teilerfremd sind",
2828:
2759:
1286:
842:
2821:
1639:
2794:
2080:
R. A. Rankin, The difference between consecutive prime numbers, J. London Math. Soc. 13 (1938), 242-247
2838:
2799:
2595:
2472:
2422:
1925:
1203:
819:
conjectured that the left-hand side of the above formula is infinite, and this was proven in 2014 by
2853:
2400:
1283:
conjectured the following asymptotic equality, stronger than Cramér's conjecture, for record gaps:
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2412:
2382:
2337:
2185:
2146:
2103:
410:
2663:
1945:
1029:
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2567:
2519:
2438:
307:
176:
2020:
Baker, R. C., Harman, G., Pintz, J. (2001), "The
Difference Between Consecutive Primes, II",
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1989:
1980:
1948:
on the numbers of primes in short intervals for which the model predicts an incorrect answer
1693:. This is again formally equivalent to the Shanks conjecture but suggests lower-order terms
1157:
1131:
1025:
36:
2579:
2494:
287:{\displaystyle \limsup _{n\rightarrow \infty }{\frac {p_{n+1}-p_{n}}{(\log p_{n})^{2}}}=1,}
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2607:
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2515:
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2606:. NATO Science Series II: Mathematics, Physics and Chemistry. Vol. 237. Dordrecht:
2476:
2426:
1010:{\displaystyle \limsup _{n\rightarrow \infty }{\frac {p_{n+1}-p_{n}}{\log ^{2}p_{n}}}=1}
2541:
2507:
1899:
1183:
497:
172:
2644:
1414:
which is formally identical to the Shanks conjecture but suggests a lower-order term.
2878:
1808:
1280:
816:
28:
2450:
1997:
2249:
605:
594:{\displaystyle \limsup _{n\to \infty }{\frac {p_{n+1}-p_{n}}{\log p_{n}}}=\infty .}
164:
2545:
2485:
2217:
254A, Supplement 4: Probabilistic models and heuristics for the primes (optional)
17:
2161:
2134:
2118:
2091:
832:
493:
48:
2434:
1973:"On the order of magnitude of the difference between consecutive prime numbers"
2562:
1670:
44:
43:: intuitively, that gaps between consecutive primes are always small, and the
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2649:
2212:
2033:
897:
180:
40:
2677:
2442:
1272:
2463:
Nicely, Thomas R. (1999), "New maximal prime gaps and first occurrences",
2401:"Nearest-neighbor-spacing distribution of prime numbers and quantum chaos"
1993:
2386:
2341:
2199:
1575:{\displaystyle G(x)\sim {\frac {x}{\pi (x)}}(2\log \pi (x)-\log x+c),}
2363:
Cadwell, J. H. (1971), "Large
Intervals Between Consecutive Primes",
2377:
2332:
2318:
Shanks, Daniel (1964), "On
Maximal Gaps between Successive Primes",
179:. While this is the statement explicitly conjectured by Cramér, his
1797:{\displaystyle G(x)\sim \log ^{2}x-2\log x\log \log x-(1-c)\log x.}
2417:
2190:
2151:
2108:
2090:
Ford, Kevin; Green, Ben; Konyagin, Sergei; Tao, Terence (2016).
2681:
2546:"Cramér vs. Cramér. On Cramér's probabilistic model for primes"
1886:{\displaystyle R={\frac {\log p_{n}}{\sqrt {p_{n+1}-p_{n}}}}.}
1336:
J.H. Cadwell has proposed the formula for the maximal gaps:
399:{\displaystyle p_{n+1}-p_{n}=O({\sqrt {p_{n}}}\,\log p_{n})}
1937:, much weaker but still unproven upper bounds on prime gaps
1689:
1679:
1087:
2276:
János Pintz, Very large gaps between consecutive primes,
1417:
Marek Wolf has proposed the formula for the maximal gaps
1074:{\displaystyle c\geq 2e^{-\gamma }\approx 1.1229\ldots }
1407:{\displaystyle G(x)\sim \log ^{2}x-\log x\log \log x,}
2231:"Harald Cramér and the distribution of prime numbers"
1902:
1820:
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60:
140:{\displaystyle p_{n+1}-p_{n}=O((\log p_{n})^{2}),\ }
2219:, section on The Cramér random model, January 2015.
