399:(i.e. the repeating digit is 1) and have a prime number of digits in its base (except trivial single-digit numbers), since, for example, the repdigit 77777 is divisible by 7, in any base > 7. In particular, as Brazilian repunits do not allow the number of digits to be exactly two, Brazilian primes must have an odd prime number of digits. Having an odd prime number of digits is not enough to guarantee that a repunit is prime; for instance, 21 = 111
3682:
597:
is true, then for every prime number of digits there would exist infinitely many repunit primes with that number of digits (and consequentially infinitely many
Brazilian primes). Alternatively, if there are infinitely many decimal repunit primes, or infinitely many Mersenne primes, then there are
283:
are numbers that can be written as a repdigit in some base, not allowing the repdigit 11, and not allowing the single-digit numbers (or all numbers will be
Brazilian). For example, 27 is a Brazilian number because 27 is the repdigit 33 in base 8, while 9 is not a Brazilian number because its only
325:), it is considered an auspicious event when the sequentially-assigned ID number of a post is a repdigit, such as 22,222,222, which is one type of "GET" (others including round numbers like 34,000,000, or sequential digits like 12,345,678).
562:
481:). This convergence implies that the Brazilian primes form a vanishingly small fraction of all prime numbers. For instance, among the 3.7×10 prime numbers smaller than 10, only 8.8×10 are Brazilian.
1092:
894:. Hence, there exist numbers that are non-Brazilian and others that are Brazilian; among these last integers, some are once Brazilian, others are twice Brazilian, or three times, or more. A number that is
288:, not allowed in the definition of Brazilian numbers. The representations of the form 11 are considered trivial and are disallowed in the definition of Brazilian numbers, because all natural numbers
471:
is a divergent series, the sum of the reciprocals of the
Brazilian prime numbers is a convergent series whose value, called the "Brazilian primes constant", is slightly larger than 0.33 (sequence
909:
are constituted with 1 and 6, together with some primes and some squares of primes. The sequence of the non-Brazilian numbers begins with 1, 2, 3, 4, 5, 6, 9, 11, 17, 19, 23, 25, ... (sequence
274:
160:
670:
192:
598:
infinitely many
Brazilian primes. Because a vanishingly small fraction of primes are Brazilian, there are infinitely many non-Brazilian primes, forming the sequence
218:
672:
is prime, it is not
Brazilian, but if it is composite, it is Brazilian. Contradicting a previous conjecture, Resta, Marcus, Grantham, and Graves found examples of
107:
468:
1784:
337:
called them "monodigit numbers". The
Brazilian numbers were introduced later, in 1994, in the 9th Iberoamerican Mathematical Olympiad that took place in
1231:
if it is a positive integer with more
Brazilian representations than any smaller positive integer has. This definition comes from the definition of
854:
Yann
Bugeaud and Maurice Mignotte conjecture that only three perfect powers are Brazilian repunits. They are 121, 343, and 400 (sequence
491:
3706:
1259:
1245:
1192:
1005:
990:
973:
916:
886:
861:
609:
570:
478:
460:
310:
1777:
1329:
1401:
2584:
1770:
2579:
223:
2594:
2574:
3287:
2867:
2589:
3373:
1710:
808:, but it is not exceptional with respect to the classification of Brazilian numbers because 20 is not prime.
112:
3039:
2689:
2358:
2151:
3215:
3074:
2905:
2719:
2709:
2363:
2343:
594:
3044:
3164:
2787:
2629:
2544:
2353:
2335:
2229:
2219:
2209:
2045:
622:
3069:
3292:
2837:
2458:
2244:
2239:
2234:
2224:
2201:
1659:
3049:
926:
is composed of other primes, the only square of prime that is
Brazilian, 121, and composite numbers
2714:
2624:
2277:
40:
20:
1190:-Brazilian numbers begins with 1, 7, 15, 24, 40, 60, 144, 120, 180, 336, 420, 360, ... and are in
3403:
3368:
3154:
3064:
2938:
2913:
2822:
2812:
2534:
2424:
2406:
2326:
1621:
1254:
1232:
994:
165:
3663:
2933:
2807:
2438:
2214:
1994:
1921:
1547:
Grantham, Jon; Graves, Hester (2019). "Brazilian primes which are also Sophie
Germain primes".
