633:
The search for
Wolstenholme primes began in the 1960s and continued over the following decades, with the latest results published in 2007. The first Wolstenholme prime 16843 was found in 1964, although it was not explicitly reported at that time. The 1964 discovery was later independently confirmed
777:
440:
328:
185:. Wolstenholme primes are also related to other special classes of numbers, studied in the hope to be able to generalize a proof for the truth of the theorem to all positive integers greater than two.
580:
634:
in the 1970s. This remained the only known example of such a prime for almost 20 years, until the discovery announcement of the second
Wolstenholme prime 2124679 in 1993. Up to 1.2
1212:
1124:
2348:
1287:
619:
1580:
1951:
2854:
1355:
195:
131:
2033:
216:
1956:
1870:
1309:
1177:
1081:
654:
It is conjectured that infinitely many
Wolstenholme primes exist. It is conjectured that the number of Wolstenholme primes ≤
1573:
1166:
2849:
1399:
684:
2207:
355:
243:
1542:
1536:
817:
2288:
1566:
2410:
2068:
2435:
519:
1901:
1254:
842:
824:–1}. By this reasoning, the probability that the remainder takes on a particular value (e.g., 0) is about 1/
2343:
1185:
1089:
342:
167:
1332:
182:
1548:
1976:
2493:
1622:
1462:
Krattenthaler, C.; Rivoal, T. (2009), "On the integrality of the Taylor coefficients of mirror maps, II",
800:
2830:
2420:
2073:
1981:
1318:
Selfridge, J. L.; Pollack, B. W. (1964), "Fermat's last theorem is true for any exponent up to 25,000",
124:
114:
2400:
1481:
1269:
1098:
852:
338:
2395:
2053:
235:
175:
171:
48:
31:
2503:
2440:
2430:
2415:
2048:
1471:
1391:
1204:
1158:
1116:
174:
satisfied by all prime numbers greater than 3. Wolstenholme primes are named after mathematician
1827:
646:
10. The latest result as of 2007 is that there are only those two
Wolstenholme primes up to 10.
2472:
2447:
2425:
2405:
2028:
2000:
1693:
1305:
886:
667:
1501:
1446:
591:
2382:
2372:
2367:
2304:
2151:
2018:
1921:
1489:
1425:
1381:
1347:
1277:
1240:
1231:
1194:
1150:
1106:
455:
1135:
2083:
2043:
1926:
1891:
1855:
1810:
1663:
1651:
1403:
1359:
1297:
837:
586:
468:
100:
1532:
1525:
1366:
1485:
1273:
1102:
2488:
2462:
2359:
2227:
2078:
2038:
2023:
1895:
1786:
1751:
1706:
1631:
1613:
890:
334:
110:
1410:
1223:
911:
2843:
2498:
2263:
2127:
2100:
1936:
1801:
1739:
1730:
1715:
1678:
1604:
1216:
1162:
1128:
498:
480:
155:
17:
1351:
2819:
2814:
2809:
2804:
2799:
2794:
2789:
2784:
2779:
2774:
2769:
2764:
2759:
2754:
2749:
2744:
2739:
2734:
2729:
2724:
2719:
2714:
2709:
2704:
2699:
2694:
2689:
2684:
2679:
2674:
2669:
2664:
2659:
2654:
2649:
2452:
2175:
2058:
1941:
1931:
1916:
1911:
1875:
1589:
847:
163:
1395:
1282:
2644:
2639:
2634:
2629:
2624:
2619:
2614:
2609:
2604:
2599:
2594:
2589:
2584:
2579:
2574:
2569:
2564:
2559:
2554:
2549:
2544:
2390:
2063:
1971:
1966:
1946:
1860:
1763:
1639:
94:
1493:
1367:"Bernoulli numbers, Wolstenholme's theorem, and p variations of Lucas' theorem"
2467:
2283:
2191:
2111:
1961:
1865:
1429:
1386:
1154:
2508:
2457:
2338:
1245:
895:
1300:(2004), "Chapter 2. How to Recognize Whether a Natural Number is a Prime",
638:
10, no further
Wolstenholme primes were found. This was later extended to 2
145:
Wolstenholme primes: primes p such that binomial(2p-1,p-1) == 1 (mod p^4)
2010:
1558:
1291:
1208:
1170:
1120:
642:
10 by McIntosh in 1995 and
Trevisan & Weber were able to reach 2.5
205:
188:
The only two known
Wolstenholme primes are 16843 and 2124679 (sequence
1199:
1111:
1543:
Wolstenholme's
Theorem, Stirling Numbers, and Binomial Coefficients
2005:
1991:
1476:
222:
Wolstenholme prime can be defined in a number of equivalent ways.
