550:
2850:
477:
Suppose that both preceding Sierpiński problems had finally been solved, showing that 78557 is the smallest Sierpiński number and that 271129 is the smallest prime Sierpiński number. This still leaves unsolved the question of the
255:
78557, 271129, 271577, 322523, 327739, 482719, 575041, 603713, 903983, 934909, 965431, 1259779, 1290677, 1518781, 1624097, 1639459, 1777613, 2131043, 2131099, 2191531, 2510177, 2541601, 2576089, 2931767, 2931991, ... (sequence
187:
534:. These are called Brier numbers. The smallest five known examples are 3316923598096294713661, 10439679896374780276373, 11615103277955704975673, 12607110588854501953787, 17855036657007596110949, ... (
516:
952:
229:
82:
376:
that 78,557 was the smallest Sierpiński number. No smaller Sierpiński numbers have been discovered, and it is now believed that 78,557 is the smallest number.
453:
Sierpiński number, and there is an ongoing "Prime Sierpiński search" which tries to prove that 271129 is the first Sierpiński number which is also a prime.
296:
However, in 1995 A. S. Izotov showed that some fourth powers could be proved to be Sierpiński numbers without establishing a covering set for all values of
2889:
2894:
2874:
263:
945:
471:
435:
358:
721:
124:
1752:
938:
1747:
1762:
1742:
855:
713:
2455:
2035:
1757:
379:
To show that 78,557 really is the smallest Sierpiński number, one must show that all the odd numbers smaller than 78,557 are
2541:
1857:
899:
2207:
1526:
1319:
2242:
2212:
1887:
1877:
2383:
1797:
1531:
1511:
301:
2073:
2237:
2332:
1955:
1712:
1521:
1503:
1397:
1387:
1377:
2217:
2460:
2005:
1626:
1412:
1407:
1402:
1392:
1369:
1445:
664:
1702:
2571:
2536:
2322:
2232:
2106:
2081:
1990:
1980:
1592:
1574:
1494:
93:
2879:
2831:
2101:
1975:
1606:
1382:
1162:
1089:
489:
2086:
1940:
1867:
1022:
2795:
2435:
880:
549:
2884:
2728:
2622:
2586:
2327:
2050:
2030:
1847:
1516:
1304:
195:
48:
36:
2450:
2314:
2309:
2277:
2040:
2015:
2010:
1985:
1915:
1911:
1842:
1732:
1564:
1360:
1329:
518:. An ongoing search is trying to prove that 271129 is the second Sierpiński number, by testing all
2849:
2853:
2607:
2602:
2516:
2490:
2388:
2367:
2139:
2020:
1970:
1892:
1862:
1802:
1569:
1549:
1480:
1193:
895:
555:
1737:
605:
2747:
2692:
2546:
2521:
2495:
2272:
1950:
1945:
1872:
1852:
1837:
1559:
1541:
1460:
1450:
1435:
1213:
1198:
877:
851:
727:
717:
687:
578:
441:
In 1976, Nathan
Mendelsohn determined that the second provable Sierpiński number is the prime
115:
2783:
2576:
2162:
2134:
2124:
2116:
2000:
1965:
1960:
1927:
1621:
1584:
1475:
1470:
1465:
1455:
1427:
1314:
1266:
1261:
1218:
1157:
677:
85:
2759:
2648:
2581:
2507:
2430:
2404:
2222:
1935:
1792:
1727:
1697:
1687:
1682:
1348:
1256:
1203:
1047:
987:
847:
2764:
2632:
2617:
2481:
2445:
2420:
2296:
2267:
2252:
2129:
2025:
1995:
1722:
1677:
1554:
1152:
1147:
1142:
1114:
1099:
1012:
997:
975:
962:
839:
705:
647:
583:
271:
39:
2868:
2687:
2671:
2612:
2566:
2262:
2247:
2157:
1882:
1440:
1309:
1271:
1228:
1109:
1094:
1084:
1042:
1032:
1007:
797:
682:
651:
643:
573:
563:
531:
369:
368:
asks for the value of the smallest Sierpiński number. In private correspondence with
279:
236:
28:
823:
771:
2723:
2712:
2627:
2465:
2440:
2357:
2257:
2227:
2202:
2186:
2091:
2058:
1807:
1781:
1692:
1631:
1208:
1104:
1037:
1017:
992:
568:
283:
17:
2682:
2557:
2362:
1826:
1717:
1672:
1667:
1417:
1324:
1223:
1052:
1027:
1002:
916:
618:
2819:
2800:
2096:
1707:
905:
745:
545:
373:
930:
731:
691:
2425:
2352:
2344:
2149:
2063:
1181:
885:
802:
410:
293:}. Most currently known Sierpiński numbers possess similar covering sets.