1321:
376:
is written with all equal digits. Prove that 1994 is Brazilian and that 1993 is not Brazilian.
1600:
Ljunggren, Wilhelm (1943). "Noen setninger om ubestemte likninger av formen (x-1)/(x-1) = y".
3627:
3267:
2918:
2772:
2699:
1854:
1276:
Some popular media publications have published articles suggesting that repunit numbers have
1165:) = (8191, 90, 2, 3, 13) corresponding to 8191 = 1111111111111
333:
The concept of a repdigit has been studied under that name since at least 1974, and earlier
303:
7, 8, 10, 12, 13, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 31, 32, 33, ... (sequence
197:
3560:
3454:
3418:
3159:
2882:
2862:
2679:
2348:
2136:
1442:
998:
819:
729:
673:
2639:
2108:
8:
3282:
3146:
3141:
3109:
2872:
2847:
2842:
2817:
2747:
2743:
2674:
2564:
2396:
2192:
2161:
1236:
56:
16:
Natural number with a decimal representation made of repeated instances of the same digit
1423:
3685:
3439:
3434:
3348:
3322:
3220:
3199:
2971:
2852:
2802:
2724:
2694:
2634:
2401:
2381:
2312:
2025:
1548:
1495:
1314:
92:
68:
2569:
1684:
341:, Brazil. The first problem in this competition, proposed by Mexico, was as follows:
3681:
3579:
3524:
3378:
3353:
3327:
2782:
2777:
2704:
2684:
2669:
2391:
2373:
2292:
2282:
2267:
2030:
1740:
1325:
823:
578:. It has been conjectured that there are infinitely many decimal repunit primes. The
3104:
3615:
3408:
2994:
2966:
2956:
2948:
2832:
2797:
2792:
2759:
2453:
2416:
2307:
2302:
2297:
2287:
2259:
2146:
2093:
2050:
1989:
1419:
1357:
2098:
3591:
3480:
3413:
3339:
3262:
3236:
3054:
2767:
2559:
2529:
2519:
2514:
2180:
2088:
2035:
1879:
1819:
1438:
1361:
1289:
583:
64:
36:
993:) consists of composites and only two primes: 31 and 8191. Indeed, according to
3596:
3464:
3449:
3313:
3277:
3252:
3128:
3099:
3084:
2961:
2857:
2827:
2554:
2509:
2386:
1984:
1979:
1974:
1946:
1931:
1844:
1829:
1807:
1794:
587:
80:
32:
1743:
1134:) = (31, 5, 2, 3, 5) corresponding to 31 = 11111
3700:
3519:
3503:
3444:
3398:
3094:
3079:
2989:
2272:
2141:
2103:
2060:
1941:
1926:
1916:
1874:
1864:
1839:
1762:
1757:
1277:
811:
765:
693:
617:
579:
386:
76:
1660:"The 333 angel number is very powerful in numerology – here's what it means"
220:
is the number of repetitions. For example, the repdigit 77777 in base 10 is
3555:
3544:
3459:
3297:
3272:
3189:
3089:
3059:
3034:
3018:
2923:
2890:
2613:
2524:
2463:
2040:
1936:
1869:
1849:
1824:
1243:
are 1, 7, 15, 24, 40, 60, 120, 180, 336, 360, 720, ... and are exactly in
689:
392:
3514:
3389:
3194:
2658:
2549:
2504:
2499:
2249:
2156:
2055:
1884:
1859:
1834:
1499:
1316:
Recreations in the Theory of Numbers: The Queen of Mathematics Entertains
788:. Therefore, the only squared prime that is Brazilian is 11 = 121 = 11111
593:
It is unknown whether there are infinitely many Brazilian primes. If the
417:(if it is prime) is a Brazilian prime. The smallest Brazilian primes are
318:
52:
48:
3651:
3632:
2928:
2539:
1346:
60:
1580:
Nagell, Trygve (1921). "Sur l'équation indéterminée (x-1)/(x-1) = y".