181:
Interest in these primes first arose due to their connection with
1009:
Selfridge and
Pollack published the first Wolstenholme prime in
1562:
985:
213:
Are there any
Wolstenholme primes other than 16843 and 2124679?
1080:
Buhler, J.; Crandall, R.; Ernvall, R.; Metsänkylä, T. (1993),
772:{\displaystyle W_{p}{=}{\frac {{2p-1 \choose p-1}-1}{p^{3}}}.}
1555:
interesting observation involving the two Wolstenholme primes
198:). There are no other Wolstenholme primes less than 10.
1082:"Irregular Primes and Cyclotomic Invariants to Four Million"
190:
1255:"A search for Fibonacci-Wieferich and Wolstenholme primes"
435:{\displaystyle {2p-1 \choose p-1}\equiv 1{\pmod {p^{3}}}.}
323:{\displaystyle {2p-1 \choose p-1}\equiv 1{\pmod {p^{4}}},}
2539:
2534:
2529:
2524:
945:
943:
941:
467:. The Wolstenholme primes therefore form a subset of the
871:
Wolstenholme primes were first described by McIntosh in
178:, who first described this theorem in the 19th century.
1447:"Demonstration of a theorem relating to prime numbers"
1411:"Wolstenholme Type Theorem for Multiple Harmonic Sums"
1506:
The Quarterly Journal of Pure and Applied Mathematics
687:
594:
522:
358:
246:
140:
2517:
2481:
2381:
2358:
2332:
2099:
2092:
1990:
1884:
1848:
1597:
130:
120:
106:
93:
81:
70:
62:
54:
44:
771:
613:
574:
434:
349: > 3 the following congruence holds:
322:
575:{\displaystyle H_{p-1}\equiv 0{\pmod {p^{3}}}\,,}
394:
362:
282:
250:
2221: = 0, 1, 2, 3, ...
1333:"Testing the Converse of Wolstenholme's Theorem"
1062:
1014:
1010:
1574:
1050:
741:
709:
8:
1320:Notices of the American Mathematical Society
1304:, New York: Springer-Verlag New York, Inc.,
1178:"Irregular Primes and Cyclotomic Invariants"
39:
1533:Wolstenholme Search Status as of March 2004
1464:Communications in Number Theory and Physics
1224:"On the converse of Wolstenholme's Theorem"
1143:Bulletin of the London Mathematical Society
932:
2096:
1581:
1567:
1559:
621:expressed in lowest terms is divisible by
38:
1475:
1385:
1281:
1253:McIntosh, R. J.; Roettger, E. L. (2007),
1244:
1198:
1110:
1026:
758:
740:
708:
706:
703:
698:
692:
686:
599:
593:
568:
558:
545:
527:
521:
419:
406:
393:
361:
359:
357:
307:
294:
281:
249:
247:
245:
1502:"On Certain Properties of Prime Numbers"
949:
872:
973:
864:
786:is a Wolstenholme prime if and only if
230:A Wolstenholme prime is a prime number
217:(more unsolved problems in mathematics)
1418:International Journal of Number Theory
803:one may assume that the remainders of
650:Expected number of Wolstenholme primes
234: > 7 that satisfies the
7:
1038:
997:
961:
226:Definition via binomial coefficients
1451:The Edinburgh Philosophical Journal
1331:Trevisan, V.; Weber, K. E. (2001),
553:
414:
302:
2855:Unsolved problems in number theory
713:
454:that divides the numerator of the
366:
254:
25:
166:satisfying a stronger version of
1957:Supersingular (moonshine theory)
1302:The Little Book of Bigger Primes
509:A Wolstenholme prime is a prime
485:A Wolstenholme prime is a prime
450:A Wolstenholme prime is a prime
446:Definition via Bernoulli numbers
546:
505:Definition via harmonic numbers
407:
295:
208:Unsolved problem in mathematics
1952:Supersingular (elliptic curve)
1134:Clarke, F.; Jones, C. (2004),
795: ≡ 0 (mod
564:
547:
475:Definition via irregular pairs
425:
408:
313:
296:
170:. Wolstenholme's theorem is a
83:
71:
1:
1733:2 ± 2 ± 1
1553:-adic Growth of Harmonic Sums
1352:10.21711/231766362001/rmc2116
1283:10.1090/S0025-5718-07-01955-2
1136:"A Congruence for Factorials"
1063:McIntosh & Roettger 2007
1015:McIntosh & Roettger 2007
1011:Selfridge & Pollack 1964
345:states that for every prime
27:Special type of prime number
2871:
1494:10.4310/CNTP.2009.v3.n3.a5
1262:Mathematics of Computation
1186:Mathematics of Computation
1090:Mathematics of Computation
585:i.e. the numerator of the
478:
29:
2828:
1500:Wolstenholme, J. (1862),
1430:10.1142/s1793042108001146
1387:10.1016/j.jnt.2006.05.005
1155:10.1112/S0024609304003194
1051:Trevisan & Weber 2001
629:Search and current status
2850:Classes of prime numbers
2339:Mega (1,000,000+ digits)
2208:Arithmetic progression (
1374:Journal of Number Theory
1340:Matemática Contemporânea
1222:McIntosh, R. J. (1995),
333:where the expression in
30:Not to be confused with
1528:from The Prime Glossary
1246:10.4064/aa-71-4-381-389
933:Clarke & Jones 2004
820:in the set {0, 1, ...,
614:{\displaystyle H_{p-1}}
2494:Industrial-grade prime
1871:Newman–Shanks–Williams
1358:6 October 2011 at the
773:
615:
576:
436:
343:Wolstenholme's theorem
324:
168:Wolstenholme's theorem
2831:List of prime numbers
2289:Sophie Germain/Safe (
1127:22 September 2021 at
913:Binomial coefficients
818:uniformly distributed
774:
676:Wolstenholme quotient
616:
577:
437:
325:
183:Fermat's Last Theorem
162:is a special type of
63:Author of publication
18:Wolstenholme quotient
2013:(10 − 1)/9
1445:Babbage, C. (1819),
1402:30 June 2010 at the
1290:10 December 2010 at
1215:28 December 2021 at
1176:Johnson, W. (1975),
891:"Wolstenholme prime"
853:Table of congruences
685:
674: ≥ 5, the
592:
520:
356:
339:binomial coefficient
244:
2322: ± 7, ...
1849:By integer sequence
1634:(2 + 1)/3
1524:Caldwell, Chris K.
1486:2009arXiv0907.2578K
1274:2007MaCom..76.2087M
1103:1993MaCom..61..151B
176:Joseph Wolstenholme
172:congruence relation
49:Joseph Wolstenholme
41:
32:Wolstenholme number
2504:Formula for primes
2137: + 2 or
2069:Smarandache–Wellin
1526:Wolstenholme prime
1268:(260): 2087–2094,
1169:2 January 2011 at
1053:, p. 283–284.
1013:, p. 97 (see
986:Buhler et al. 1993
887:Weisstein, Eric W.
843:Wall–Sun–Sun prime
769:
611:
572:
432:
320:
160:Wolstenholme prime
121:Largest known term
40:Wolstenholme prime
2837:
2836:
2448:Carmichael number
2383:Composite numbers
2318: ± 3, 8
2314: ± 1, 4
2277: ± 1, …
2273: ± 1, 4
2269: ± 1, 2
2259:
2258:
1804:3·2 − 1
1709:2·3 + 1
1623:Double Mersenne (
1409:Zhao, J. (2008),
1365:Zhao, J. (2007),
1311:978-0-387-20169-6
764:
739:
670:. For each prime
668:natural logarithm
392:
341:. In comparison,
280:
152:
151:
16:(Redirected from
2862:
2368:Eisenstein prime
2323:
2299:
2278:
2250:
2222:
2202:
2186:
2170:
2165: + 6,
2161: + 2,
2146:
2141: + 4,
2122:
2097:
2014:
1977:Highly cototient
1839:
1838:
1832:
1822:
1805:
1796:
1781:
1758:
1757:·2 − 1
1746:
1745:·2 + 1
1734:
1725:
1710:
1701:
1688:
1673:
1658:
1646:
1645:·2 + 1
1635:
1626:
1617:
1608:
1583:
1576:
1569:
1560:
1531:McIntosh, R. J.
1513:
1496:
1479:
1458:
1432:
1415:
1398:
1389:
1371:
1354:
1337:
1327:
1314:
1286:
1285:
1259:
1249:
1248:
1232:Acta Arithmetica
1228:
1211:
1202:
1193:(129): 113–120,
1182:
1165:
1140:
1123:
1114:
1097:(203): 151–153,
1086:
1066:
1060:
1054:
1048:
1042:
1036:
1030:
1024:
1018:
1017:, p. 2092).