839:. The two sets of authors improved the result by a
2550:Functiones et Approximatio Commentarii Mathematici
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286:
139:
2598:(2007). "The distribution of prime numbers". In
2048:Commentationes Physico-Mathematicae Helsingsfors
1896:He writes, "For the largest known maximal gaps,
926:
900:—in which the probability that a number of size
619:
514:
194:
2326:(88), American Mathematical Society: 646–651,
2092:"Large gaps between consecutive prime numbers"
2022:Proceedings of the London Mathematical Society
482:{\displaystyle p_{n+1}-p_{n}=O(p_{n}^{0.525})}
2693:
51:just how small they must be. It states that
8:
2178:Journal of the American Mathematical Society
1966:
1964:
1962:
2700:
2686:
2678:
1629:{\displaystyle c=\log(C_{2})=0.2778769...}
35:, formulated by the Swedish mathematician
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754:
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266:
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235:
216:
209:
197:
191:
183:actually supports the stronger statement
122:
112:
84:
65:
59:
1271:
413:. The best known unconditional bound is
302:Conditional proven results on prime gaps
39:in 1936, is an estimate for the size of
1958:
41:gaps between consecutive prime numbers
1115:. János Pintz has suggested that the
7:
1327:{\displaystyle G(x)\sim \log ^{2}x.}
877:{\displaystyle \log \log \log p_{n}}
1180:such that there is a prime between
2895:Unsolved problems in number theory
2512:Unsolved problems in number theory
1662:{\displaystyle C_{2}=1.3203236...}
1268:Related conjectures and heuristics
936:
892:Cramér's conjecture is based on a
629:
585:
524:
204:
25:
2308:, University of Cambridge (2020).
2890:Conjectures about prime numbers
1264:(internal references removed).
1253:Similarly, Robin Visser writes
1241:{\displaystyle x+d(\log x)^{c}}
1119:may be infinite, and similarly
912:or Cramér model of the primes.
2250:10.1080/03461238.1995.10413946
2238:Scandinavian Actuarial Journal
1779:
1767:
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476:
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128:
119:
99:
96:
1:
2664:"Cramér-Granville Conjecture"
2486:10.1090/S0025-5718-99-01065-0
2283::2 (April 1997), pp. 286–301.
1024:. However, as pointed out by
915:In the Cramér random model,
2162:10.4007/annals.2016.183.3.3
2135:"Large gaps between primes"
2119:10.4007/annals.2016.183.3.4
604:His result was improved by
2911:
2465:Mathematics of Computation
2435:10.1103/physreve.89.022922
2365:Mathematics of Computation
2320:Mathematics of Computation
1446:expressed in terms of the
2713:
2306:Large Gaps Between Primes
1916:has remained near 1.13."
1123:and Kevin McCurley write
1113:Euler–Mascheroni constant
409:on the assumption of the
2708:Prime number conjectures
2602:; Rudnick, Zeév (eds.).
2278:Journal of Number Theory
1941:Firoozbakht's conjecture
884:factor later that year.
2859:Schinzel's hypothesis H
2563:10.7169/facm/1229619660
2133:Maynard, James (2016).
