358:
119:
2215:. In this way, successive primes in a Cunningham chain are essentially shifted left in binary with ones filling in the least significant digits. For example, here is a complete length 6 chain which starts at 141361469:
2682:
124:
2755:
1843:
2607:
627:
2436:
2325:
1941:
1892:
1719:
678:
409:
2380:
2003:
1774:
478:
727:
2816:
3795:
1950:. (Note that, as with all bases, multiplying by the base "shifts" the digits to the left; e.g. in decimal we have 314 × 10 = 3140.) When we consider
2849:
2489:
2183:
2150:
2090:
353:{\displaystyle {\begin{aligned}p_{2}&=2p_{1}+1,\\p_{3}&=4p_{1}+3,\\p_{4}&=8p_{1}+7,\\&{}\ \vdots \\p_{i}&=2^{i-1}p_{1}+(2^{i-1}-1),\end{aligned}}}
2213:
2782:
2516:
2117:
2057:
2030:
1670:
3027:
3398:
2456:
429:
3480:
2966:
884:
There are computing competitions for the longest
Cunningham chain or for the one built up of the largest primes, but unlike the breakthrough of
3403:
896:, that there are arithmetic progressions of primes of arbitrary length – there is no general result known on large Cunningham chains to date.
3317:
953:
3020:
2518:. As with Cunningham chains of the first kind, the bits left of the pattern shift left by one position with each successive prime.
36:
849:
Cunningham chains are now considered useful in cryptographic systems since "they provide two concurrent suitable settings for the
3654:
2871:
3735:
3013:
2612:
1633:
As of 2018, the longest known
Cunningham chain of either kind is of length 19, discovered by Jaroslaw Wroblewski in 2014.
3857:
3515:
2691:
2438:. In binary notation, the primes in a Cunningham chain of the second kind end with a pattern "0...01", where, for each
2119:
is odd—that is, the least significant digit is 1 in base 2–we know that the secondmost least significant digit of
1779:
3882:
3348:
2524:
542:
3790:
2685:
870:
854:
2385:
2975:
sequence A005602 (Smallest prime beginning a complete
Cunningham chain of length n (of the first kind))
2962:
Primecoin discoveries (primes.zone): online database of primecoin findings with list of records and visualization
804:
if it cannot be further extended, i.e., if the previous and the next terms in the chain are not prime numbers.
2281:
1897:
1848:
1675:
3423:
866:
635:
366:
2922:
4296:
3940:
3069:
893:
4277:
3867:
3520:
3428:
2330:
1953:
1724:
3847:
1642:
434:
103:
683:
3842:
3500:
3950:
3887:
3877:
3862:
3495:
3353:
850:
3274:
2994:-- the first term of the lowest complete Cunningham chains of the second kind with length
2956:
2787:
3919:
3894:
3872:
3852:
3475:
3447:
3140:
2821:
2461:
2155:
2122:
2062:
3829:
3819:
3814:
3598:
3465:
3368:
2977:-- the first term of the lowest complete Cunningham chains of the first kind of length
2934:
2278:
A similar result holds for
Cunningham chains of the second kind. From the observation that
2188:
1383:
106680560818292299253267832484567360951928953599522278361651385665522443588804123392×61# − 1
25:
2760:
2494:
2095:
2035:
2008:
1648:
1404:
38249410745534076442242419351233801191635692835712219264661912943040353398995076864×47# + 1
3530:
3490:
3373:
3338:
3302:
3257:
3110:
3098:
1449:
5819411283298069803200936040662511327268486153212216998535044251830806354124236416×47# + 1
1672:
be the first prime of a
Cunningham chain of the first kind. The first prime is odd, thus
1342:
288320466650346626888267818984974462085357412586437032687304004479168536445314040×83# − 1
1301:
73853903764168979088206401473739410396455001112581722569026969860983656346568919×151# − 1
3935:
3909:
3806:
3674:
3525:
3485:
3470:
3342:
3233:
3198:
3153:
3078:
3060:
2441:
1428:
4631673892190914134588763508558377441004250662630975370524984655678678526944768×47# − 1
414:
2939:
1946:
The above property can be informally observed by considering the primes of a chain in
4290:
3945:
3710:
3574:
3547:
3383:
3248:
3186:
3177:
3162:
3125:
3051:
2866:
828:
89, 179, 359, 719, 1439, 2879 (The next number would be 5759, but that is not prime.)
33:
4266:
4261:
4256:
4251:
4246:
4241:
4236:
4231:
4226:
4221:
4216:
4211:
4206:
4201:
4196:
4191:
4186:
4181:
4176:
4171:
4166:
4161:
4156:
4151:
4146:
4141:
4136:
4131:
4126:
4121:
4116:
4111:
4106:
4101:
4096:
3899:
3622:
3505:
3388:
3378:
3363:
3358:
3322:
3036:
2906:
885:
29:
1626:
2 × 3 × 5 × 7 × ... ×
4091:
4086:
4081:
4076:
4071:
4066:
4061:
4056:
4051:
4046:
4041:
4036:
4031:
4026:
4021:
4016:
4011:
4006:
4001:
3996:
3991:
3837:
3510:
3418:
3413:
3393:
3307:
3210:
3086:
889:
732:
Cunningham chains are also sometimes generalized to sequences of prime numbers (
17:
3914:
3730:
3638:
3558:
3408:
3312:
107:
3955:
3904:
3785:
2860:
1623:
975:
816:
2, 5, 11, 23, 47 (The next number would be 95, but that is not prime.)
881:. There are, however, no known direct methods of generating such chains.
832:
Examples of complete
Cunningham chains of the second kind include these:
2992:
sequence A005603 (Chains of length n of nearly doubled primes)
812:
Examples of complete
Cunningham chains of the first kind include these:
3457:
3005:
783:
780:
1947:
431:
is not part of the sequence and need not be a prime number), we have
3452:
3438:
2891:
Algorithmic Number Theory: Third
International Symposium, ANTS-III
2217:
845:
31, 61 (The next number would be 121 = 11, but that is not prime.)
825:
41, 83, 167 (The next number would be 335, but that is not prime.)
2988:
2971:
842:
19, 37, 73 (The next number would be 145, but that is not prime.)
3009:
1459:
1438:
1414:
1393:
1352:
1311:
822:
29, 59 (The next number would be 119, but that is not prime.)
836:
2, 3, 5 (The next number would be 9, but that is not prime.)
2991:
2974:
839:
7, 13 (The next number would be 25, but that is not prime.)
819:
3, 7 (The next number would be 15, but that is not prime.)
3986:
3981:
3976:
3971:
2863:, which uses Cunningham chains as a proof-of-work system
2851:). Thus, no Cunningham chain can be of infinite length.