2526:
673:
97:
870:
2531:
2190:
460:
424:
347:
101:
289:}. For another known Sierpiński number, 271129, the covering set is
182:{\displaystyle \left\{\,k\cdot 2^{n}+1:n\in \mathbb {N} \,\right\}.}
482:
Sierpinski number; there could exist a composite Sierpiński number
2817:
2781:
2745:
2709:
2669:
2294:
2183:
1909:
1824:
1779:
1656:
1346:
1293:
1245:
1179:
1131:
1069:
973:
934:
333:
give rise to a composite, and so it remains to eliminate only
535:
258:
270:
The number 78557 was proved to be a Sierpiński number by
413:
is attempting to eliminate all the remaining values of
387:
below 78,557, there needs to exist a positive integer
492:
198:
127:
114:
is a Sierpiński number, all members of the following
51:
2641:
2595:
2555:
2506:
2480:
2413:
2397:
2376:
2343:
2308:
2148:
2115:
2072:
2049:
1926:
1614:
1605:
1583:
1540:
1502:
1493:
1426:
1368:
1359:
510:
223:
181:
76:
274:in 1962, who showed that all numbers of the form
432:Is 271,129 the smallest prime Sierpiński number?
522:values between 78557 and 271129, prime or not.
530:A number may be simultaneously Sierpiński and
946:
871:The Sierpinski problem: definition and status
8:
798:"Welcome to the Extended Sierpinski Problem"
652:"On the density of odd integers of the form
409:The distributed volunteer computing project
383:Sierpiński numbers. That is, for every odd
2814:
2778:
2742:
2706:
2666:
2340:
2305:
2291:
2180:
1923:
1906:
1821:
1776:
1653:
1611:
1499:
1365:
1356:
1343:
1290:
1247:Possessing a specific set of other numbers
1242:
1176:
1128:
1066:
970:
953:
939:
931:
681:
491:
355:Is 78,557 the smallest Sierpiński number?
209:
197:
170:
166:
165:
144:
133:
126:
62:
50:
468:Is 271,129 the second Sierpiński number?
405:= 21181, 22699, 24737, 55459, and 67607.
881:"Sierpinski's composite number theorem"
606:Sierpinski number at The Prime Glossary
595:
472:(more unsolved problems in mathematics)
436:(more unsolved problems in mathematics)
359:(more unsolved problems in mathematics)
716:. pp. B21:119–121, F13:383–385.
7:
772:"About the Prime Sierpinski Problem"
601:
599:
526:Simultaneously Sierpiński and Riesel
511:{\displaystyle 78557<k<271129}
449:asks for the value of the smallest
2890:Unsolved problems in number theory
844:Unsolved Problems in Number Theory
710:Unsolved Problems in Number Theory
25:
2895:Science and technology in Poland
2875:Eponymous numbers in mathematics
2848:
2456:Perfect digit-to-digit invariant
770:Goetz, Michael (July 10, 2008).
548:
251:Sierpiński numbers begins with:
796:Goetz, Michael (6 April 2018).
463:Unsolved problem in mathematics
427:Unsolved problem in mathematics
350:Unsolved problem in mathematics
224:{\displaystyle k\times 2^{n}-1}
77:{\displaystyle k\times 2^{n}+1}
746:"Seventeen or Bust statistics"
1:
1295:Expressible via specific sums
683:10.1016/0022-314X(79)90043-X
619:"Note on Sierpinski Numbers"
326:. This establishes that all
2384:Multiplicative digital root
457:Extended Sierpiński problem
302:aurifeuillean factorization
300:. His proof depends on the
2911:
824:Problem 29.- Brier Numbers
617:Anatoly S. Izotov (1995).
247:The sequence of currently
107:which have this property.
2844:
2827:
2813:
2791:
2777:
2755:
2741:
2719:
2705:
2678:
2665:
2461:Perfect digital invariant
2304:
2290:
2198:
2179:
2036:Superior highly composite
1922:
1905:
1833:
1820:
1788:
1775:
1663:
1652:
1355:
1342:
1300:
1289:
1252:
1241:
1189:
1175:
1138:
1127:
1080:
1065:
983:
969:
2074:Euler's totient function
1858:Euler–Jacobi pseudoprime
1133:Other polynomial numbers
906:"78557 and Proth Primes"
665:Journal of Number Theory
447:prime Sierpiński problem
421:Prime Sierpiński problem
287:{3, 5, 7, 13, 19, 37, 73
243:Known Sierpiński numbers
88:for all natural numbers
1888:Somer–Lucas pseudoprime
1878:Lucas–Carmichael number
1713:Lazy caterer's sequence
648:Odlyzko, Andrew Michael
192:If the form is instead
1763:Wedderburn–Etherington
1163:Lucky numbers of Euler
660:and related questions"
512:
340:using a covering set.