1971:Cramér, Harald (1936),
1471:{\displaystyle \pi (x)}
1448:prime-counting function
1104:{\displaystyle \gamma }
908:. This is known as the
888:Heuristic justification
835:, and independently by
2885:Analytic number theory
2229:Granville, A. (1995),
1910:
1887:
1798:
1663:
1630:
1576:
1472:
1440:
1408:
1328:
1277:
1242:
1194:
1174:
1173:{\displaystyle d>0}
1154:, there is a constant
1148:
1147:{\displaystyle c>2}
1105:
1075:
1011:
878:
807:
595:
483:
400:
288:
141:
2864:Waring's prime number
2139:Annals of Mathematics
2096:Annals of Mathematics
2034:10.1112/plms/83.3.532
2028:(3), Wiley: 532–562,
1931:Legendre's conjecture
1911:
1888:
1799:
1664:
1631:
1577:
1473:
1441:
1409:
1329:
1275:
1243:
1195:
1175:
1149:
1106:
1076:
1012:
879:
808:
596:
484:
401:
289:
142:
2399:Wolf, Marek (2014),
1994:10.4064/aa-2-1-23-46
1935:Andrica's conjecture
1926:Prime number theorem
1900:
1818:
1700:
1640:
1589:
1485:
1453:
1439:{\displaystyle G(x)}
1421:
1340:
1287:
1204:
1184:
1158:
1132:
1095:
1037:
922:
896:model—essentially a
843:
615:
510:
420:
321:
190:
58:
2829:Legendre's constant
2645:"Cramér Conjecture"
2477:1999MaCom..68.1311N
2427:2014PhRvE..89b2922W
910:Cramér random model
475:
175:, and "log" is the
33:Cramér's conjecture
2780:Elliott–Halberstam
2765:Chinese hypothesis
2661:Weisstein, Eric W.
2642:Weisstein, Eric W.
2610:. pp. 59–83.
2471:(227): 1311–1315,
1906:
1883:
1794:
1659:
1626:
1572:
1468:
1436:
1404:
1324:
1278:
1276:Prime gap function
1238:
1190:
1170:
1144:
1101:
1071:
1007:
940:
904:is prime is 1/log
874:
803:
633:
608:, who proved that
591:
528:
479:
461:
411:Riemann hypothesis
396:
284:
208:
137:
2872:
2871:
2800:Landau's problems
2617:978-1-4020-5403-7
2600:Granville, Andrew
2596:Soundararajan, K.
2525:978-0-387-20860-2
2141:. Second series.
2098:. Second series.
1909:{\displaystyle R}
1878:
1877:
1522:
1193:{\displaystyle x}
999:
925:
795:
685:
618:
580:
513:
374:
308:conditional proof
273:
193:
177:natural logarithm
136:
18:Cramer conjecture
16:(Redirected from
2902:
2718:Hardy–Littlewood
2702:
2695:
2688:
2679:
2674:
2673:
2655:
2654:
2629:
2591:
2565:
2537:
2514:(3rd ed.).
2499:
2497:
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2420:
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2390:
2389:
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2371:(116): 909–913,
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2309:
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2290:
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2264:
2258:
2252:, archived from
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2200:10.1090/jams/876
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2011:
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2009:
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2002:
1996:, archived from
1981:Acta Arithmetica
1977:
1968:
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1026:Andrew Granville
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2608:Springer-Verlag
2594:
2540:
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2516:Springer-Verlag
2508:Guy, Richard K.
2506:
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2378:10.2307/2004355
2362:
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2333:10.2307/2002951
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2293:Leonard Adleman
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1946:Maier's theorem
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1121:Leonard Adleman
1093:
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1030:Maier's theorem
1022:probability one
988:
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829:Sergei Konyagin
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2634:External links
2632:
2631:
2630:
2616:
2592:
2556:(2): 361–376.
2538:
2524:
2501:
2500:
2455:
2391:
2355:
2310:
2304:Robin Visser,
2297:
2285:
2269:
2221:
2205:
2168:
2145:(3): 915–933.
2125:
2102:(3): 935–974.