2059:
becomes the secondmost least significant digit of
102:. (Hence each term of such a chain except the last is a
2961:
877:
there are infinitely many
Cunningham chains of length
2824:
2790:
2763:
2694:
2615:
2527:
2497:
2464:
2444:
2388:
2333:
2284:
2191:
2158:
2125:
2098:
2065:
2038:
2011:
1956:
1900:
1851:
1782:
1727:
1678:
1651:
686:
638:
545:
437:
417:
369:
122:
2677:{\displaystyle p_{i}\equiv 2^{i-1}-1{\pmod {p_{1}}}}
3964:
3928:
3828:
3805:
3779:
3546:
3539:
3437:
3331:
3295:
3044:
2843:
2810:
2776:
2750:{\displaystyle 2^{p_{1}-1}\equiv 1{\pmod {p_{1}}}}
2749:
2676:
2601:
2510:
2483:
2450:
2430:
2374:
2319:
2207:
2177:
2144:
2111:
2084:
2051:
2024:
1997:
1935:
1886:
1838:{\displaystyle p_{i}\equiv 2^{i}-1{\pmod {2^{i}}}}
1837:
1768:
1713:
1664:
873:, both widely believed to be true, that for every
721:
672:
621:
472:
423:
403:
352:
3668: = 0, 1, 2, 3, ...
2602:{\displaystyle p_{i}=2^{i-1}p_{1}+(2^{i-1}-1)\,}
2152:is also 1. And, finally, we can see that
853:... can be implemented in any field where the
622:{\displaystyle p_{i}=2^{i-1}p_{1}-(2^{i-1}-1).}
1721:. Since each successive prime in the chain is
3021:
2005:in base 2, we see that, by multiplying
8:
2431:{\displaystyle p_{i}\equiv 1{\pmod {2^{i}}}}
2032:by 2, the least significant digit of
3543:
3028:
3014:
3006:
898:
2938:
2901:
2899:
2835:
2823:
2800:
2795:
2789:
2768:
2762:
2737:
2724:
2704:
2699:
2693:
2664:
2651:
2633:
2620:
2614:
2598:
2577:
2561:
2545:
2532:
2526:
2502:
2496:
2491:is one more than the number of zeros for
2469:
2463:
2458:, the number of zeros in the pattern for
2443:
2418:
2405:
2393:
2387:
2360:
2338:
2332:
2301:
2289:
2283:
2199:
2190:
2163:
2157:
2130:
2124:
2103:
2097:
2070:
2064:
2043:
2037:
2016:
2010:
1983:
1961:
1955:
1917:
1905:
1899:
1868:
1856:
1850:
1825:
1812:
1800:
1787:
1781:
1754:
1732:
1726:
1695:
1683:
1677:
1656:
1650:
900:Largest known Cunningham chain of length
704:
691:
685:
652:
645:
637:
595:
579:
563:
550:
544:
455:
442:
436:
416:
383:
376:
368:
322:
306:
290:
273:
257:
237:
217:
194:
174:
151:
131:
123:
121:
2320:{\displaystyle p_{1}\equiv 1{\pmod {2}}}
2185:will be odd due to the addition of 1 to
1936:{\displaystyle p_{3}\equiv 7{\pmod {8}}}
1887:{\displaystyle p_{2}\equiv 3{\pmod {4}}}
1714:{\displaystyle p_{1}\equiv 1{\pmod {2}}}
2882:
2923:"Long chains of nearly doubled primes"
673:{\displaystyle a={\frac {p_{1}-1}{2}}}
404:{\displaystyle a={\frac {p_{1}+1}{2}}}
106:, and each term except the first is a
1473:14354792166345299956567113728×43# - 1
525: − 1 for all 1 ≤
94: + 1 for all 1 ≤
7:
2957:The Prime Glossary: Cunningham chain
1363:906644189971753846618980352×233# + 1
536:It follows that the general term is
32:. Cunningham chains are named after
2732:
2659:
2413:
2309:
1925:
1876:
1820:
1703:
484:Cunningham chain of the second kind
1527:2×1540797425367761006138858881 − 1
1281:2044300700000658875613184×311# + 1
1264:3696772637099483023015936×311# − 1
1051:Michael Angel & Dirk Augustin
1031:Michael Angel & Dirk Augustin
793:; the resulting chains are called
53:Cunningham chain of the first kind
14:
2940:10.1090/S0025-5718-1989-0979939-8
2905:Norman Luhn & Dirk Augustin,
2269:100001101101000000010011110111111
1322:341841671431409652891648×311# + 1
1244:173129832252242394185728×401# + 1
1133:52992297065385779421184×1531# + 1
994:Ryan Propper & Serge Batalov
3404:Supersingular (moonshine theory)
2893:. New York: Springer (1998): 290
2872:Primes in arithmetic progression
2375:{\displaystyle p_{i+1}=2p_{i}-1}
2261:10000110110100000001001111011111
1998:{\displaystyle p_{i+1}=2p_{i}+1}
1769:{\displaystyle p_{i+1}=2p_{i}+1}
1637:Congruences of Cunningham chains
1190:89628063633698570895360×593# − 1
490:is a sequence of prime numbers (
59:is a sequence of prime numbers (
2725:
2652:
2406:
2302:
2253:1000011011010000000100111101111
1918:
1869:
1813:
1696:
1096:181439827616655015936×4673# + 1
861:Largest known Cunningham chains
473:{\displaystyle p_{i}=2^{i}a-1.}
41:chains of nearly doubled primes
3399:Supersingular (elliptic curve)
2998:, for 1 ≤
2981:, for 1 ≤
2967:PrimeLinks++: Cunningham chain
2743:
2726:
2670:
2653:
2595:
2570:
2424:
2407:
2313:
2303:
2245:100001101101000000010011110111
1929:
1919:
1880:
1870:
1831:
1814:
1707:
1697:
1170:25802590081726373888×1033# + 1
1079:31017701152691334912×4091# − 1
722:{\displaystyle p_{i}=2^{i}a+1}
613:
588:
340:
315:
1:
3180:2 ± 2 ± 1
2237:10000110110100000001001111011
800:A Cunningham chain is called
795:generalized Cunningham chains
2921:Löh, Günter (October 1989).
2229:1000011011010000000100111101
1564:1540797425367761006138858881
1490:67040002730422542592×53# + 1
1207:2373007846680317952×761# + 1
2267:
2259:
2251:
2243:
2235:
2227:
1606:
1586:
1584:658189097608811942204322721
1566:
1549:
1529:
1512:
1492:
1475:
1451:
1430:
1406:
1385:
1365:
1344:
1324:
1303:
1283:
1266:
1246:
1229:
1227:553374939996823808×593# − 1
1209:
1192:
1172:
1155:
1153:82466536397303904×1171# − 1
1135:
1118:
1098:
1081:
1061:
1044:
1024:
1007:
987:
968:
945:
4313:
2927:Mathematics of Computation
2911:. Retrieved on 2018-06-08.