1007:
1001:
995:
989:
983:
977:
971:
965:
959:
953:
947:
936:
930:
924:
923:
922:
920:
907:
901:
900:
899:
882:
876:
869:
778:
776:
775:
770:
765:
763:
762:
753:
746:
745:
744:
738:
727:
712:
704:
702:
697:
696:
645:
641:
637:
620:
618:
617:
612:
610:
609:
581:
579:
578:
573:
567:
563:
562:
538:
537:
469:irregular primes
456:Bernoulli number
441:
439:
438:
433:
428:
424:
423:
399:
398:
397:
391:
380:
365:
329:
327:
326:
321:
316:
312:
311:
287:
286:
285:
279:
268:
253:
209:
193:
101:Irregular primes
85:
73:
55:Publication year
42:
21:
2870:
2869:
2865:
2864:
2863:
2861:
2860:
2859:
2840:
2839:
2838:
2833:
2824:
2518:First 60 primes
2513:
2477:
2377:
2360:Complex numbers
2354:
2328:
2306:
2290:
2265:
2264:Bi-twin chain (
2255:
2229:
2209:
2193:
2177:
2153:
2129:
2113:
2088:
2074:Strobogrammatic
2012:
1986:
1880:
1844:
1836:
1830:
1829:
1812:
1803:
1788:
1765:
1753:
1741:
1732:
1717:
1708:
1695:
1687:# + 1
1685:
1680:
1672:# ± 1
1670:
1665:
1657:! ± 1
1653:
1641:
1633:
1625:2 − 1
1624:
1616:2 − 1
1615:
1607:2 + 1
1606:
1593:
1587:
1537:Paul Zimmermann
1521:
1516:
1499:
1461:
1444:
1440:
1438:Further reading
1435:
1413:
1408:
1404:Wayback Machine
1369:
1364:
1360:Wayback Machine
1346:(16): 275–286,
1335:
1330:
1317:
1312:
1296:
1257:
1252:
1226:
1221:
1200:10.2307/2005468
1180:
1175:
1138:
1133:
1112:10.2307/2152942
1084:
1079:
1075:
1070:
1069:
1065:, p. 2092.
1061:
1057:
1049:
1045:
1037:
1033:
1025:
1021:
1008:
1004:
996:
992:
984:
980:
972:
968:
960:
956:
948:
939:
931:
927:
918:
916:
909:
908:
904:
885:
884:
883:
879:
870:
866:
861:
838:Wieferich prime
834:
811:
794:
754:
728:
714:
707:
705:
688:
683:
682:
652:
643:
639:
635:
631:
595:
590:
589:
587:harmonic number
554:
523:
518:
517:
507:
483:
481:Irregular prime
477:
466:
448:
415:
381:
367:
360:
354:
353:
303:
269:
255:
248:
242:
241:
228:
220:
219:
214:
211:
207:
204:
189:
148:
66:McIntosh, R. J.
35:
28:
23:
22:
15:
12:
11:
5:
2868:
2866:
2858:
2857:
2852:
2842:
2841:
2835:
2834:
2829:
2826:
2825:
2823:
2822:
2817:
2812:
2807:
2802:
2797:
2792:
2787:
2782:
2777:
2772:
2767:
2762:
2757:
2752:
2747:
2742:
2737:
2732:
2727:
2722:
2717:
2712:
2707:
2702:
2697:
2692:
2687:
2682:
2677:
2672:
2667:
2662:
2657:
2652:
2647:
2642:
2637:
2632:
2627:
2622:
2617:
2612:
2607:
2602:
2597:
2592:
2587:
2582:
2577:
2572:
2567:
2562:
2557:
2552:
2547:
2542:
2537:
2532:
2527:
2521:
2519:
2515:
2514:
2512:
2511:
2506:
2501:
2496:
2491:
2489:Probable prime
2485:
2483:
2482:Related topics
2479:
2478:
2476:
2475:
2470:
2465:
2463:Sphenic number
2460:
2455:
2450:
2445:
2444:
2443:
2438:
2433:
2428:
2423:
2418:
2413:
2408:
2403:
2398:
2387:
2385:
2379:
2378:
2376:
2375:
2373:Gaussian prime
2370:
2364:
2362:
2356:
2355:
2353:
2352:
2351:
2341:
2336:
2334:
2330:
2329:
2327:
2326:
2302:
2298: + 1
2286:
2281:
2260:
2257:
2256:
2254:
2253:
2225:
2205:
2201: + 6
2189:
2185: + 4
2173:
2169: + 8
2149:
2145: + 6
2125:
2121: + 2
2108:
2106:
2094:
2090:
2089:
2087:
2086:
2081:
2076:
2071:
2066:
2061:
2056:
2051:
2046:
2041:
2036:
2031:
2026:
2021:
2016:
2008:
2003:
1997:
1995:
1988:
1987:
1985:
1984:
1979:
1974:
1969:
1964:
1959:
1954:
1949:
1944:
1939:
1934:
1929:
1924:
1919:
1914:
1909:
1904:
1899:
1888:
1886:
1882:
1881:
1879:
1878:
1873:
1868:
1863:
1858:
1852:
1850:
1846:
1845:
1843:
1842:
1825:
1821: − 1
1808:
1799:
1784:
1761:
1749:
1737:
1728:
1713:
1704:
1700: + 1
1691:
1683:
1676:
1668:
1661:
1649:
1637:
1629:
1620:
1611:
1601:
1599:
1595:
1594:
1588:
1586:
1585:
1578:
1571:
1563:
1557:
1556:
1545:
1539:
1529:
1520:
1519:External links
1517:
1515:
1514:
1497:
1470:(3): 555–591,
1459:
1441:
1439:
1436:
1434:
1433:
1406:
1362:
1328:
1315:
1310:
1294:
1250:
1239:(4): 381–389,
1219:
1173:
1149:(4): 553–558,
1131:
1076:
1074:
1071:
1068:
1067:
1055:
1043:
1031:
1027:Ribenboim 2004
1019:
1002:
990:
988:, p. 152.