1504:
3257:
3184:
3176:
2981:
2895:
2013:
1748:
338:
1375:
696:
of the primes, for every other number is the product of two factors
1553:
688:
The only positive integers that can be non-Brazilian are 1, 6, the
557:{\displaystyle R_{n}={\tfrac {10^{n}-1}{9}}\ {\mbox{with }}n\geq 3}
792:. There is also one more nontrivial repunit square, the solution (
3363:
3022:
3016:
1253:. From 360 to 321253732800 (maybe more), there are 80 successive
1186:, there exists a smallest term. The sequence with these smallest
1174:
485:
396:
72:
44:
1516:
1087:{\displaystyle p={\frac {x^{m}-1}{x-1}}={\frac {y^{n}-1}{y-1}}}
2078:
1528:
683:
322:
87:
930:
that are the product of only two distinct factors such that
1561:
1263:
1249:
1196:
985:
968:
911:
890:
856:
604:
602:
2, 3, 5, 11, 17, 19, 23, 29, 37, 41, 47, 53, ... (sequence
574:
473:
455:
411:, every repunit prime in that base with the exception of 11
305:
1476:
1464:
864:), the two squares listed above and the cube 343 = 7 = 111
814:
that are repunits with three digits or more in some base
768:
has proved that this equation has only one solution when
1400:
Palau, AdriĂ Salvador; Roozenbeek, Jon (March 7, 2017).
1402:"How an ancient Egyptian god spurred the rise of Trump"
997:, these two primes are the only known solutions of the
1424:"Infinite sequences of palindromic triangular numbers"
539:
509:
2742:
1008:
676:
that are Brazilian, the first one is 28792661 = 11111
625:
494:
226:
200:
168:
115:
95:
3127:
1320:(2 ed.). New York: Dover Publications. p.
1758:
Problemas IX OlimpĂada Iberoamericana de Matemática
1646:Dictionnaire de (presque) tous les nombres entiers
1313:
1280:significance, describing them as "angel numbers".
1225:Dictionnaire de (presque) tous les nombres entiers
1086:
664:
556:
268:
212:
186:
154:
101:
83:(which are repdigits when represented in binary).
2126:
1738:
1711:"Everything You Need to Know About Angel Numbers"
1685:"Everything You Need to Know About Angel Numbers"
1455:
352:is called "Brazilian" if there exists an integer
3698:
1622:"L'Ă©quation de Nagell-Ljunggren (x-1)/(x-1) = y"
269:{\displaystyle 7\times {\frac {10^{5}-1}{10-1}}}
2012:
1619:
1399:
804:) = (20, 7, 4) corresponding to 20 = 400 = 1111
469:the sum of the reciprocals of the prime numbers
1806:
1792:
1546:
1778:
1643:
1257:that are also highly Brazilian numbers, see
3614:
1964:
1227:, Daniel Lignon proposes that an integer is
684:Non-Brazilian composites and repunit powers
292:greater than two have the representation 11
35:composed of repeated instances of the same
2079:Possessing a specific set of other numbers
1902:
1785:
1771:
1460:. Paris: Vuibert. p. 7, exercice a35.
407:= 3 Ă— 37 are not prime. In any given base
3542:
2489:
1620:Bugeaud, Yann; Mignotte, Maurice (2002).
1599:
1552:
299:. The first twenty Brazilian numbers are
75:. Other well-known repdigits include the
1307:
1305:
51:of "repeated" and "digit". Examples are
155:{\displaystyle x{\frac {B^{y}-1}{B-1}}}
3699:
3650:
1579:
1567:
1534:
1522:
1510:
1482:
1470:
1344:
1311:
880:The number of ways such that a number
871:
586:and the binary repunit primes are the
380:
334:
3649:
3613:
3577:
3541:
3501:
3126:
3015:
2741:
2656:
2611:
2488:
2178:
2125:
2077:
2011:
1963:
1901:
1805:
1766:
1739:
1418:
1368:
1338:
1302:
2179:
1393:
86:Repdigits are the representation in
3578:
1582:Norsk Matematisk Forenings Skrifter
1200:. For instance, 40 is the smallest
13:
3502:
720:− 1. If a square of a prime
14:
3718:
1732:
712:− 1, and can be written as
665:{\displaystyle F_{n}=2^{2^{n}}+1}
3707:Base-dependent integer sequences
3680:
3288:Perfect digit-to-digit invariant
2657:
368:for which the representation of
279:A variation of repdigits called
1703:
1677:
1652:
1648:. Paris: Ellipses. p. 420.