2082:
2073:
2038:
2012:
1957:
1956:
1954:
1951:
1950:
1949:
1943:
1938:
1928:
1921:
1918:
1905:
1894:
1893:
1882:
1874:
1870:
1866:
1861:
1858:
1855:
1851:
1843:
1839:
1835:
1832:
1826:
1823:
1806:
1805:
1793:
1790:
1787:
1784:
1781:
1778:
1775:
1772:
1769:
1766:
1763:
1760:
1757:
1754:
1751:
1748:
1745:
1742:
1739:
1736:
1733:
1730:
1725:
1721:
1717:
1714:
1711:
1708:
1705:
1673:constant; see
1658:
1655:
1650:
1646:
1625:
1622:
1619:
1614:
1610:
1606:
1603:
1600:
1597:
1594:
1583:
1582:
1571:
1568:
1565:
1562:
1559:
1556:
1553:
1550:
1547:
1544:
1541:
1538:
1535:
1532:
1529:
1526:
1520:
1517:
1514:
1511:
1507:
1502:
1499:
1496:
1493:
1490:
1467:
1464:
1461:
1458:
1435:
1432:
1429:
1426:
1403:
1400:
1397:
1394:
1391:
1388:
1385:
1382:
1379:
1376:
1373:
1370:
1365:
1361:
1357:
1354:
1351:
1348:
1345:
1323:
1320:
1317:
1312:
1308:
1304:
1301:
1298:
1295:
1292:
1269:
1266:
1262:
1261:
1251:
1250:
1235:
1231:
1227:
1224:
1221:
1218:
1215:
1212:
1209:
1189:
1169:
1166:
1163:
1143:
1140:
1137:
1100:
1070:
1067:
1064:
1059:
1056:
1052:
1048:
1045:
1042:
1018:
1017:
1006:
1003:
995:
991:
987:
982:
978:
970:
966:
962:
957:
954:
951:
947:
938:
935:
932:
928:
927:lim sup
889:
886:
871:
867:
863:
860:
857:
854:
851:
848:
814:
813:
802:
799:
791:
787:
783:
780:
777:
774:
771:
768:
765:
762:
757:
753:
749:
746:
743:
740:
734:
729:
723:
719:
715:
712:
709:
706:
703:
700:
696:
689:
681:
677:
673:
670:
663:
659:
655:
650:
647:
644:
640:
631:
628:
625:
621:
620:lim sup
602:
601:
590:
587:
584:
576:
572:
568:
565:
558:
554:
550:
545:
542:
539:
535:
526:
523:
520:
516:
515:lim sup
492:due to Baker,
490:
489:
478:
473:
468:
464:
460:
457:
454:
449:
445:
441:
436:
433:
430:
426:
407:
406:
395:
390:
386:
382:
379:
371:
367:
361:
358:
355:
350:
346:
342:
337:
334:
331:
327:
306:Cramér gave a
303:
300:
295:
294:
283:
280:
277:
269:
265:
259:
255:
251:
248:
245:
238:
234:
230:
225:
222:
219:
215:
206:
203:
200:
196:
195:lim sup
173:big O notation
154:
148:
147:
133:
130:
125:
121:
115:
111:
107:
104:
101:
98:
95:
92:
87:
83:
79:
74:
71:
68:
64:
49:asymptotically
24:
14:
13:
10:
9:
6:
4:
3:
2:
2907:
2896:
2893:
2891:
2888:
2886:
2883:
2882:
2880:
2865:
2862:
2860:
2857:
2855:
2852:
2850:
2847:
2845:
2842:
2840:
2837:
2835:
2832:
2830:
2827:
2823:
2820:
2818:
2815:
2811:
2808:
2807:
2806:
2803:
2802:
2801:
2798:
2796:
2793:
2791:
2788:
2786:
2785:Firoozbakht's
2783:
2781:
2778:
2776:
2773:
2771:
2768:
2766:
2763:
2761:
2758:
2756:
2753:
2751:
2748:
2746:
2743:
2741:
2738:
2736:
2733:
2729:
2726:
2724:
2721:
2720:
2719:
2716:
2715:
2712:
2703:
2698:
2696:
2691:
2689:
2684:
2683:
2680:
2671:
2670:
2665:
2662:
2657:
2652:
2651:
2646:
2643:
2638:
2637:
2633:
2627:
2623:
2619:
2613:
2609:
2605:
2601:
2597:
2593:
2589:
2585:
2581:
2577:
2573:
2569:
2564:
2559:
2555:
2551:
2547:
2543:
2539:
2535:
2531:
2527:
2521:
2517:
2513:
2509:
2505:
2504:
2496:
2492:
2487:
2482:
2478:
2474:
2470:
2466:
2459:
2456:
2452:
2448:
2444:
2440:
2436:
2432:
2428:
2424:
2419:
2414:
2411:(2): 022922,
2410:
2406:
2402:
2395:
2392:
2388:
2384:
2379:
2374:
2370:
2366:
2359:
2356:
2351:
2347:
2343:
2339:
2334:
2329:
2325:
2321:
2314:
2311:
2307:
2301:
2298:
2294:
2289:
2286:
2282:
2279:
2273:
2270:
2259:on 2015-09-23
2255:
2251:
2247:
2243:
2239:
2232:
2225:
2222:
2218:
2214:
2209:
2206:
2201:
2197:
2192:
2187:
2183:
2179:
2172:
2169:
2163:
2158:
2153:
2148:
2144:
2140:
2136:
2129:
2126:
2120:
2115:
2110:
2105:
2101:
2097:
2093:
2086:
2083:
2077:
2074:
2069:
2065:
2061:
2057:
2053:
2050:(in German),
2049:
2042:
2039:
2035:
2031:
2027:
2023:
2016:
2013:
2003:on 2018-07-23
1999:
1995:
1991:
1987:
1983:
1982:
1974:
1967:
1965:
1963:
1959:
1952:
1947:
1944:
1942:
1939:
1936:
1932:
1929:
1927:
1924:
1923:
1919:
1917:
1903:
1880:
1872:
1868:
1864:
1859:
1856:
1853:
1849:
1841:
1837:
1833:
1830:
1824:
1821:
1814:
1813:
1812:
1810:
1809:Thomas Nicely
1791:
1788:
1785:
1782:
1776:
1773:
1770:
1764:
1761:
1758:
1755:
1752:
1749:
1746:
1743:
1740:
1737:
1734:
1731:
1728:
1723:
1719:
1715:
1709:
1703:
1696:
1695:
1694:
1691:
1687:
1681:
1677:
1672:
1669:is twice the
1656:
1653:
1648:
1644:
1623:
1620:
1612:
1608:
1601:
1598:
1595:
1592:
1569:
1563:
1560:
1557:
1554:
1551:
1548:
1542:
1536:
1533:
1530:
1527:
1515:
1509:
1505:
1500:
1494:
1488:
1481:
1480:
1479:
1462:
1456:
1449:
1430:
1424:
1415:
1401:
1398:
1395:
1392:
1389:
1386:
1383:
1380:
1377:
1374:
1371:
1368:
1363:
1359:
1355:
1349:
1343:
1334:
1321:
1318:
1315:
1310:
1306:
1302:
1296:
1290:
1282:
1281:Daniel Shanks
1274:
1267:
1265:
1260:
1256:
1255:
1254:
1249:
1233:
1225:
1222:
1219:
1213:
1210:
1207:
1187:
1167:
1164:
1161:
1141:
1138:
1135:
1126:
1125:
1124:
1122:
1118:
1114:
1098:
1089:
1085:
1068:
1065:
1062:
1057:
1054:
1050:
1046:
1043:
1040:
1031:
1027:
1023:
1004:
1001:
993:
989:
985:
980:
976:
968:
964:
960:
955:
952:
949:
945:
930:
918:
917:
916:
913:
911:
907:
903:
899:
895:
894:probabilistic
887:
885:
869:
865:
861:
858:
855:
852:
849:
846:
838:
837:James Maynard
834:
830:
826:
822:
818:
800:
797:
789:
785:
781:
778:
775:
772:
769:
766:
763:
760:
755:
751:
747:
744:
741:
738:
732:
727:
721:
717:
713:
710:
707:
704:
701:
698:
694:
687:
679:
675:
671:
668:
661:
657:
653:
648:
645:
642:
638:
623:
611:
610:
609:
607:
588:
582:
574:
570:
566:
563:
556:
552:
548:
543:
540:
537:
533:
518:
506:
505:
504:
501:
499:
495:
471:
466:
462:
455:
452:
447:
443:
439:
434:
431:
428:
424:
416:
415:
414:
412:
388:
384:
380:
377:
369:
365:
356:
353:
348:
344:
340:
335:
332:
329:
325:
317:
316:
315:
313:
309:
301:
299:
281:
278:
275:
267:
257:
253:
249:
246:
236:
232:
228:
223:
220:
217:
213:
198:
186:
185:
184:
182:
178:
174:
170:
166:
162:
157:
153:
131:
123:
113:
109:
105:
102:
93:
90:
85:
81:
77:
72:
69:
66:
62:
54:
53:
52:
50:
46:
42:
38:
37:Harald Cramér
34:
30:
29:number theory
19:
2769:
2750:Bateman–Horn
2667:
2648:
2603:
2553:
2549:
2542:Pintz, János
2511:
2468:
2464:
2458:
2408:
2405:Phys. Rev. E
2404:
2394:
2368:
2364:
2358:
2323:
2319:
2313:
2300:
2288:
2280:
2277:
2272:
2261:, retrieved
2254:the original
2241:
2237:
2224:
2208:
2181:
2177:
2171:
2142:
2138:
2128:
2099:
2095:
2085:
2076:
2051:
2047:
2041:
2025:
2021:
2015:
2005:, retrieved
1998:the original
1985:
1979:
1895:
1807:
1657:1.3203236...