978:
976:, p. 114.
966:
954:
952:, p. 387.
937:
935:, p. 553.
925:
902:
877:
863:
862:
860:
857:
856:
855:
850:
845:
840:
833:
830:
807:
790:
780:
779:
768:
761:
757:
752:
749:
743:
737:
734:
731:
726:
723:
720:
717:
711:
701:
695:
691:
678:is defined as
651:
648:
630:
627:
608:
605:
602:
598:
583:
582:
571:
566:
561:
557:
552:
549:
544:
541:
536:
533:
530:
526:
506:
503:
499:irregular pair
479:Main article:
476:
473:
461:
447:
444:
443:
442:
431:
427:
422:
418:
413:
410:
405:
402:
396:
390:
387:
384:
379:
376:
373:
370:
364:
335:left-hand side
331:
330:
319:
315:
310:
306:
301:
298:
293:
290:
284:
278:
275:
272:
267:
264:
261:
258:
252:
227:
224:
215:
212:
206:
203:
200:
150:
149:
147:
146:
143:
137:
135:
128:
127:
122:
118:
117:
108:
104:
103:
98:
91:
90:
87:
79:
78:
75:
74:of known terms
68:
67:
64:
60:
59:
56:
52:
51:
46:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2867:
2856:
2853:
2851:
2848:
2847:
2845:
2832:
2827:
2821:
2818:
2816:
2813:
2811:
2808:
2806:
2803:
2801:
2798:
2796:
2793:
2791:
2788:
2786:
2783:
2781:
2778:
2776:
2773:
2771:
2768:
2766:
2763:
2761:
2758:
2756:
2753:
2751:
2748:
2746:
2743:
2741:
2738:
2736:
2733:
2731:
2728:
2726:
2723:
2721:
2718:
2716:
2713:
2711:
2708:
2706:
2703:
2701:
2698:
2696:
2693:
2691:
2688:
2686:
2683:
2681:
2678:
2676:
2673:
2671:
2668:
2666:
2663:
2661:
2658:
2656:
2653:
2651:
2648:
2646:
2643:
2641:
2638:
2636:
2633:
2631:
2628:
2626:
2623:
2621:
2618:
2616:
2613:
2611:
2608:
2606:
2603:
2601:
2598:
2596:
2593:
2591:
2588:
2586:
2583:
2581:
2578:
2576:
2573:
2571:
2568:
2566:
2563:
2561:
2558:
2556:
2553:
2551:
2548:
2546:
2543:
2541:
2538:
2536:
2533:
2531:
2528:
2526:
2523:
2522:
2520:
2516:
2510:
2507:
2505:
2502:
2500:
2499:Illegal prime
2497:
2495:
2492:
2490:
2487:
2486:
2484:
2480:
2474:
2471:
2469:
2466:
2464:
2461:
2459:
2456:
2454:
2451:
2449:
2446:
2442:
2439:
2437:
2434:
2432:
2429:
2427:
2424:
2422:
2419:
2417:
2414:
2412:
2409:
2407:
2404:
2402:
2399:
2397:
2394:
2393:
2392:
2389:
2388:
2386:
2384:
2380:
2374:
2371:
2369:
2366:
2365:
2363:
2361:
2357:
2350:
2347:
2346:
2345:
2344:Largest known
2342:
2340:
2337:
2335:
2331:
2325:
2321:
2317:
2313:
2309:
2303:
2301:
2297:
2293:
2287:
2285:
2282:
2280:
2276:
2272:
2268:
2262:
2261:
2252:
2249:
2246: +
2245:
2241:
2237:
2234: −
2233:
2226:
2224:
2220:
2216:
2213: +
2212:
2206:
2204:
2200:
2196:
2190:
2188:
2184:
2180:
2174:
2172:
2168:
2164:
2160:
2156:
2150:
2148:
2144:
2140:
2136:
2132:
2126:
2124:
2120:
2116:
2110:
2109:
2107:
2105:
2103:
2098:
2095:
2091:
2085:
2082:
2080:
2077:
2075:
2072:
2070:
2067:
2065:
2062:
2060:
2057:
2055:
2052:
2050:
2047:
2045:
2042:
2040:
2037:
2035:
2032:
2030:
2027:
2025:
2022:
2020:
2017:
2015:
2009:
2007:
2004:
2002:
1999:
1998:
1996:
1993:
1989:
1983:
1980:
1978:
1975:
1973:
1970:
1968:
1965:
1963:
1960:
1958:
1955:
1953:
1950:
1948:
1945:
1943:
1940:
1938:
1935:
1933:
1930:
1928:
1925:
1923:
1920:
1918:
1915:
1913:
1910:
1908:
1905:
1903:
1900:
1897:
1893:
1890:
1889:
1887:
1883:
1877:
1874:
1872:
1869:
1867:
1864:
1862:
1859:
1857:
1854:
1853:
1851:
1847:
1841:
1835:
1826:
1824:
1820:
1816:
1809:
1807:
1800:
1798:
1795:
1792: +
1791:
1785:
1783:
1780:
1777: −
1776:
1772:
1769: −
1768:
1762:
1760:
1756:
1750:
1748:
1744:
1738:
1736:
1729:
1727:
1724:
1721: +
1720:
1714:
1712:
1705:
1703:
1699:
1694:Pythagorean (
1692:
1690:
1686:
1677:
1675:
1671:
1662:
1660:
1656:
1650:
1648:
1644:
1638:
1636:
1630:
1628:
1621:
1619:
1612:
1610:
1603:
1602:
1600:
1596:
1591:
1584:
1579:
1577:
1572:
1570:
1565:
1564:
1561:
1554:
1552:
1546:
1544:
1540:
1538:
1534:
1530:
1527:
1523:
1522:
1518:
1511:
1507:
1503:
1498:
1495:
1491:
1487:
1483:
1478:
1473:
1469:
1465:
1460:
1456:
1452:
1448:
1443:
1442:
1437:
1431:
1427:
1424:(1): 73–106,
1423:
1419:
1412:
1407:
1405:
1401:
1397:
1393:
1388:
1383:
1379:
1375:
1368:
1363:
1361:
1357:
1353:
1349:
1345:
1341:
1334:
1329:
1325:
1321:
1316:
1313:
1307:
1303:
1299:
1298:Ribenboim, P.
1295:
1293:
1289:
1284:
1279:
1275:
1271:
1267:
1263:
1256:
1251:
1247:
1242:
1238:
1234:
1233:
1225:
1220:
1218:
1217:archive.today
1214:
1210:
1206:
1201:
1196:
1192:
1188:
1187:
1179:
1174:
1172:
1168:
1164:
1160:
1156:
1152:
1148:
1144:
1137:
1132:
1130:
1129:archive.today
1126:
1122:
1118:
1113:
1108:
1104:
1100:
1096:
1092:
1091:
1083:
1078:
1077:
1072:
1064:
1059:
1056:
1052:
1047:
1044:
1041:, p. 25.
1040:
1035:
1032:
1029:, p. 23.
1028:
1023:
1020:
1016:
1012:
1006:
1003:
1000:, p. 18.
999:
994:
991:
987:
982:
979:
975:
970:
967:
964:, p. 25.