1637:
1613:
1593:
1573:
1540:
1488:
1449:
1412:
1345:Schott, Bernard (March 2010).
1:
2127:Expressible via specific sums
1295:
1271:
488:repunit primes have the form
284:repdigit representation is 11
317:On some websites (including
7:
3216:Multiplicative digital root
1626:L'Enseignement Mathématique
1602:Norsk Matematisk Tidsskrift
1496:The Prime Glossary: repunit
1283:
1239:in 1915. The first numbers
187:{\displaystyle 0<x<B}
10:
3723:
2612:
1456:Pierre Bornsztein (2001).
1362:10.1051/quadrature/2010005
1173:, with 11111111111 is the
905:Non-Brazilian numbers or 0
898:times Brazilian is called
772:is prime corresponding to
384:
328:
194:is the repeated digit and
3676:
3659:
3645:
3623:
3609:
3587:
3573:
3551:
3537:
3510:
3497:
3473:
3427:
3387:
3338:
3312:
3293:Perfect digital invariant
3245:
3229:
3208:
3175:
3140:
3136:
3122:
3030:
3011:
2980:
2947:
2904:
2881:
2868:Superior highly composite
2758:
2754:
2737:
2665:
2652:
2620:
2607:
2495:
2484:
2446:
2437:
2415:
2372:
2334:
2325:
2258:
2200:
2191:
2187:
2174:
2132:
2121:
2084:
2073:
2021:
2007:
1970:
1959:
1912:
1897:
1815:
1801:
1356:(in French) (76): 30–38.
724:is Brazilian, then prime
2906:Euler's totient function
2690:Euler–Jacobi pseudoprime
1965:Other polynomial numbers
1347:"Les nombres brésiliens"
1255:highly composite numbers
1233:highly composite numbers
1110: > 2 :
764:Norwegian mathematician
41:positional number system
21:recreational mathematics
2720:Somer–Lucas pseudoprime
2710:Lucas–Carmichael number
2545:Lazy caterer's sequence
1513:, Sections V.1 and V.2.
1431:The Fibonacci Quarterly
1312:Beiler, Albert (1966).
1177:with thirteen digits 1.
995:Goormaghtigh conjecture
595:Bateman–Horn conjecture
2595:Wedderburn–Etherington
1995:Lucky numbers of Euler
1644:Daniel Lignon (2012).
1102: > 1 and
1088:
666:
558:
378:
270:
214:
213:{\displaystyle 1<y}
188:
156:
103:
79:and in particular the
2883:Prime omega functions
2700:Frobenius pseudoprime
2490:Combinatorial numbers
2359:Centered dodecahedral
2152:Primary pseudoperfect
1182:For each sequence of
1089:
818:are described by the
674:Sophie Germain primes
667:
559:
403:= 3 Ă— 7 and 111 = 111
391:For a repdigit to be
343:
271:
215:
189:
157:
104:
71:and are multiples of
3342:-composition related
3142:Arithmetic functions
2744:Arithmetic functions
2680:Elliptic pseudoprime
2364:Centered icosahedral
2344:Centered tetrahedral
1006:
999:Diophantine equation
820:Diophantine equation
730:Diophantine equation
623:
492:
224:
198:
166:
113:
93:
67:. All repdigits are
3268:Kaprekar's constant
2788:Colossally abundant
2675:Catalan pseudoprime
2575:Schröder–Hipparchus
2354:Centered octahedral
2230:Centered heptagonal
2220:Centered pentagonal
2210:Centered triangular
1810:and related numbers
1237:Srinivasa Ramanujan
1184:k-Brazilian numbers
884:is Brazilian is in
381:Primes and repunits
69:palindromic numbers
3686:Mathematics portal
3628:Aronson's sequence
3374:Smarandache–Wellin
3131:-dependent numbers
2838:Primitive abundant
2725:Strong pseudoprime
2715:Perrin pseudoprime
2695:Fermat pseudoprime
2635:Wolstenholme prime
2459:Squared triangular
2245:Centered decagonal
2240:Centered nonagonal
2235:Centered octagonal
2225:Centered hexagonal
1741:Weisstein, Eric W.