1624:0.2778769...
1584:
1416:
1335:
1279:
1263:
1257:
1252:
1127:
1019:
914:
909:
905:
901:
891:
815:
606:R. A. Rankin
603:
502:
491:
408:
310:of the much
305:
296:
168:
165:prime number
160:
159:denotes the
155:
151:
149:
32:
26:
2844:Oppermann's
2790:Gilbreath's
2760:Bunyakovsky
1671:twin primes
833:Terence Tao
47:quantifies
2879:Categories
2849:Polignac's
2822:Twin prime
2817:Legendre's
2805:Goldbach's
2735:Agoh–Giuga
2626:1141.11043
2588:1226.11096
2534:1058.11001
2350:0128.04203
2263:2007-06-05
2184:: 65–105.
2068:0003.24601
2060:57.0186.02
2007:2012-03-12
1953:References
821:Kevin Ford
817:Paul Erdős
45:conjecture
2834:Lemoine's
2775:Dickson's
2755:Brocard's
2740:Andrica's
2669:MathWorld
2650:MathWorld
2572:0208-6573
2418:1212.3841
2244:: 12–28,
2213:Terry Tao
2191:1412.5029
2152:1408.5110
2109:1408.4505
1988:: 23–46,
1865:−
1834:
1786:
1774:−
1765:−
1759:
1753:
1744:
1735:−
1729:
1716:∼
1602:
1555:
1549:−
1537:π
1534:
1510:π
1501:∼
1457:π
1396:
1390:
1381:
1375:−
1369:
1356:∼
1316:
1303:∼
1223:
1117:limit sup
1099:γ
1091:), where
1069:…
1063:≈
1058:γ
1055:−
1044:≥
986:
961:−
937:∞
934:→
898:heuristic
862:
856:
850:
825:Ben Green
782:
776:
770:
764:
748:
742:
714:
708:
702:
688:⋅
672:
654:−
630:∞
627:→
586:∞
567:
549:−
525:∞
522:→
440:−
381:
341:−
250:
229:−
205:∞
202:→
181:heuristic
106:
78:−
2839:Mersenne
2770:Cramér's
2544:(2007).
2510:(2004).
2451:25003349
2443:25353560
2054:: 1–37,
1920:See also
2795:Grimm's
2745:Artin's
2580:2363833
2495:1627813
2473:Bibcode
2423:Bibcode
2387:2004355
2342:2002951
1690:A114907
1688::
1680:A005597
1678::
1111:is the
1088:A125313
1086::
2624:
2614:
2586:
2578:
2570:
2532:
2522:
2518:. A8.
2493:
2449:
2441:
2385:
2348:
2340:
2066:
2058:
1585:where
1259:model.