963:
958:
955:
951:
950:McIntosh 1995
946:
944:
942:
938:
934:
929:
926:
915:
914:
910:Cook, J. D.,
906:
903:
898:
897:
892:
888:
881:
878:
875:, p. 385
874:
873:McIntosh 1995
868:
865:
858:
854:
851:
849:
846:
844:
841:
839:
836:
835:
831:
829:
827:
823:
819:
815:
810:
806:
802:
798:
793:
789:
785:
766:
759:
755:
750:
747:
735:
732:
729:
724:
721:
718:
715:
699:
693:
689:
681:
680:
679:
677:
673:
669:
665:
661:
657:
649:
647:
628:
626:
624:
606:
603:
600:
596:
588:
569:
559:
555:
550:
542:
539:
534:
531:
528:
524:
516:
515:
514:
512:
504:
502:
500:
496:
492:
488:
482:
474:
472:
470:
464:
460:
457:
453:
445:
429:
420:
416:
411:
403:
400:
388:
385:
382:
377:
374:
371:
368:
352:
351:
350:
348:
344:
340:
336:
317:
308:
304:
299:
291:
288:
276:
273:
270:
265:
262:
259:
256:
240:
239:
238:
237:
233:
225:
223:
218:
201:
199:
197:
192:
186:
184:
179:
177:
173:
169:
165:
161:
157:
156:number theory
144:
142:
139:
138:
136:
133:
129:
126:
123:
119:
116:
112:
109:
105:
102:
99:
96:
92:
88:
80:
76:
69:
65:
61:
57:
53:
50:
47:
43:
37:
33:
19:
2453:Almost prime
2411:Euler–Jacobi
2319:
2315:
2311:
2307:
2305:Cunningham (
2295:
2291:
2274:
2270:
2266:
2247:
2243:
2239:
2235:
2231:
2230:consecutive
2218:
2214:
2210:
2198:
2194:
2182:
2178:
2166:
2162:
2158:
2154:
2152:Quadruplet (
2142:
2138:
2134:
2130:
2118:
2114:
2101:
2049:Full reptend
1907:Wolstenholme
1906:
1902:Wall–Sun–Sun
1833:
1818:
1814:
1793:
1789:
1778:
1774:
1770:
1766:
1754:
1742:
1722:
1718:
1697:
1681:
1666:
1654:
1642:
1590:Prime number
1550:
1509:
1505:
1467:
1463:
1454:
1450:
1421:
1417:
1377:
1373:
1343:
1339:
1323:
1319:
1301:
1265:
1261:
1236:
1230:
1190:
1184:
1146:
1142:
1094:
1088:
1058:
1046:
1034:
1022:
1005:
993:
981:
974:Johnson 1975
969:
957:
928:
917:, retrieved
912:
905:
894:
880:
867:
848:Wilson prime
825:
821:
813:
808:
804:
796:
791:
787:
783:
781:
675:
671:
666:denotes the
663:
659:
655:
653:
632:
622:
584:
510:
508:
494:
490:
486:
484:
462:
458:
451:
449:
346:
332:
231:
229:
221:
187:
180:
164:prime number
159:
153:
82:Conjectured
36:
2436:Somer–Lucas
2391:Pseudoprime
2029:Truncatable
2001:Palindromic
1885:By property
1664:Primorial (
1652:Factorial (
1547:Conrad, K.
919:21 December
801:Empirically
489:such that (
107:First terms
95:Subsequence
45:Named after
2844:Categories
2473:Pernicious
2468:Interprime
2228:Balanced (
2019:Permutable
1994:-dependent
1811:Williams (
1707:Pierpont (
1632:Wagstaff
1614:Mersenne (
1598:By formula
1541:Bruck, R.
1535:e-mail to
1073:References
513:such that
497:–3) is an
337:denotes a
236:congruence
202:Definition
2509:Prime gap
2458:Semiprime
2421:Frobenius
2128:Triplet (
1927:Ramanujan
1922:Fortunate
1892:Wieferich
1856:Fibonacci
1787:Leyland (
1752:Woodall (
1731:Solinas (
1716:Quartan (
1477:0907.2578
1380:: 18–26,
1163:120202453
1039:Zhao 2007
998:Zhao 2007
962:Zhao 2008
896:MathWorld
782:Clearly,
748:−
733:−
722:−
658:is about
604:−
540:≡
532:−
401:≡
386:−
375:−
289:≡
274:−
263:−
2401:Elliptic
2176:Cousin (
2093:Patterns
2084:Tetradic
2079:Dihedral
2044:Primeval
2039:Delicate
2024:Circular
2011:Repunit
1802:Thabit (
1740:Cullen (
1679:Euclid (
1605:Fermat (
1400:Archived
1356:Archived
1288:Archived
1213:Archived
1167:Archived
1125:Archived
832:See also
662:, where
191:A088164
89:Infinite
86:of terms