1613:Chermoni & Wroblewski
1604:79910197721667870187016101
1593:Chermoni & Wroblewski
1573:Chermoni & Wroblewski
1536:Chermoni & Wroblewski
855:discrete logarithm problem
4275:
2811:{\displaystyle p_{p_{1}}}
1540:
1503:
1466:
1421:
1376:
1335:
1294:
1257:
1220:
1183:
1146:
1109:
1072:
1035:
998:
959:
3786:Mega (1,000,000+ digits)
3655:Arithmetic progression (
2908:Cunningham Chain records
2844:{\displaystyle i=p_{1}}
2686:Fermat's little theorem
2484:{\displaystyle p_{i+1}}
2178:{\displaystyle p_{i+1}}
2145:{\displaystyle p_{i+1}}
2085:{\displaystyle p_{i+1}}
1510:91304653283578934559359
1116:2799873605326×2371# - 1
871:Schinzel's hypothesis H
39:. They are also called
3941:Industrial-grade prime
3318:Newman–Shanks–Williams
2845:
2812:
2778:
2751:
2678:
2603:
2512:
2485:
2452:
2432:
2376:
2321:
2209:
2208:{\displaystyle 2p_{i}}
2179:
2146:
2113:
2086:
2053:
2026:
1999:
1937:
1888:
1839:
1770:
1715:
1666:
1547:2759832934171386593519
904:(as of 17 March 2023)
771:for all 1 ≤
723:
674:
623:
474:
425:
405:
354:
4278:List of prime numbers
3736:Sophie Germain/Safe (
2846:
2813:
2779:
2777:{\displaystyle p_{1}}
2752:
2679:
2604:
2513:
2511:{\displaystyle p_{i}}
2486:
2453:
2433:
2377:
2322:
2210:
2180:
2147:
2114:
2112:{\displaystyle p_{i}}
2087:
2054:
2052:{\displaystyle p_{i}}
2027:
2025:{\displaystyle p_{i}}
2000:
1938:
1889:
1840:
1771:
1716:
1667:
1665:{\displaystyle p_{1}}
1059:49325406476×9811# + 1
1042:13720852541×7877# − 1
724:
675:
624:
475:
426:
406:
355:
3460:(10 − 1)/9
2822:
2788:
2761:
2692:
2613:
2525:
2495:
2462:
2442:
2386:
2331:
2282:
2189:
2156:
2123:
2096:
2063:
2036:
2009:
1954:
1898:
1849:
1780:
1725:
1676:
1649:
1556:Jaroslaw Wroblewski
1519:Jaroslaw Wroblewski
867:Dickson's conjecture
851:ElGamal cryptosystem
684:
636:
543:
435:
415:
367:
120:
104:Sophie Germain prime
3769: ± 7, ...
3296:By integer sequence
3081:(2 + 1)/3
2521:Similarly, because
1005:1128330746865×2 − 1
966:2618163402417×2 − 1
905:
37:A. J. C. Cunningham
3951:Formula for primes
3584: + 2 or
3516:Smarandache–Wellin
2841:
2808:
2774:
2747:
2674:
2599:
2508:
2481:
2448:
2428:
2372:
2317:
2205:
2175:
2142:
2109:
2082:
2049:
2022:
1995:
1933:
1884:
1835:
1766:
1711:
1662:
1022:742478255901×2 + 1
985:213778324725×2 + 1
899:
719:
670:
619:
470:
421:
401:
350:
348:
4284:
4283:
3895:Carmichael number
3830:Composite numbers
3765: ± 3, 8
3761: ± 1, 4
3724: ± 1, …
3720: ± 1, 4
3716: ± 1, 2
3706:
3705:
3251:3·2 − 1
3156:2·3 + 1
3070:Double Mersenne (
2451:{\displaystyle i}
2327:and the relation
2276:
2275:
1617:
1616:
952:Patrick Laroche,
894:Green–Tao theorem
668:
424:{\displaystyle a}
399:
261:
4304:
3815:Eisenstein prime
3770:
3746:
3725:
3697:
3669:
3649:
3633:
3617:
3612: + 6,
3608: + 2,
3593:
3588: + 4,
3569:
3544:
3461:
3424:Highly cototient
3286:
3285:
3279:
3269:
3252:
3243:
3228:
3205:
3204:·2 − 1
3193:
3192:·2 + 1
3181:
3172:
3157:
3148:
3135:
3120:
3105:
3093:
3092:·2 + 1
3082:
3073:
3064:
3055:
3030:
3023:
3016:
3007:
2990:
2973:
2945:
2944:
2942:
2933:(188): 751–759.
2918:
2912:
2903:
2894:
2887:
2850:
2848:
2847:
2842:
2840:
2839:
2817:
2815:
2814:
2809:
2807:
2806:
2805:
2804:
2783:
2781:
2780:
2775:
2773:
2772:
2756:
2754:
2753:
2748:
2746:
2742:
2741:
2717:
2716:
2709:
2708:
2683:
2681:
2680:
2675:
2673:
2669:
2668:
2644:
2643:
2625:
2624:
2609:it follows that
2608:
2606:
2605:
2600:
2588:
2587:
2566:
2565:
2556:
2555:
2537:
2536:
2517:
2515:
2514:
2509:
2507:
2506:
2490:
2488:
2487:
2482:
2480:
2479:
2457:
2455:
2454:
2449:
2437:
2435:
2434:
2429:
2427:
2423:
2422:
2398:
2397:
2382:it follows that
2381:
2379:
2378:
2373:
2365:
2364:
2349:
2348:
2326:
2324:
2323:
2318:
2316:
2294:
2293:
2218:
2214:
2212:
2211:
2206:
2204:
2203:
2184:
2182:
2181:
2176:
2174:
2173:
2151:
2149:
2148:
2143:
2141:
2140:
2118:
2116:
2115:
2110:
2108:
2107:
2091:
2089:
2088:
2083:
2081:
2080:
2058:
2056:
2055:
2050:
2048:
2047:
2031:
2029:
2028:
2023:
2021:
2020:
2004:
2002:
2001:
1996:
1988:
1987:
1972:
1971:
1943:, and so forth.