1066:1.1229
831:, and
496:, and
494:Harman
312:weaker
150:where
135:
2854:Pólya
2447:S2CID
2413:arXiv
2383:JSTOR
2338:JSTOR
2257:(PDF)
2234:(PDF)
2186:arXiv
2147:arXiv
2104:arXiv
2001:(PDF)
1976:(PDF)
1020:with
498:Pintz
472:0.525
2810:weak
2612:ISBN
2568:ISSN
2520:ISBN
2439:PMID
1933:and
1686:OEIS
1676:OEIS
1636:and
1200:and
1165:>
1139:>
1084:OEIS
798:>
2728:2nd
2723:1st
2622:Zbl
2584:Zbl
2558:doi
2530:Zbl
2481:doi
2431:doi
2373:doi
2346:Zbl
2328:doi
2246:doi
2196:doi
2157:doi
2143:183
2114:doi
2100:183
2064:Zbl
2056:JFM
2030:doi
1990:doi
1831:log
1783:log
1756:log
1750:log
1741:log
1720:log
1599:log
1552:log
1531:log
1393:log
1387:log
1378:log
1360:log
1307:log
1220:log
977:log
859:log
853:log
847:log
779:log
773:log
767:log
761:log
745:log
739:log
711:log
705:log
699:log
669:log
564:log
378:log
247:log
171:is
163:th
103:log
27:In
2881::
2666:.
2647:.
2620:.
2582:.
2576:MR
2574:.
2566:.
2554:37
2552:.
2548:.
2528:.
2491:MR
2489:,
2479:,
2469:68
2467:,
2445:,
2437:,
2429:,
2421:,
2409:89
2407:,
2403:,
2381:,
2369:25
2367:,
2344:,
2336:,
2324:18
2322:,
2281:63
2240:,
2236:,
2215:,
2194:.
2182:31
2180:.
2155:.
2137:.
2112:.
2094:.
2062:,
2026:83
2024:,
1984:,
1978:,
1961:^
1683:,
1478::
1028:,
827:,
823:,
801:0.
500:.
167:,
31:,
2701:e
2694:t
2687:v
2672:.
2653:.
2628:.
2590:.
2560::
2536:.
2498:.
2483::
2475::
2433::
2425::
2415::
2375::
2353:.
2330::
2267:.
2248::
2242:1
2202:.
2198::
2188::
2165:.
2159::
2149::
2122:.
2116::
2106::
2071:.
2052:5
2032::
1992::
1986:2
1904:R
1881:.
1873:n
1869:p
1860:1
1857:+
1854:n
1850:p
1842:n
1838:p
1825:=
1822:R
1804:.
1792:.
1789:x
1780:)
1777:c
1771:1
1768:(
1762:x
1747:x
1738:2
1732:x
1724:2
1713:)
1710:x
1707:(
1704:G
1654:=
1649:2
1645:C
1621:=
1618:)
1613:2
1609:C
1605:(
1596:=
1593:c
1570:,
1567:)
1564:c
1561:+
1558:x
1546:)
1543:x
1540:(
1528:2
1525:(
1519:)
1516:x
1513:(
1506:x
1498:)
1495:x
1492:(
1489:G
1466:)
1463:x
1460:(
1434:)
1431:x
1428:(
1425:G
1402:,
1399:x
1384:x
1372:x
1364:2
1353:)
1350:x
1347:(
1344:G
1322:.
1319:x
1311:2
1300:)
1297:x
1294:(
1291:G
1248:.
1234:c
1230:)
1226:x
1217:(
1214:d
1211:+
1208:x
1188:x
1168:0
1162:d
1142:2
1136:c
1081:(
1051:e
1047:2
1041:c
1005:1
1002:=
994:n
990:p
981:2
969:n
965:p
956:1
953:+
950:n
946:p
931:n
906:x
902:x
870:n
866:p
790:n
786:p
756:n
752:p
733:2
728:)
722:n
718:p
695:(
680:n
676:p
662:n
658:p
649:1
646:+
643:n
639:p
624:n
589:.
583:=
575:n
571:p
557:n
553:p
544:1
541:+
538:n
534:p
519:n
477:)
467:n
463:p
459:(
456:O
453:=
448:n
444:p
435:1
432:+
429:n
425:p
394:)
389:n
385:p
370:n
366:p
360:(
357:O
354:=
349:n
345:p
336:1
333:+
330:n
326:p
282:,
279:1
276:=
268:2
264:)
258:n
254:p
244:(
237:n
233:p
224:1
221:+
218:n
214:p
199:n
169:O
161:n
156:n
152:p
132:,
129:)
124:2
120:)
114:n
110:p
100:(
97:(
94:O
91:=
86:n
82:p
73:1
70:+
67:n
63:p
20:)
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