2396:Catalan
2333:By size
2104:-tuples
2034:Minimal
1937:Regular
1828:Mills (
1764:Cuban (
1640:Proth (
1592:classes
1512:: 35–39
1482:Bibcode
1457:: 46–49
1292:WebCite
1270:Bibcode
1209:2005468
1171:WebCite
1121:2152942
1099:Bibcode
812:modulo
660:ln ln x
194:in the
141:A088164
125:2124679
115:2124679
2441:Strong
2431:Perrin
2416:Fermat
2192:Sexy (
2112:Twin (
2054:Unique
1982:Unique
1942:Strong
1932:Pillai
1912:Wilson
1876:Perrin
1396:937685
1394:
1308:
1207:
1161:
1119:
2426:Lucas
2406:Euler
2059:Happy
2006:Emirp
1972:Higgs
1967:Super
1947:Stern
1917:Lucky
1861:Lucas
1472:arXiv
1414:(PDF)
1392:S2CID
1370:(PDF)
1336:(PDF)
1258:(PDF)
1227:(PDF)
1205:JSTOR
1181:(PDF)
1159:S2CID
1139:(PDF)
1117:JSTOR
1085:(PDF)
859:Notes
134:index
111:16843
2349:list
2284:Chen
2064:Self
1992:Base
1962:Good
1896:pair
1866:Pell
1817:−1)·
1549:The
1326:: 97
1306:ISBN
921:2010
816:are
196:OEIS
158:, a
132:OEIS
58:1995
2820:281
2815:277
2810:271
2805:269
2800:263
2795:257
2790:251
2785:241
2780:239
2775:233
2770:229
2765:227
2760:223
2755:211
2750:199
2745:197
2740:193
2735:191
2730:181
2725:179
2720:173
2715:167
2710:163
2705:157
2700:151
2695:149
2690:139
2685:137
2680:131
2675:127
2670:113
2665:109
2660:107
2655:103
2650:101
2310:, 2
2294:, 2
2215:a·n
1773:)/(
1490:doi
1426:doi
1382:doi
1378:123
1348:doi
1278:doi
1241:doi
1195:doi
1151:doi
1107:doi
799:).
551:mod
412:mod
300:mod
154:In
84:no.
72:No.
2846::
2645:97
2640:89
2635:83
2630:79
2625:73
2620:71
2615:67
2610:61
2605:59
2600:53
2595:47
2590:43
2585:41
2580:37
2575:31
2570:29
2565:23
2560:19
2555:17
2550:13
2545:11
2242:,
2238:,
2217:,
2197:,
2181:,
2157:,
2133:,
2117:,
1508:,
1504:,
1488:,
1480:,
1466:,
1453:,
1449:,
1420:,
1416:,
1390:,
1376:,
1372:,
1344:21
1342:,
1338:,
1324:11
1322:,
1276:,
1266:76
1264:,
1260:,
1237:71
1235:,
1229:,
1203:,
1191:29
1189:,
1183:,
1157:,
1147:36
1145:,
1141:,
1115:,
1105:,
1095:61
1093:,
1087:,
940:^
893:,
889:,
828:.
664:ln
625:.
501:.
493:,
471:.
465:−3
113:,
97:of
2540:7
2535:5
2530:3
2525:2
2324:)
2320:p
2316:p
2312:p
2308:p
2300:)
2296:p
2292:p
2279:)
2275:n
2271:n
2267:n
2251:)
2248:n
2244:p
2240:p
2236:n
2232:p
2223:)
2219:n
2211:p
2203:)
2199:p
2195:p
2187:)
2183:p
2179:p
2171:)
2167:p
2163:p
2159:p
2155:p
2147:)
2143:p
2139:p
2135:p
2131:p
2123:)
2119:p
2115:p
2102:k
1898:)
1894:(
1840:)
1837:⌋
1834:A
1831:⌊
1823:)
1819:b
1815:b
1813:(
1806:)
1797:)
1794:y
1790:x
1782:)
1779:y
1775:x
1771:y
1767:x
1759:)
1755:n
1747:)
1743:n
1735:)
1726:)
1723:y
1719:x
1711:)
1702:)
1698:n
1696:4
1689:)
1684:n
1682:p
1674:)
1669:n
1667:p
1659:)
1655:n
1647:)
1643:k
1627:)
1618:)
1609:)
1582:e
1575:t
1568:v
1551:p
1510:5
1492::
1484::
1474::
1468:3
1455:1
1428::
1422:4
1384::
1350::
1280::
1272::
1243::
1197::
1153::
1109::
1101::
826:p
822:p
814:p
809:p
805:W
797:p
792:p
788:W
784:p
767:.
760:3
756:p
751:1
742:)
736:1
730:p
725:1
719:p
716:2
710:(
700:=
694:p
690:W
672:p
656:x
644:×
640:×
636:×
623:p
607:1
601:p
597:H
570:,
565:)
560:3
556:p
548:(
543:0
535:1
529:p
525:H
511:p
495:p
491:p
487:p
463:p
459:B
452:p
430:.
426:)
421:3
417:p
409:(
404:1
395:)
389:1
383:p
378:1
372:p
369:2
363:(
347:p
318:,
314:)
309:4
305:p
297:(
292:1
283:)
277:1
271:p
266:1
260:p
257:2
251:(
232:p
210::
77:2
34:.
20:)
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