1691:. 24 December 2021
1202:4-Brazilian number
1084:
981:-Brazilian numbers
924:-Brazilian numbers
907:-Brazilian numbers
900:k-Brazilian number
875:-Brazilian numbers
662:
564:for the values of
554:
543:
533:
266:
210:
184:
152:
99:
47:). The word is a
43:(often implicitly
3694:
3693:
3672:
3671:
3641:
3640:
3605:
3604:
3569:
3568:
3533:
3532:
3493:
3492:
3489:
3488:
3308:
3307:
3118:
3117:
3007:
3006:
3003:
3002:
2949:Aliquot sequences
2760:Divisor functions
2733:
2732:
2705:Lucas pseudoprime
2685:Euler pseudoprime
2670:Carmichael number
2648:
2647:
2603:
2602:
2480:
2479:
2476:
2475:
2472:
2471:
2433:
2432:
2321:
2320:
2278:Square triangular
2170:
2169:
2117:
2116:
2069:
2068:
2003:
2002:
1955:
1954:
1893:
1892:
1494:Chris Caldwell, "
1420:Trigg, Charles W.
1331:978-0-486-21096-4
1082:
1046:
922:The sequence of 1
728:must satisfy the
582:repunits are the
542:
537:
532:
281:Brazilian numbers
264:
150:
102:{\displaystyle B}
3714:
3684:
3647:
3646:
3616:Natural language
3611:
3610:
3575:
3574:
3543:Generated via a
3539:
3538:
3499:
3498:
3404:Digit-reassembly
3369:Self-descriptive
3173:
3172:
3138:
3137:
3124:
3123:
3075:Lucas–Carmichael
3065:Harmonic divisor
3013:
3012:
2939:Sparsely totient
2914:Highly cototient
2823:Multiply perfect
2813:Highly composite
2756:
2755:
2739:
2738:
2654:
2653:
2609:
2608:
2590:Telephone number
2486:
2485:
2444:
2443:
2425:Square pyramidal
2407:Stella octangula
2332:
2331:
2198:
2197:
2189:
2188:
2181:Figurate numbers
2176:
2175:
2123:
2122:
2075:
2074:
2009:
2008:
1961:
1960:
1899:
1898:
1803:
1802:
1787:
1780:
1773:
1764:
1763:
1754:
1753:
1726:
1725:
1723:
1722:
1707:
1701:
1700:
1698:
1696:
1681:
1675:
1674:
1672:
1671:
1656:
1650:
1649:
1641:
1635:
1633:
1617:
1611:
1609:
1604:(in Norwegian).
1597:
1591:
1589:
1577:
1571:
1565:
1559:
1558:
1556:
1544:
1538:
1532:
1526:
1525:, Proposition 3.
1520:
1514:
1508:
1502:
1492:
1486:
1480:
1474:
1468:
1462:
1461:
1453:
1447:
1446:
1428:
1416:
1410:
1409:
1406:The Conversation
1397:
1391:
1390:
1388:
1386:
1372:
1366:
1365:
1351:
1342:
1336:
1335:
1319:
1309:
1266:
1252:
1241:highly Brazilian
1229:highly Brazilian
1199:
1093:
1091:
1090:
1085:
1083:
1081:
1070:
1063:
1062:
1052:
1047:
1045:
1034:
1027:
1026:
1016:
988:
971:
965:
953:
929:
914:
893:
859:
787:
671:
669:
668:
663:
655:
654:
653:
652:
635:
634:
607:
584:Mersenne numbers
577:
563:
561:
560:
555:
544:
540:
535:
534:
528:
521:
520:
510:
504:
503:
476:
458:
453:, ... (sequence
367:
351:
308:
275:
273:
272:
267:
265:
263:
252:
245:
244:
234:
219:
217:
216:
211:
193:
191:
190:
185:
161:
159:
158:
153:
151:
149:
138:
131:
130:
120:
108:
106:
105:
100:
3722:
3721:
3717:
3716:
3715:
3713:
3712:
3711:
3697:
3696:
3695:
3690:
3668:
3664:Strobogrammatic
3655:
3637:
3619:
3601:
3583:
3565:
3547:
3529:
3506:
3485:
3469:
3428:Divisor-related
3423:
3383:
3334:
3304:
3241:
3225:
3204:
3171:
3144:
3132:
3114:
3026:
3025:related numbers
2999:
2976:
2943:
2934:Perfect totient
2900:
2877:
2808:Highly abundant
2750:
2729:
2661:
2644:
2616:
2599:
2585:Stirling second
2491:
2468:
2429:
2411:
2368:
2317:
2254:
2215:Centered square
2183:
2166:
2128:
2113:
2080:
2065:
2017:
2016:defined numbers
1999:
1966:
1951:
1922:Double Mersenne
1908:
1889:
1811:
1797:
1795:natural numbers
1791:
1735:
1730:
1729:
1720:
1718:
1709:
1708:
1704:
1694:
1692:
1683:
1682:
1678:
1669:
1667:
1658:
1657:
1653:
1642:
1638:
1618:
1614:
1598:
1594:
1578:
1574:
1566:
1562:
1545:
1541:
1537:, Conjecture 1.