225:
183:
96:proved that there are
78:
2051:Prime omega functions
1868:Frobenius pseudoprime
1658:Combinatorial numbers
1527:Centered dodecahedral
1320:Primary pseudoperfect
513:
291:{3, 5, 7, 13, 17, 241
226:
184:
110:In other words, when
79:
2510:-composition related
2310:Arithmetic functions
1912:Arithmetic functions
1848:Elliptic pseudoprime
1532:Centered icosahedral
1512:Centered tetrahedral
706:Guy, Richard Kenneth
490:
196:
125:
49:
2436:Kaprekar's constant
1956:Colossally abundant
1843:Catalan pseudoprime
1743:Schröder–Hipparchus
1522:Centered octahedral
1398:Centered heptagonal
1388:Centered pentagonal
1378:Centered triangular
978:and related numbers
626:Fibonacci Quarterly
2854:Mathematics portal
2796:Aronson's sequence
2542:Smarandache–Wellin
2299:-dependent numbers
2006:Primitive abundant
1893:Strong pseudoprime
1883:Perrin pseudoprime
1863:Fermat pseudoprime
1803:Wolstenholme prime
1627:Squared triangular
1413:Centered decagonal
1408:Centered nonagonal
1403:Centered octagonal
1393:Centered hexagonal
904:Grime, Dr. James.
878:Weisstein, Eric W.
556:Mathematics portal
508:
366:Sierpiński problem
344:Sierpiński problem
221:
179:
74:
18:Sierpinski numbers
2862:
2861:
2840:
2839:
2809:
2808:
2773:
2772:
2737:
2736:
2701:
2700:
2661:
2660:
2657:
2656:
2476:
2475:
2286:
2285:
2175:
2174:
2171:
2170:
2117:Aliquot sequences
1928:Divisor functions
1901:
1900:
1873:Lucas pseudoprime
1853:Euler pseudoprime
1838:Carmichael number
1816:
1815:
1771:
1770:
1648:
1647:
1644:
1643:
1640:
1639:
1601:
1600:
1489:
1488:
1446:Square triangular
1338:
1337:
1285:
1284:
1237:
1236:
1171:
1170:
1123:
1122:
1061:
1060:
723:978-0-387-20860-2
579:Seventeen or Bust
338:≡ 0, 1, 3 (mod 4)
94:Wacław Sierpiński
33:Sierpiński number
16:(Redirected from
2902:
2852:
2815:
2784:Natural language
2779:
2743:
2711:Generated via a
2707:
2667:
2572:Digit-reassembly
2537:Self-descriptive
2341:
2306:
2292:
2243:Lucas–Carmichael
2233:Harmonic divisor
2181:
2107:Sparsely totient
2082:Highly cototient
1991:Multiply perfect
1981:Highly composite
1924:
1907:
1822:
1777:
1758:Telephone number
1654:
1612:
1593:Square pyramidal
1575:Stella octangula
1500:
1366:
1357:
1349:Figurate numbers
1344:
1291:
1243:
1177:
1129:
1067:
971:
955:
948:
941:
932:
927:
925:
923:
910:
891:
890:
860:
826:
821:
815:
814:
812:
810:
793:
787:
786:
784:
782:
767:
761:
760:
758:
756:
742:
736:
735:
702:
696:
695:
685:
659:
640:
634:
633:
623:
614:
608:
603:
558:
553:
552:
517:
515:
514:
509:
464:
428:
397:
351:
339:
332:
325:
292:
288:
277:
261:
230:
228:
227:
222:
214:
213:
188:
186:
185:
180:
175:
171:
169:
149:
148:
83:
81:
80:
75:
67:
66:
21:
2910:
2909:
2905:
2904:
2903:
2901:
2900:
2899:
2865:
2864:
2863:
2858:
2836:
2832:Strobogrammatic
2823:
2805:
2787:
2769:
2751:
2733:
2715:
2697:
2674:
2653:
2637:
2596:Divisor-related
2591:
2551:
2502:
2472:
2409:
2393:
2372:
2339:
2312:
2300:
2282:
2194:
2193:related numbers
2167:
2144:
2111:
2102:Perfect totient
2068:
2045:
1976:Highly abundant
1918:
1897:
1829:
1812:
1784:
1767:
1753:Stirling second
1659:
1636:
1597:
1579:
1536:
1485:
1422:
1383:Centered square
1351:
1334:
1296:
1281:
1248:
1233:
1185:
1184:defined numbers
1167:
1134:
1119:
1090:Double Mersenne
1076:
1057:
979:
965:
963:natural numbers
959:
921:
919:
908:
903:
900:Wayback Machine
876:
875:
867:
858:
850:, p. 120,
848:Springer-Verlag
840:Guy, Richard K.