1942:
1940:
1939:
1934:
1932:
1910:
1909:
1893:
1891:
1890:
1885:
1883:
1861:
1860:
1844:
1842:
1841:
1836:
1834:
1830:
1829:
1805:
1804:
1792:
1791:
1776:it follows that
1775:
1773:
1772:
1767:
1759:
1758:
1743:
1742:
1720:
1718:
1717:
1712:
1710:
1688:
1687:
1671:
1669:
1668:
1663:
1661:
1660:
1499:Andrey Balyakin
1482:Andrey Balyakin
1372:Andrey Balyakin
1331:Andrey Balyakin
1290:Andrey Balyakin
1273:Andrey Balyakin
1253:Andrey Balyakin
1236:Andrey Balyakin
1216:Andrey Balyakin
1199:Andrey Balyakin
1179:Andrey Balyakin
1162:Andrey Balyakin
1142:Andrey Balyakin
1105:Andrey Balyakin
1088:Andrey Balyakin
1014:Michael Paridon
923:(starting prime)
906:
869:and the broader
865:It follows from
728:
726:
725:
720:
709:
708:
696:
695:
679:
677:
676:
671:
669:
664:
657:
656:
646:
632:Now, by setting
628:
626:
625:
620:
606:
605:
584:
583:
574:
573:
555:
554:
529: <
479:
477:
476:
471:
460:
459:
447:
446:
430:
428:
427:
422:
410:
408:
407:
402:
400:
395:
388:
387:
377:
359:
357:
356:
351:
349:
333:
332:
311:
310:
301:
300:
278:
277:
259:
258:
255:
242:
241:
222:
221:
199:
198:
179:
178:
156:
155:
136:
135:
113:It follows that
98: <
22:Cunningham chain
4312:
4311:
4307:
4306:
4305:
4303:
4302:
4301:
4287:
4286:
4285:
4280:
4271:
3965:First 60 primes
3960:
3924:
3824:
3807:Complex numbers
3801:
3775:
3753:
3737:
3712:
3711:Bi-twin chain (
3702:
3676:
3656:
3640:
3624:
3600:
3576:
3560:
3535:
3521:Strobogrammatic
3459:
3433:
3327:
3291:
3283:
3277:
3276:
3259:
3250:
3235:
3212:
3200:
3188:
3179:
3164:
3155:
3142:
3134:# + 1
3132:
3127:
3119:# ± 1
3117:
3112:
3104:! ± 1
3100:
3088:
3080:
3072:2 − 1
3071:
3063:2 − 1
3062:
3054:2 + 1
3053:
3040:
3034:
3002: ≤ 15
2985: ≤ 14
2953:
2948:
2920:
2919:
2915:
2904:
2897:
2888:
2884:
2880:
2857:
2831:
2820:
2819:
2796:
2791:
2786:
2785:
2764:
2759:
2758:
2733:
2700:
2695:
2690:
2689:
2660:
2629:
2616:
2611:
2610:
2573:
2557:
2541:
2528:
2523:
2522:
2498:
2493:
2492:
2465:
2460:
2459:
2440:
2439:
2414:
2389:
2384:
2383:
2356:
2334:
2329:
2328:
2285:
2280:
2279:
2195:
2187:
2186:
2159:
2154:
2153:
2126:
2121:
2120:
2099:
2094:
2093:
2066:
2061:
2060:
2039:
2034:
2033:
2012:
2007:
2006:
1979:
1957:
1952:
1951:
1901:
1896:
1895:
1852:
1847:
1846:
1821:
1796:
1783:
1778:
1777:
1750:
1728:
1723:
1722:
1679:
1674:
1673:
1652:
1647:
1646:
1639:
922:
863:
857:is difficult."
810:
766:
757:
747:
738:
700:
687:
682:
681:
648:
647:
634:
633:
591:
575:
559:
546:
541:
540:
524:
515:
505:
496:
451:
438:
433:
432:
413:
412:
379:
378:
365:
364:
363:or, by setting
347:
346:
318:
302:
286:
279:
269:
266:
265:
253:
252:
233:
223:
213:
210:
209:
190:
180:
170:
167:
166:
147:
137:
127:
118:
117:
93:
84:
74:
65:
49:
12:
11:
5:
4310:
4308:
4300:
4299:
4289:
4288:
4282:
4281:
4276:
4273:
4272:
4270:
4269:
4264:
4259:
4254:
4249:
4244:
4239:
4234:
4229:
4224:
4219:
4214:
4209:
4204:
4199:
4194:
4189:
4184:
4179:
4174:
4169:
4164:
4159:
4154:
4149:
4144:
4139:
4134:
4129:
4124:
4119:
4114:
4109:
4104:
4099:
4094:
4089:
4084:
4079:
4074:
4069:
4064:
4059:
4054:
4049:
4044:
4039:
4034:
4029:
4024:
4019:
4014:
4009:
4004:
3999:
3994:
3989:
3984:
3979:
3974:
3968:
3966:
3962:
3961:
3959:
3958:
3953:
3948:
3943:
3938:
3936:Probable prime
3932:
3930:
3929:Related topics
3926:
3925:
3923:
3922:
3917:
3912:
3910:Sphenic number
3907:
3902:
3897:
3892:
3891:
3890:
3885:
3880:
3875:
3870:
3865:
3860:
3855:
3850:
3845:
3834:
3832:
3826:
3825:
3823:
3822:
3820:Gaussian prime
3817:
3811:
3809:
3803:
3802:
3800:
3799:
3798:
3788:
3783:
3781:
3777:
3776:
3774:
3773:
3749:
3745: + 1
3733:
3728:
3707:
3704:
3703:
3701:
3700:
3672:
3652:
3648: + 6
3636:
3632: + 4
3620:
3616: + 8
3596:
3592: + 6
3572:
3568: + 2
3555:
3553:
3541:
3537:
3536:
3534:
3533:
3528:
3523:
3518:
3513:
3508:
3503:
3498:
3493:
3488:
3483:
3478:
3473:
3468:
3463:
3455:
3450:
3444:
3442:
3435:
3434:
3432:
3431:
3426:
3421:
3416:
3411:
3406:
3401:
3396:
3391:
3386:
3381:
3376:
3371:
3366:
3361:
3356:
3351:
3346:
3335:
3333:
3329:
3328:
3326:
3325:
3320:
3315:
3310:
3305:
3299:
3297:
3293:
3292:
3290:
3289:
3272:
3268: − 1
3255:
3246:
3231:
3208:
3196:
3184:
3175:
3160:
3151:
3147: + 1
3138:
3130:
3123:
3115:
3108:
3096:
3084:
3076:
3067:
3058:
3048:
3046:
3042:
3041:
3035:
3033:
3032:
3025:
3018:
3010:
3004:
3003:
2986:
2969:
2964:
2959:
2952:
2951:External links
2949:
2947:
2946:
2913:
2895:
2881:
2879:
2876:
2875:
2874:
2869:
2864:
2856:
2853:
2838:
2834:
2830:
2827:
2803:
2799:
2794:
2771:
2767:
2745:
2740:
2736:
2731:
2728:
2723:
2720:
2715:
2712:
2707:
2703:
2698:
2672:
2667:
2663:
2658:
2655:
2650:
2647:
2642:
2639:
2636:
2632:
2628:
2623:
2619:
2597:
2594:
2591:
2586:
2583:
2580:
2576:
2572:
2569:
2564:
2560:
2554:
2551:
2548:
2544:
2540:
2535:
2531:
2505:
2501:
2478:
2475:
2472:
2468:
2447:
2426:
2421:
2417:
2412:
2409:
2404:
2401:
2396:
2392:
2371:
2368:
2363:
2359:
2355:
2352:
2347:
2344:
2341:
2337:
2315:
2312:
2308:
2305:
2300:
2297:
2292:
2288:
2274:
2273:
2270:
2266:
2265:
2262:
2258:
2257:
2254:
2250:
2249:
2246:
2242:
2241:
2238:
2234:
2233:
2230:
2226:
2225:
2222:
2202:
2198:
2194:
2172:
2169:
2166:
2162:
2139:
2136:
2133:
2129:
2106:
2102:
2079:
2076:
2073:
2069:
2046:
2042:
2019:
2015:
1994:
1991:
1986:
1982:
1978:
1975:
1970:
1967:
1964:
1960:
1931:
1928:
1924:
1921:
1916:
1913:
1908:
1904:
1882:
1879:
1875:
1872:
1867:
1864:
1859:
1855:
1833:
1828:
1824:
1819:
1816:
1811:
1808:
1803:
1799:
1795:
1790:
1786:
1765:
1762:
1757:
1753:
1749:
1746:
1741:
1738:
1735:
1731:
1709:
1706:
1702:
1699:
1694:
1691:
1686:
1682:
1659:
1655:
1638:
1635:
1622:# denotes the
1615:
1614:
1611:
1608:
1605:
1602:
1599:
1595:
1594:
1591:
1588:
1585:
1582:
1579:
1575:
1574:
1571:
1568:
1565:
1562:
1558:
1557:
1554:
1551:
1548:
1545:
1542:
1538:
1537:
1534:
1531:
1528:
1525:
1521:
1520:
1517:
1514:
1511:
1508:
1505:
1501:
1500:
1497:
1494:
1491:
1488:
1484:
1483:
1480:
1477:
1474:
1471:
1468:
1464:
1463:
1456:
1453:
1450:
1447:
1443:
1442:
1435:
1432:
1429:
1426:
1423:
1419:
1418:
1411:
1408:
1405:
1402:
1398:
1397:
1390:
1387:
1384:
1381:
1378:
1374:
1373:
1370:
1367:
1364:
1361:
1357:
1356:
1349:
1346:
1343:
1340:
1337:
1333:
1332:
1329:
1326:
1323:
1320:
1316:
1315:
1308:
1305:
1302:
1299:
1296:
1292:
1291:
1288:
1285:
1282:
1279:
1275:
1274:
1271:
1268:
1265:
1262:
1259:
1255:
1254:
1251:
1248:
1245:
1242:
1238:
1237:
1234:
1231:
1228:
1225:
1222:
1218:
1217:
1214:
1211:
1208:
1205:
1201:
1200:
1197:
1194:
1191:
1188:
1185:
1181:
1180:
1177:
1174:
1171:
1168:
1164:
1163:
1160:
1157:
1154:
1151:
1148:
1144:
1143:
1140:
1137:
1134:
1131:
1127:
1126:
1125:Serge Batalov
1123:
1120:
1117:
1114:
1111:
1107:
1106:
1103:
1100:
1097:
1094:
1090:
1089:
1086:
1083:
1080:
1077:
1074:
1070:
1069:
1066:
1063:
1060:
1057:
1053:
1052:
1049:
1046:
1043:
1040:
1037:
1033:
1032:
1029:
1026:
1023:
1020:
1016:
1015:
1012:
1009:
1006:
1003:
1000:
996:
995:
992:
989:
986:
983:
979:
978:
973:
970:
967:
964:
961:
957:
956:
950:
947:
944:
941:
938:
934:
933:
930:
927:
924:
920:
915:
912:
862:
859:
847:
846:
843:
840:
837:
830:
829:
826:
823:
820:
817:
809:
806:
762:
752:
743:
736:
718:
715:
712:
707:
703:
699:
694:
690:
667:
663:
660:
655:
651:
644:
641:
630:
629:
618:
615:
612:
609:
604:
601:
598:
594:
590:
587:
582:
578:
572:
569:
566:
562:
558:
553:
549:
520:
516: = 2
510:
501:
494:
469:
466:
463:
458:
454:
450:
445:
441:
420:
398:
394:
391:
386:
382:
375:
372:
361:
360:
345:
342:
339:
336:
331:
328:
325:
321:
317:
314:
309:
305:
299:
296:
293:
289:
285:
282:
280:
276:
272:
268:
267:
264:
256:
254:
251:
248:
245:
240:
236:
232:
229:
226:
224:
220:
216:
212:
211:
208:
205:
202:
197:
193:
189:
186:
183:
181:
177:
173:
169:
168:
165:
162:
159:
154:
150:
146:
143:
140:
138:
134:
130:
126:
125:
89:
85: = 2
79:
70:
63:
48:
45:
13:
10:
9:
6:
4:
3:
2:
4309:
4298:
4297:Prime numbers
4295:
4294:
4292:
4279:
4274:
4268:
4265:
4263:
4260:
4258:
4255:
4253:
4250:
4248:
4245:
4243:
4240:
4238:
4235:
4233:
4230:
4228:
4225:
4223:
4220:
4218:
4215:
4213:
4210:
4208:
4205:
4203:
4200:
4198:
4195:
4193:
4190:
4188:
4185:
4183:
4180:
4178:
4175:
4173:
4170:
4168:
4165:
4163:
4160:
4158:
4155:
4153:
4150:
4148:
4145:
4143:
4140:
4138:
4135:
4133:
4130:
4128:
4125:
4123:
4120:
4118:
4115:
4113:
4110:
4108:
4105:
4103:
4100:
4098:
4095:
4093:
4090:
4088:
4085:
4083:
4080:
4078:
4075:
4073:
4070:
4068:
4065:
4063:
4060:
4058:
4055:
4053:
4050:
4048:
4045:
4043:
4040:
4038:
4035:
4033:
4030:
4028:
4025:
4023:
4020:
4018:
4015:
4013:
4010:
4008:
4005:
4003:
4000:
3998:
3995:
3993:
3990:
3988:
3985:
3983:
3980:
3978:
3975:
3973:
3970:
3969:
3967:
3963:
3957:
3954:
3952:
3949:
3947:
3946:Illegal prime
3944:
3942:
3939:
3937:
3934:
3933:
3931:
3927:
3921:
3918:
3916:
3913:
3911:
3908:
3906:
3903:
3901:
3898:
3896:
3893:
3889:
3886:
3884:
3881:
3879:
3876:
3874:
3871:
3869:
3866:
3864:
3861:
3859:
3856:
3854:
3851:
3849:
3846:
3844:
3841:
3840:
3839:
3836:
3835:
3833:
3831:
3827:
3821:
3818:
3816:
3813:
3812:
3810:
3808:
3804:
3797:
3794:
3793:
3792:
3791:Largest known
3789:
3787:
3784:
3782:
3778:
3772:
3768:
3764:
3760:
3756:
3750:
3748:
3744:
3740:
3734:
3732:
3729:
3727:
3723:
3719:
3715:
3709:
3708:
3699:
3696:
3693: +
3692:
3688:
3684:
3681: −