1533:
1529:
1521:
1517:
1509:
1505:
1493:
1489:
1481:
1477:
1469:
1465:
1454:
1450:
1426:
1417:
1413:
1398:
1394:
1384:
1382:
1374:
1373:
1369:
1349:
1343:
1339:
1332:
1310:
1303:
1298:
1290:Six nines in pi
1286:
1274:
1258:
1244:
1219:
1215:
1211:
1207:
1204:with 40 = 1111
1191:
1172:
1168:
1141:
1137:
1111:
1071:
1058:
1054:
1053:
1051:
1035:
1022:
1018:
1017:
1015:
1007:
1004:
1003:
984:
967:
955:
952:
931:
927:
910:
885:
877:
867:
855:
852:
807:
791:
773:
762:
757:≥ 3 primes and
686:
679:
648:
644:
643:
639:
630:
626:
624:
621:
620:
603:
588:Mersenne primes
569:
538:
516:
512:
511:
508:
499:
495:
493:
490:
489:
472:
454:
452:
448:
445:, 127 = 1111111
444:
440:
436:
432:
428:
424:
416:
406:
402:
395:, it must be a
389:
383:
357:
346:
331:
304:
298:
287:
253:
240:
236:
235:
233:
225:
222:
221:
199:
196:
195:
167:
164:
163:
139:
126:
122:
121:
119:
114:
111:
110:
94:
91:
90:
81:Mersenne primes
17:
12:
11:
5:
3720:
3710:
3709:
3692:
3691:
3689:
3688:
3677:
3674:
3673:
3670:
3669:
3667:
3666:
3660:
3657:
3656:
3643:
3642:
3639:
3638:
3636:
3635:
3630:
3624:
3621:
3620:
3607:
3606:
3603:
3602:
3600:
3599:
3597:Sorting number
3594:
3592:Pancake number
3588:
3585:
3584:
3571:
3570:
3567:
3566:
3564:
3563:
3558:
3552:
3549:
3548:
3535:
3534:
3531:
3530:
3528:
3527:
3522:
3517:
3511:
3508:
3507:
3504:Binary numbers
3495:
3494:
3491:
3490:
3487:
3486:
3484:
3483:
3477:
3475:
3471:
3470:
3468:
3467:
3462:
3457:
3452:
3447:
3442:
3437:
3431:
3429:
3425:
3424:
3422:
3421:
3416:
3411:
3406:
3401:
3395:
3393:
3385:
3384:
3382:
3381:
3376:
3371:
3366:
3361:
3356:
3351:
3345:
3343:
3336:
3335:
3333:
3332:
3331:
3330:
3319:
3317:
3314:P-adic numbers
3310:
3309:
3306:
3305:
3303:
3302:
3301:
3300:
3290:
3285:
3280:
3275:
3270:
3265:
3260:
3255:
3249:
3247:
3243:
3242:
3240:
3239:
3233:
3231:
3230:Coding-related
3227:
3226:
3224:
3223:
3218:
3212:
3210:
3206:
3205:
3203:
3202:
3197:
3192:
3187:
3181:
3179:
3170:
3169:
3168:
3167:
3165:Multiplicative
3162:
3151:
3149:
3134:
3133:
3129:Numeral system
3120:
3119:
3116:
3115:
3113:
3112:
3107:
3102:
3097:
3092:
3087:
3082:
3077:
3072:
3067:
3062:
3057:
3052:
3047:
3042:
3037:
3031:
3028:
3027:
3009:
3008:
3005:
3004:
3001:
3000:
2998:
2997:
2992:
2986:
2984:
2978:
2977:
2975:
2974:
2969:
2964:
2959:
2953:
2951:
2945:
2944:
2942:
2941:
2936:
2931:
2926:
2921:
2919:Highly totient