838:
835:
833:Further reading
830:
829:
822:
818:
808:
806:
795:
794:
790:
780:
778:
769:
768:
764:
754:
752:
744:
743:
739:
724:
714:Springer-Verlag
704:
703:
699:
653:
650:(May 1, 1979).
642:
641:
637:
621:
616:
615:
611:
604:
597:
592:
554:
547:
544:
528:
488:
487:
475:
474:
469:
466:
462:
459:
439:
438:
433:
430:
426:
423:
392:
362:
361:
356:
353:
349:
346:
334:
327:
304:
290:
286:
275:
257:
245:
205:
194:
193:
140:
132:
128:
123:
122:
118:are composite:
98:infinitely many
58:
47:
46:
23:
22:
15:
12:
11:
5:
2908:
2906:
2898:
2897:
2892:
2887:
2882:
2877:
2867:
2866:
2860:
2859:
2857:
2856:
2845:
2842:
2841:
2838:
2837:
2835:
2834:
2828:
2825:
2824:
2818:
2811:
2810:
2807:
2806:
2804:
2803:
2798:
2792:
2789:
2788:
2782:
2775:
2774:
2771:
2770:
2768:
2767:
2765:Sorting number
2762:
2760:Pancake number
2756:
2753:
2752:
2746:
2739:
2738:
2735:
2734:
2732:
2731:
2726:
2720:
2717:
2716:
2710:
2703:
2702:
2699:
2698:
2696:
2695:
2690:
2685:
2679:
2676:
2675:
2672:Binary numbers
2670:
2663:
2662:
2659:
2658:
2655:
2654:
2652:
2651:
2645:
2643:
2639:
2638:
2636:
2635:
2630:
2625:
2620:
2615:
2610:
2605:
2599:
2597:
2593:
2592:
2590:
2589:
2584:
2579:
2574:
2569:
2563:
2561:
2553:
2552:
2550:
2549:
2544:
2539:
2534:
2529:
2524:
2519:
2513:
2511:
2504:
2503:
2501:
2500:
2499:
2498:
2487:
2485:
2482:P-adic numbers
2478:
2477:
2474:
2473:
2471:
2470:
2469:
2468:
2458:
2453:
2448:
2443:
2438:
2433:
2428:
2423:
2417:
2415:
2411:
2410:
2408:
2407:
2401:
2399:
2398:Coding-related
2395:
2394:
2392:
2391:
2386:
2380:
2378:
2374:
2373:
2371:
2370:
2365:
2360:
2355:
2349:
2347:
2338:
2337:
2336:
2335:
2333:Multiplicative
2330:
2319:
2317:
2302:
2301:
2297:Numeral system
2295:
2288:
2287:
2284:
2283:
2281:
2280:
2275:
2270:
2265:
2260:
2255:
2250:
2245:
2240:
2235:
2230:
2225:
2220:
2215:
2210:
2205:
2199:
2196:
2195:
2184:
2177:
2176:
2173:
2172:
2169:
2168:
2166:
2165:
2160:
2154:
2152:
2146:
2145:
2143:
2142:
2137:
2132:
2127:
2121:
2119:
2113:
2112:
2110:
2109:
2104:
2099:
2094:
2089:
2087:Highly totient
2084:
2078:
2076:
2070:
2069:
2067:
2066:
2061:
2055:
2053:
2047:
2046:
2044:
2043:
2038:
2033:
2028:
2023:
2018:
2013:
2008:
2003:
1998:
1993:
1988:
1983:
1978:
1973:
1968:
1963:
1958:
1953:
1948:
1943:
1941:Almost perfect
1938:
1932:
1930:
1920:
1919:
1910:
1903:
1902:
1899:
1898:
1896:
1895:
1890:
1885:
1880:
1875:
1870:
1865:
1860:
1855:
1850:
1845:
1840:
1834:
1831:
1830:
1825:
1818:
1817:
1814:
1813:
1811:
1810:
1805:
1800:
1795:
1789:
1786:
1785:
1780:
1773:
1772:
1769:
1768:
1766:
1765:
1760:
1755:
1750:
1748:Stirling first
1745:
1740:
1735:
1730:
1725:
1720:
1715:
1710:
1705:
1700:
1695:
1690:
1685:
1680:
1675:
1670:
1664:
1661:
1660:
1657:
1650:
1649:
1646:
1645:
1642:
1641:
1638:
1637:
1635:
1634:
1629:
1624:
1618:
1616:
1609:
1603:
1602:
1599:
1598:
1596:
1595:
1589:
1587:
1581:
1580:
1578:
1577:
1572:
1567:
1562:
1557:
1552:
1546:
1544:
1538:
1537:
1535:
1534:
1529:
1524:
1519:
1514:
1508:
1506:
1497:
1491:
1490:
1487:
1486:
1484:
1483:
1478:
1473:
1468:
1463:
1458:
1453:
1448:
1443:
1438:
1432:
1430:
1424:
1423:
1421:
1420:
1415:
1410:
1405:
1400:
1395:
1390:
1385:
1380:
1374:
1372:
1363:
1353:
1352:
1347:
1340:
1339:
1336:
1335:
1333:
1332:
1327:
1322:
1317:
1312:
1307:
1301:
1298:
1297:
1294:
1287:
1286:
1283:
1282:
1280:
1279:
1274:
1269:
1264:
1259:
1253:
1250:
1249:
1246:
1239:
1238:
1235:
1234:
1232:
1231:
1226:
1221:
1216:
1211:
1206:
1201:
1196:
1190:
1187:
1186:
1180:
1173:
1172:
1169:
1168:
1166:
1165:
1160:
1155:
1150:
1145:
1139:
1136:
1135:
1132:
1125:
1124:
1121:
1120:
1118:
1117:
1112:
1107:
1102:
1097:
1092:
1087:
1081:
1078:
1077:
1070:
1063:
1062:
1059:
1058:
1056:
1055:
1050:
1045:
1040:
1035:
1030:
1025:
1020:
1015:
1010:
1005:
1000:
995:
990:
984:
981:
980:
974:
967:
966:
960:
958:
957:
950:
943:
935:
929:
928:
892:
873:
866:
865:External links
863:
862:
861:
856:
834:
831:
828:
827:
816:
788:
762:
737:
722:
697:
635:
609:
594:
593:
591:
588:
587:
586:
584:Woodall number
581:
576:
571:
566:
560:
559:
543:
540:
527:
524:
507:
504:
501:
498:
495:
470:
467:
461:
458:
455:
445:= 271129. The
434:
431:
425:
422:
419:
407:
406:
357:
354:
348:
345:
342:
272:John Selfridge
268:
267:
244:
241:
220:
217:
212:
208:
204:
201:
190:
189:
178:
174:
168:
164:
161:
158:
155:
152:
147:
143:
139:
136:
131:
73:
70:
65:
61:
57:
54:
40:natural number
24:
14:
13:
10:
9:
6:
4:
3:
2:
2907:
2896:
2893:
2891:
2888:
2886:
2883:
2881:
2880:Prime numbers
2878:
2876:
2873:
2872:
2870:
2855:
2851:
2847:
2846:
2843:
2833:
2830:
2829:
2826:
2821:
2816:
2812:
2802:
2799:
2797:
2794:
2793:
2790:
2785:
2780:
2776:
2766:
2763:
2761:
2758:
2757:
2754:
2749:
2744:
2740:
2730:
2727:
2725:
2722:
2721:
2718:
2714:
2708:
2704:
2694:
2691:
2689:
2686:
2684:
2681:
2680:
2677:
2673:
2668:
2664:
2650:
2647:
2646:
2644:
2640:
2634:
2631:
2629:
2626:
2624:
2623:Polydivisible
2621:
2619:
2616:
2614:
2611:
2609:
2606:
2604:
2601:
2600:
2598:
2594:
2588:
2585:
2583:
2580:
2578:
2575:
2573:
2570:
2568:
2565:
2564:
2562:
2559:
2554:
2548:
2545:
2543:
2540:
2538:
2535:
2533:
2530:
2528:
2525:
2523:
2520:
2518:
2515:
2514:
2512:
2509:
2505:
2497:
2494:
2493:
2492:
2489:
2488:
2486:
2483:
2479:
2467:
2464:
2463:
2462:
2459:
2457:
2454:
2452:
2449:
2447:
2444:
2442:
2439:
2437:
2434:
2432:
2429:
2427:
2424:
2422:
2419:
2418:
2416:
2412:
2406:
2403:
2402:
2400:
2396:
2390:
2387:
2385:
2382:
2381:
2379:
2377:Digit product
2375:
2369:
2366:
2364:
2361:
2359:
2356:
2354:
2351:
2350:
2348:
2346:
2342:
2334:
2331:
2329:
2326:
2325:
2324:
2321:
2320:
2318:
2316:
2311:
2307:
2303:
2298:
2293:
2289:
2279:
2276:
2274:
2271:
2269:
2266:
2264:
2261:
2259:
2256:
2254:
2251:
2249:
2246:
2244:
2241:
2239:
2236:
2234:
2231:
2229:
2226:
2224:
2221:
2219:
2216:
2214:
2213:Erdős–Nicolas
2211:
2209:
2206:
2204:
2201:
2200:
2197:
2192:
2188:
2182:
2178:
2164:
2161:
2159:
2156:
2155:
2153:
2151:
2147:
2141:
2138:
2136:
2133:
2131:
2128:
2126:
2123:
2122:
2120:
2118:
2114:
2108:
2105:
2103:
2100:
2098:
2095:
2093:
2090:
2088:
2085:
2083:
2080:
2079:
2077:
2075:
2071:
2065:
2062:
2060:
2057:
2056:
2054:
2052:
2048:
2042:
2039:
2037:
2034:
2032:
2031:Superabundant
2029:
2027:
2024:
2022:
2019:
2017:
2014:
2012:
2009:
2007:
2004:
2002:
1999:
1997:
1994:
1992:
1989:
1987:
1984:
1982:
1979:
1977:
1974:
1972:
1969:
1967:
1964:
1962:
1959:
1957:
1954:
1952:
1949:
1947:
1944:
1942:
1939:
1937:
1934:
1933:
1931:
1929:
1925:
1921:
1917:
1913:
1908:
1904:
1894:
1891:
1889:
1886:
1884:
1881:
1879:
1876:
1874:
1871:
1869:
1866:
1864:
1861:
1859:
1856:
1854:
1851:
1849:
1846:
1844:
1841:
1839:
1836:
1835:
1832:
1828:
1823:
1819:
1809:
1806:
1804:
1801:
1799:
1796:
1794:
1791:
1790:
1787:
1783:
1778:
1774:
1764:
1761:
1759:
1756:
1754:
1751:
1749:
1746:
1744:
1741:
1739:
1736:
1734:
1731:
1729:
1726:
1724:
1721:
1719:
1716:
1714:
1711:
1709:
1706:
1704:
1701:
1699:
1696:
1694:
1691:
1689:
1686:
1684:
1681:
1679:
1676:
1674:
1671:
1669:
1666:
1665:
1662:
1655:
1651:
1633:
1630:
1628:
1625:
1623:
1620:
1619:
1617:
1613:
1610:
1608:
1607:4-dimensional
1604:
1594:
1591:
1590:
1588:
1586:
1582:
1576:
1573:
1571:
1568:
1566:
1563:
1561:
1558:
1556:
1553:
1551:
1548:
1547:
1545:
1543:
1539:
1533:
1530:
1528:
1525:
1523:
1520:
1518:
1517:Centered cube
1515:
1513:
1510:
1509:
1507:
1505:
1501:
1498:
1496:
1495:3-dimensional
1492:
1482:
1479:
1477:
1474:
1472:
1469:
1467:
1464:
1462:
1459:
1457:
1454:
1452:
1449:
1447:
1444:
1442:
1439:
1437:
1434:
1433:
1431:
1429:
1425:
1419:
1416:
1414:
1411:
1409:
1406:
1404:
1401:
1399:
1396:
1394:
1391:
1389:
1386:
1384:
1381:
1379:
1376:
1375:
1373:
1371:
1367:
1364:
1362:
1361:2-dimensional
1358:
1354:
1350:
1345:
1341:
1331:
1328:
1326:
1323:
1321:
1318:
1316:
1313:
1311:
1308:
1306:
1305:Nonhypotenuse
1303:
1302:
1299:
1292:
1288:
1278:
1275:
1273:
1270:
1268:
1265:
1263:
1260:
1258:
1255:
1254:
1251:
1244:
1240:
1230:
1227:
1225:
1222:
1220:
1217:
1215:
1212:
1210:
1207:
1205:
1202:
1200:
1197:
1195:
1192:
1191:
1188:
1183:
1178:
1174:
1164:
1161:
1159:
1156:
1154:
1151:
1149:
1146:
1144:
1141:
1140:
1137:
1130:
1126:
1116:
1113:
1111:
1108:
1106:
1103:
1101:
1098:
1096:
1093:
1091:
1088:
1086:
1083:
1082:
1079:
1074:
1068:
1064:
1054:
1051:
1049:
1046:
1044:
1043:Perfect power
1041:
1039:
1036:
1034:
1033:Seventh power
1031:
1029:
1026:
1024:
1021:
1019:
1016:
1014:
1011:
1009:
1006:
1004:
1001:
999:
996:
994:
991:
989:
986:
985:
982:
977:
972:
968:
964:
956:
951:
949:
944:
942:
937:
936:
933:
918:
914:
907:
901:
897:
893:
888:
887:
882:
879:
874:
872:
869:
868:
864:
859:
857:0-387-20860-7
853:
849:
845:
841:
837:
836:
832:
825:
820:
817:
805:
804:
799:
792:
789:
781:September 12,
777:
773:
766:
763:
751:
747:
741:
738:
733:
729:
725:
719:
715:
711:
707:
701:
698:
693:
689:
684:
679:
675:
671:
667:
666:
661:
657:
649:
645:
639:
636:
631:
627:
620:
613:
610:
607:
602:
600:
596:
589:
585:
582:
580:
577:
575:
574:Riesel number
572:
570:
567:
565:
564:Cullen number
562:
561:
557:
551:
546:
541:
539:
537:
533:
525:
523:
521:
505:
502:
499:
496:
493:
485:
481:
473:
456:
454:
452:
448:
444:
437:
420:
418:
416:
412:
404:
401:
400:
399:
395:
390:
386:
382:
377:
375:
371:
367:
360:
343:
341:
337:
330:
323:
319:
315:
311:
307:
303:
299:
294:
285:
281:
273:
265:
260:
254:
253:
252:
250:
242:
240:
238:
237:Riesel number
234:
218:
215:
210:
206:
202:
199:
176:
172:
162:
159:
156:
153:
150:
145:
141:
137:
134:
129:
121:
120:
119:
117:
113:
108:
106:
103:
99:
95:
91:
87:
71:
68:
63:
59:
55:
52:
44:
41:
38:
34:
30:
29:number theory
19:
2587:Transposable
2451:Narcissistic
2358:Digital root
2278:Super-Poulet
2238:Jordan–Pólya
2187:prime factor
2092:Noncototient
2059:Almost prime
2041:Superperfect
2016:Refactorable
2011:Quasiperfect
1986:Hyperperfect
1827:Pseudoprimes
1798:Wall–Sun–Sun
1733:Ordered Bell
1703:Fuss–Catalan
1615:non-centered
1565:Dodecahedral
1542:non-centered
1428:non-centered
1330:Wolstenholme
1276:
1075:× 2 ± 1
1072:
1071:Of the form
1038:Eighth power
1018:Fourth power
920:. Retrieved
912:
896:Ghostarchive
894:Archived at
884:
846:, New York:
843:
819:
807:. Retrieved
801:
791:
779:. Retrieved
775:
765:
755:November 21,
753:. Retrieved
749:
740:
712:. New York:
709:
700:
669:
663:
655:
638:
629:
625:
612:
569:Proth number
529:
519:
483:
479:
476:
450:
446:
442:
440:
414:
408:
402:
393:
388:
384:
380:
378:
372:, Selfridge
365:
363:
335:
328:
321:
317:
313:
309:
305:
297:
295:
284:covering set
269:
248:
246:
232:
191:
111:
109:
104:
89:
42:
32:
26:
2885:Conjectures
2608:Extravagant
2603:Equidigital
2558:permutation
2517:Palindromic
2491:Automorphic
2389:Sum-product
2368:Sum-product
2323:Persistence
2218:Erdős–Woods
2140:Untouchable
2021:Semiperfect
1971:Hemiperfect
1632:Tesseractic
1570:Icosahedral
1550:Tetrahedral
1481:Dodecagonal
1182:Recursively
1053:Prime power
1028:Sixth power
1023:Fifth power
1003:Power of 10
961:Classes of
922:13 November
917:Brady Haran
644:Erdős, Paul
374:conjectured
331:≡ 2 (mod 4)
276:78557⋅2 + 1
92:. In 1960,
2869:Categories
2820:Graphemics
2693:Pernicious
2547:Undulating
2522:Pandigital
2496:Trimorphic
2097:Nontotient
1946:Arithmetic
1560:Octahedral
1461:Heptagonal
1451:Pentagonal
1436:Triangular
1277:Sierpiński
1199:Jacobsthal
998:Power of 3
993:Power of 2
590:References
486:such that
398:is prime.