3680:
3673:
3671:
3667:
3663:
3660: +
3659:
3653:
3651:
3647:
3643:
3637:
3635:
3631:
3627:
3621:
3619:
3615:
3611:
3607:
3603:
3597:
3595:
3591:
3587:
3583:
3579:
3573:
3571:
3567:
3563:
3557:
3556:
3554:
3552:
3550:
3545:
3542:
3538:
3532:
3529:
3527:
3524:
3522:
3519:
3517:
3514:
3512:
3509:
3507:
3504:
3502:
3499:
3497:
3494:
3492:
3489:
3487:
3484:
3482:
3479:
3477:
3474:
3472:
3469:
3467:
3464:
3462:
3456:
3454:
3451:
3449:
3446:
3445:
3443:
3440:
3436:
3430:
3427:
3425:
3422:
3420:
3417:
3415:
3412:
3410:
3407:
3405:
3402:
3400:
3397:
3395:
3392:
3390:
3387:
3385:
3382:
3380:
3377:
3375:
3372:
3370:
3367:
3365:
3362:
3360:
3357:
3355:
3352:
3350:
3347:
3344:
3340:
3337:
3336:
3334:
3330:
3324:
3321:
3319:
3316:
3314:
3311:
3309:
3306:
3304:
3301:
3300:
3298:
3294:
3288:
3282:
3273:
3271:
3267:
3263:
3256:
3254:
3247:
3245:
3242:
3239: +
3238:
3232:
3230:
3227:
3224: −
3223:
3219:
3216: −
3215:
3209:
3207:
3203:
3197:
3195:
3191:
3185:
3183:
3176:
3174:
3171:
3168: +
3167:
3161:
3159:
3152:
3150:
3146:
3141:Pythagorean (
3139:
3137:
3133:
3124:
3122:
3118:
3109:
3107:
3103:
3097:
3095:
3091:
3085:
3083:
3077:
3075:
3068:
3066:
3059:
3057:
3050:
3049:
3047:
3043:
3038:
3031:
3026:
3024:
3019:
3017:
3012:
3011:
3008:
3001:
2997:
2993:
2987:
2984:
2980:
2976:
2970:
2968:
2965:
2963:
2960:
2958:
2955:
2954:
2950:
2941:
2936:
2932:
2928:
2924:
2917:
2914:
2910:
2909:
2902:
2900:
2896:
2892:
2886:
2883:
2877:
2873:
2870:
2868:
2867:Bi-twin chain
2865:
2862:
2859:
2858:
2854:
2852:
2836:
2832:
2828:
2825:
2801:
2797:
2792:
2769:
2765:
2738:
2734:
2729:
2721:
2718:
2713:
2710:
2705:
2701:
2696:
2687:
2665:
2661:
2656:
2648:
2645:
2640:
2637:
2634:
2630:
2626:
2621:
2617:
2592:
2589:
2584:
2581:
2578:
2574:
2567:
2562:
2558:
2552:
2549:
2546:
2542:
2538:
2533:
2529:
2519:
2503:
2499:
2476:
2473:
2470:
2466:
2445:
2419:
2415:
2410:
2402:
2399:
2394:
2390:
2369:
2366:
2361:
2357:
2353:
2350:
2345:
2342:
2339:
2335:
2310:
2306:
2298:
2295:
2290:
2286:
2271:
2268:
2263:
2260:
2255:
2252:
2247:
2244:
2239:
2236:
2231:
2228:
2223:
2220:
2219:
2216:
2200:
2196:
2192:
2170:
2167:
2164:
2160:
2137:
2134:
2131:
2127:
2104:
2100:
2077:
2074:
2071:
2067:
2044:
2040:
2017:
2013:
1992:
1989:
1984:
1980:
1976:
1973:
1968:
1965:
1962:
1958:
1949:
1944:
1926:
1922:
1914:
1911:
1906:
1902:
1877:
1873:
1865:
1862:
1857:
1853:
1826:
1822:
1817:
1809:
1806:
1801:
1797:
1793:
1788:
1784:
1763:
1760:
1755:
1751:
1747:
1744:
1739:
1736:
1733:
1729:
1704:
1700:
1692:
1689:
1684:
1680:
1657:
1653:
1644:
1636:
1634:
1631:
1629:
1625:
1621:
1612:
1609:
1603:
1600:
1597:
1596:
1592:
1589:
1583:
1580:
1577:
1576:
1572:
1569:
1563:
1560:
1559:
1555:
1552:
1546:
1543:
1539:
1535:
1532:
1526:
1523:
1522:
1518:
1515:
1509:
1506:
1502:
1498:
1495:
1489:
1486:
1485:
1481:
1478:
1472:
1469:
1465:
1461:
1457:
1454:
1448:
1445:
1444:
1440:
1439:block 2659167
1436:
1433:
1427:
1424:
1420:
1416:
1412:
1409:
1403:
1400:
1399:
1395:
1391:
1388:
1382:
1379:
1375:
1371:
1368:
1362:
1359:
1358:
1354:
1350:
1347:
1341:
1338:
1334:
1330:
1327:
1321:
1318:
1317:
1313:
1309:
1306:
1300:
1297:
1293:
1289:
1286:
1280:
1277:
1276:
1272:
1269:
1263:
1260:
1256:
1252:
1249:
1243:
1240:
1239:
1235:
1232:
1226:
1223:
1219:
1215:
1212:
1206:
1203:
1202:
1198:
1195:
1189:
1186:
1182:
1178:
1175:
1169:
1166:
1165:
1161:
1158:
1152:
1149:
1145:
1141:
1138:
1132:
1129:
1128:
1124:
1121:
1115:
1112:
1108:
1104:
1101:
1095:
1092:
1091:
1087:
1084:
1078:
1075:
1071:
1068:Oscar Östlin
1067:
1064:
1058:
1055:
1054:
1050:
1047:
1041:
1038:
1034:
1030:
1027:
1021:
1018:
1017:
1013:
1010:
1004:
1001:
997:
993:
990:
984:
981:
980:
977:
974:
971:
965:
962:
958:
955:
951:
948:
942:
939:
936:
935:
931:
928:
925:
919:
916:
913:
911:
908:
907:
903:
897:
895:
891:
887:
882:
880:
876:
872:
868:
860:
858:
856:
852:
844:
841:
838:
835:
834:
833:
827:
824:
821:
818:
815:
814:
813:
807:
805:
803:
798:
796:
792:
788:
785:
782:
778:
775: ≤
774:
770:
767: +
765:
761:
755:
751:
746:
742:
735:
730:
716:
713:
710:
705:
701:
697:
692:
688:
665:
661:
658:
653:
649:
642:
639:
616:
610:
607:
602:
599:
596:
592:
585:
580:
576:
570:
567:
564:
560:
556:
551:
547:
539:
538:
537:
534:
532:
528:
523:
519:
513:
509:
504:
500:
493:
489:
485:
482:Similarly, a
480:
467:
464:
461:
456:
452:
448:
443:
439:
418:
396:
392:
389:
384:
380:
373:
370:
343:
337:
334:
329:
326:
323:
319:
312:
307:
303:
297:
294:
291:
287:
283:
281:
274:
270:
262:
249:
246:
243:
238:
234:
230:
227:
225:
218:
214:
206:
203:
200:
195:
191:
187:
184:
182:
175:
171:
163:
160:
157:
152:
148:
144:
141:
139:
132:
128:
116:
115:
114:
111:
109:
105:
101:
97:
92:
88:
82:
78:
73:
69:
62:
58:
54:
46:
44:
42:
38:
35:
34:mathematician
31:
30:prime numbers
27:
24:is a certain
23:
19:
3900:Almost prime
3858:Euler–Jacobi
3766:
3762:
3758:
3754:
3752:Cunningham (
3751:
3742:
3738:
3721:
3717:
3713:
3694:
3690:
3686:
3682:
3678:
3677:consecutive
3665:
3661:
3657:
3645:
3641:
3629:
3625:
3613:
3609:
3605:
3601:
3599:Quadruplet (
3589:
3585:
3581:
3577:
3565:
3561:
3548:
3496:Full reptend
3354:Wolstenholme
3349:Wall–Sun–Sun
3280:
3265:
3261:
3240:
3236:
3225:
3221:
3217:
3213:
3201:
3189:
3169:
3165:
3144:
3128:
3113:
3101:
3089:
3037:Prime number
2999:
2995:
2982:
2978:
2930:
2926:
2916:
2907:
2890:
2889:Joe Buhler,
2885:
2520:
2277:
1945:
1640:
1632:
1627:
1619:
1618:
1460:block 547276
1415:block 539977
1394:block 368051
1353:block 558800
917:
909:
901:
886:Ben J. Green
883:
878:
874:
864:
848:
831:
811:
801:
799:
794:
790:
786:
776:
772:
768:
763:
759:
753:
749:
748:) such that
744:
740:
733:
731:
631:
535:
530:
526:
521:
517:
511:
507:
506:) such that
502:
498:
491:
487:
483:
481:
411:(the number
362:
112:
99:
95:
90:
86:
80:
76:
75:) such that
71:
67:
60:
56:
52:
50:
40:
21:
15:
3883:Somer–Lucas
3838:Pseudoprime
3476:Truncatable
3448:Palindromic
3332:By property
3111:Primorial (
3099:Factorial (
2818:(i.e. with
2272:4523567039
2264:2261783519
2256:1130891759
1458:Primecoin (
1437:Primecoin (
1413:Primecoin (
1392:Primecoin (
1351:Primecoin (
1312:block 95569
1310:Primecoin (
932:Discoverer
890:Terence Tao
18:mathematics
3920:Pernicious
3915:Interprime
3675:Balanced (
3466:Permutable
3441:-dependent
3258:Williams (
3154:Pierpont (
3079:Wagstaff
3061:Mersenne (
3045:By formula
2878:References
2684:. But, by
2248:565445879
2240:282722939
2232:141361469
2092:. Because
779:for fixed
680:, we have
486:of length
108:safe prime
55:of length
47:Definition
3956:Prime gap
3905:Semiprime
3868:Frobenius
3575:Triplet (
3374:Ramanujan
3369:Fortunate
3339:Wieferich
3303:Fibonacci
3234:Leyland (
3199:Woodall (
3178:Solinas (
3163:Quartan (
2861:Primecoin
2719:≡
2711:−
2646:−
2638:−
2627:≡
2590:−
2582:−
2550:−
2400:≡
2367:−
2296:≡
1912:≡
1863:≡
1807:−
1794:≡
1690:≡
1624:primorial
976:PrimeGrid
940:1st / 2nd
659:−
608:−
600:−
586:−
568:−
465:−
335:−
327:−
295:−
263:⋮
4291:Category
3848:Elliptic
3623:Cousin (
3540:Patterns
3531:Tetradic
3526:Dihedral
3491:Primeval
3486:Delicate
3471:Circular
3458:Repunit
3249:Thabit (
3187:Cullen (
3126:Euclid (
3052:Fermat (
2855:See also
2784:divides
2224:Decimal
1845:. Thus,
1641:Let the
946:24862048
808:Examples
802:complete
784:integers
26:sequence
3843:Catalan
3780:By size
3551:-tuples
3481:Minimal
3384:Regular
3275:Mills (
3211:Cuban (
3087:Proth (
3039:classes
781:coprime
758:=
739:, ...,
497:, ...,
66:, ...,
3888:Strong
3878:Perrin
3863:Fermat
3639:Sexy (
3559:Twin (
3501:Unique
3429:Unique
3389:Strong
3379:Pillai
3359:Wilson
3323:Perrin
2221:Binary
1948:base 2
1645:prime
988:169015
969:388342
926:Digits
892:– the
260:
3873:Lucas
3853:Euler
3506:Happy
3453:Emirp
3419:Higgs
3414:Super
3394:Stern
3364:Lucky
3308:Lucas
2757:, so
1025:12074
1008:20013
954:GIMPS
943:2 − 1
3796:list
3731:Chen
3511:Self
3439:Base
3409:Good
3343:pair
3313:Pell
3264:−1)·
2989:OEIS
2972:OEIS
1610:2014
1590:2014
1570:2014
1553:2008
1533:2014
1516:2008
1496:2016
1479:2016
1455:2014
1434:2018
1410:2014
1389:2014
1369:2015
1348:2014
1328:2016
1307:2013
1287:2016
1270:2016
1250:2015
1233:2016
1213:2016
1196:2015
1176:2015
1159:2016
1139:2015
1122:2015
1119:1016
1102:2016
1099:2018
1085:2016
1082:1765
1065:2019
1062:4234
1048:2016
1045:3384
1028:2016
1011:2020
991:2023
972:2016
949:2018
929:Year
914:Kind
888:and
789:and
20:, a
4267:281
4262:277
4257:271
4252:269
4247:263
4242:257
4237:251
4232:241
4227:239
4222:233
4217:229
4212:227
4207:223
4202:211
4197:199
4192:197
4187:193
4182:191
4177:181
4172:179
4167:173
4162:167
4157:163
4152:157
4147:151
4142:149
4137:139
4132:137
4127:131
4122:127
4117:113
4112:109
4107:107
4102:103
4097:101
3757:, 2
3741:, 2
3662:a·n
3220:)/(
2935:doi
2730:mod
2657:mod
2411:mod
2307:mod
1923:mod
1874:mod
1818:mod
1701:mod
1643:odd
1601:2nd
1581:2nd
1561:2nd
1544:1st
1524:2nd
1507:1st
1487:2nd
1470:1st
1452:100
1446:2nd
1425:1st
1407:101
1401:2nd
1386:107
1380:1st
1366:121
1360:2nd
1345:113
1339:1st
1325:149
1319:2nd
1304:140
1298:1st
1284:150
1278:2nd
1267:150
1261:1st
1247:187
1241:2nd
1230:260
1224:1st
1210:337
1204:2nd
1193:265
1187:1st
1173:453
1167:2nd
1156:509
1150:1st
1136:668
1130:2nd
1113:1st
1093:2nd
1076:1st
1056:2nd
1039:1st
1019:2nd
1002:1st
982:2nd
963:1st
110:).
