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