391:such that
370:Paul Erdős
308:⋅2 + 1 = (
45:such that
2577:Parasitic
2426:Factorion
2353:Digit sum
2345:Digit sum
2163:Fortunate
2150:Primorial
2064:Semiprime
2001:Practical
1966:Descartes
1961:Deficient
1951:Betrothed
1793:Wieferich
1622:Pentatope
1585:pyramidal
1476:Decagonal
1471:Nonagonal
1466:Octagonal
1456:Hexagonal
1315:Practical
1262:Congruent
1194:Fibonacci
1158:Loeschian
886:MathWorld
809:21 August
803:PrimeGrid
776:PrimeGrid
750:PrimeGrid
732:634701581
692:0022-314X
632:(3): 206.
411:PrimeGrid
316:⋅2 + 1)⋅(
216:−
203:×
163:∈
138:⋅
86:composite
56:×
2649:Friedman
2582:Primeval
2527:Repdigit
2484:-related
2431:Kaprekar
2405:Meertens
2328:Additive
2315:dynamics
2223:Friendly
2135:Sociable
2125:Amicable
1936:Abundant
1916:dynamics
1738:Schröder
1728:Narayana
1698:Eulerian
1688:Delannoy
1683:Dedekind
1504:centered
1370:centered
1257:Amenable
1214:Narayana
1204:Leonardo
1100:Mersenne
1048:Powerful
988:Achilles
898:and the
842:(2004),
708:(2005).
674:Elsevier
542:See also
102:integers
2822:related
2786:related
2750:related
2748:Sorting
2633:Vampire
2618:Harshad
2560:related
2532:Repunit
2446:Lychrel
2421:Dudeney
2273:Størmer
2268:Sphenic
2253:Regular
2191:divisor
2130:Perfect
2026:Sublime
1996:Perfect
1723:Motzkin
1678:Catalan
1219:Padovan
1153:Leyland
1148:Idoneal
1143:Hilbert
1115:Woodall
913:YouTube
909:(video)
676:: 258.
536:A076335
324:⋅2 + 1)
282:in the
278:have a
262:in the
259:A076336
231:, then
2688:Odious
2613:Frugal
2567:Cyclic
2556:Digit-
2263:Smooth
2248:Pronic
2208:Cyclic
2185:Other
2158:Euclid
1808:Wilson
1782:Primes
1441:Square
1310:Polite
1272:Riesel
1267:Knödel
1229:Perrin
1110:Thabit
1095:Fermat
1085:Cullen
1008:Square
976:Powers
854:
730:
720:
690:
532:Riesel
506:271129
480:second
280:factor
35:is an
2729:Prime
2724:Lucky
2713:sieve
2642:Other
2628:Smith
2508:Digit
2466:Happy
2441:Keith
2414:Other
2258:Rough
2228:Giuga
1693:Euler
1555:Cubic
1209:Lucas
1105:Proth
672:(2).
658:− 1)2
622:(PDF)
494:78557
451:prime
396:2 + 1
320:⋅2 −
312:⋅2 +
249:known
235:is a
2683:Evil
2363:Self
2313:and
2203:Blum
1914:and
1718:Lobb
1673:Cake
1668:Bell
1418:Star
1325:Ulam
1224:Pell
1013:Cube
924:2017
852:ISBN
811:2019
783:2019
757:2019
728:OCLC
718:ISBN
688:ISSN
503:<
497:<
364:The
264:OEIS
100:odd
31:, a
2801:Ban
2189:or
1708:Lah
678:doi
538:).
381:not
116:set
84:is
37:odd
27:In
2871::
915:.
911:.
902::
883:.
800:.
774:.
748:.
726:.
686:.
670:11
668:.
662:.
646:;
630:33
628:.
624:.
598:^
417:.
266:).
239:.
1073:a
954:e
947:t
940:v
926:.
889:.
813:.
785:.
759:.
734:.
694:.
680::
656:p
654:(
520:k
500:k
484:k
465::
443:k
429::
415:k
403:k
394:k
389:n
385:k
352::
336:n
329:n
322:t
318:t
314:t
310:t
306:t
298:n
233:k
219:1
211:n
207:2
200:k
177:.
173:}
167:N
160:n
157::
154:1
151:+
146:n
142:2
135:k
130:{
112:k
105:k
90:n
72:1
69:+
64:n
60:2
53:k
43:k
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