43:.
28:of
16:In
4293::
4092:97
4087:89
4082:83
4077:79
4072:73
4067:71
4062:67
4057:61
4052:59
4047:53
4042:47
4037:43
4032:41
4027:37
4022:31
4017:29
4012:23
4007:19
4002:17
3997:13
3992:11
3689:,
3685:,
3664:,
3644:,
3628:,
3604:,
3580:,
3564:,
2931:53
2929:.
2925:.
2898:^
2688:,
1894:,
1630:.
1607:26
1598:19
1587:27
1578:18
1567:28
1550:22
1541:17
1530:28
1513:23
1504:16
1493:40
1476:45
1467:15
1462:)
1441:)
1431:97
1422:14
1417:)
1396:)
1377:13
1355:)
1336:12
1314:)
1295:11
1258:10
797:.
760:ap
756:+1
729:.
533:.
514:+1
468:1.
83:+1
51:A
3987:7
3982:5
3977:3
3972:2
3771:)
3767:p
3763:p
3759:p
3755:p
3747:)
3743:p
3739:p
3726:)
3722:n
3718:n
3714:n
3698:)
3695:n
3691:p
3687:p
3683:n
3679:p
3670:)
3666:n
3658:p
3650:)
3646:p
3642:p
3634:)
3630:p
3626:p
3618:)
3614:p
3610:p
3606:p
3602:p
3594:)
3590:p
3586:p
3582:p
3578:p
3570:)
3566:p
3562:p
3549:k
3345:)
3341:(
3287:)
3284:⌋
3281:A
3278:⌊
3270:)
3266:b
3262:b
3260:(
3253:)
3244:)
3241:y
3237:x
3229:)
3226:y
3222:x
3218:y
3214:x
3206:)
3202:n
3194:)
3190:n
3182:)
3173:)
3170:y
3166:x
3158:)
3149:)
3145:n
3143:4
3136:)
3131:n
3129:p
3121:)
3116:n
3114:p
3106:)
3102:n
3094:)
3090:k
3074:)
3065:)
3056:)
3029:e
3022:t
3015:v
3000:n
2996:n
2983:n
2979:n
2943:.
2937::
2837:1
2833:p
2829:=
2826:i
2802:1
2798:p
2793:p
2770:1
2766:p
2744:)
2739:1
2735:p
2727:(
2722:1
2714:1
2706:1
2702:p
2697:2
2671:)
2666:1
2662:p
2654:(
2649:1
2641:1
2635:i
2631:2
2622:i
2618:p
2596:)
2593:1
2585:1
2579:i
2575:2
2571:(
2568:+
2563:1
2559:p
2553:1
2547:i
2543:2
2539:=
2534:i
2530:p
2504:i
2500:p
2477:1
2474:+
2471:i
2467:p
2446:i
2425:)
2420:i
2416:2
2408:(
2403:1
2395:i
2391:p
2370:1
2362:i
2358:p
2354:2
2351:=
2346:1
2343:+
2340:i
2336:p
2314:)
2311:2
2304:(
2299:1
2291:1
2287:p
2201:i
2197:p
2193:2
2171:1
2168:+
2165:i
2161:p
2138:1
2135:+
2132:i
2128:p
2105:i
2101:p
2078:1
2075:+
2072:i
2068:p
2045:i
2041:p
2018:i
2014:p
1993:1
1990:+
1985:i
1981:p
1977:2
1974:=
1969:1
1966:+
1963:i
1959:p
1930:)
1927:8
1920:(
1915:7
1907:3
1903:p
1881:)
1878:4
1871:(
1866:3
1858:2
1854:p
1832:)
1827:i
1823:2
1815:(
1810:1
1802:i
1798:2
1789:i
1785:p
1764:1
1761:+
1756:i
1752:p
1748:2
1745:=
1740:1
1737:+
1734:i
1730:p
1708:)
1705:2
1698:(
1693:1
1685:1
1681:p
1658:1
1654:p
1628:q
1620:q
1221:9
1184:8
1147:7
1110:6
1073:5
1036:4
999:3
960:2
937:1
921:1
918:p
910:k
902:k
879:k
875:k
791:b
787:a
777:n
773:i
769:b
764:i
754:i
750:p
745:n
741:p
737:1
734:p
717:1
714:+
711:a
706:i
702:2
698:=
693:i
689:p
666:2
662:1
654:1
650:p
643:=
640:a
617:.
614:)
611:1
603:1
597:i
593:2
589:(
581:1
577:p
571:1
565:i
561:2
557:=
552:i
548:p
531:n
527:i
522:i
518:p
512:i
508:p
503:n
499:p
495:1
492:p
488:n
462:a
457:i
453:2
449:=
444:i
440:p
419:a
397:2
393:1
390:+
385:1
381:p
374:=
371:a
344:,
341:)
338:1
330:1
324:i
320:2
316:(
313:+
308:1
304:p
298:1
292:i
288:2
284:=
275:i
271:p
250:,
247:7
244:+
239:1
235:p
231:8
228:=
219:4
215:p
207:,
204:3
201:+
196:1
192:p
188:4
185:=
176:3
172:p
164:,
161:1
158:+
153:1
149:p
145:2
142:=
133:2
129:p
100:n
96:i
91:i
87:p
81:i
77:p
72:n
68:p
64:1
61:p
57:n
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