2916:
2910:
2908:
2902:
2901:
2899:
2898:
2893:
2887:
2885:
2879:
2878:
2876:
2875:
2870:
2865:
2860:
2855:
2850:
2845:
2840:
2835:
2830:
2825:
2820:
2815:
2810:
2805:
2800:
2795:
2790:
2785:
2780:
2775:
2773:Almost perfect
2770:
2764:
2762:
2752:
2751:
2735:
2734:
2731:
2730:
2728:
2727:
2722:
2717:
2712:
2707:
2702:
2697:
2692:
2687:
2682:
2677:
2672:
2666:
2663:
2662:
2650:
2649:
2646:
2645:
2643:
2642:
2637:
2632:
2627:
2621:
2618:
2617:
2605:
2604:
2601:
2600:
2598:
2597:
2592:
2587:
2582:
2580:Stirling first
2577:
2572:
2567:
2562:
2557:
2552:
2547:
2542:
2537:
2532:
2527:
2522:
2517:
2512:
2507:
2502:
2496:
2493:
2492:
2482:
2481:
2478:
2477:
2474:
2473:
2470:
2469:
2467:
2466:
2461:
2456:
2450:
2448:
2441:
2435:
2434:
2431:
2430:
2428:
2427:
2421:
2419:
2413:
2412:
2410:
2409:
2404:
2399:
2394:
2389:
2384:
2378:
2376:
2370:
2369:
2367:
2366:
2361:
2356:
2351:
2346:
2340:
2338:
2329:
2323:
2322:
2319:
2318:
2316:
2315:
2310:
2305:
2300:
2295:
2290:
2285:
2280:
2275:
2270:
2264:
2262:
2256:
2255:
2253:
2252:
2247:
2242:
2237:
2232:
2227:
2222:
2217:
2212:
2206:
2204:
2195:
2185:
2184:
2172:
2171:
2168:
2167:
2165:
2164:
2159:
2154:
2149:
2144:
2139:
2133:
2130:
2129:
2119:
2118:
2115:
2114:
2112:
2111:
2106:
2101:
2096:
2091:
2085:
2082:
2081:
2071:
2070:
2067:
2066:
2064:
2063:
2058:
2053:
2048:
2043:
2038:
2033:
2028:
2022:
2019:
2018:
2005:
2004:
2001:
2000:
1998:
1997:
1992:
1987:
1982:
1977:
1971:
1968:
1967:
1957:
1956:
1953:
1952:
1950:
1949:
1944:
1939:
1934:
1929:
1924:
1919:
1913:
1910:
1909:
1895:
1894:
1891:
1890:
1888:
1887:
1882:
1877:
1872:
1867:
1862:
1857:
1852:
1847:
1842:
1837:
1832:
1827:
1822:
1816:
1813:
1812:
1799:
1798:
1790:
1789:
1782:
1775:
1767:
1761:
1760:
1755:
1734:
1733:External links
1731:
1728:
1727:
1717:. 21 July 2021
1702:
1676:
1651:
1636:
1612:
1592:
1572:
1560:
1539:
1527:
1515:
1503:
1487:
1475:
1463:
1448:
1411:
1392:
1367:
1337:
1330:
1300:
1299:
1297:
1294:
1293:
1292:
1285:
1282:
1273:
1270:
1269:
1268:
1221:
1217:
1213:
1209:
1205:
1180:
1179:
1178:
1170:
1166:
1143:
1139:
1135:
1080:
1077:
1074:
1069:
1066:
1061:
1057:
1050:
1044:
1041:
1038:
1033:
1030:
1025:
1021:
1014:
1011:
1002:
977:
947:
920:
903:
876:
870:
865:
827:
822:of Nagell and
812:Perfect powers
805:
789:
786:) = (11, 3, 5)
733:
685:
682:
677:
661:
658:
651:
647:
642:
638:
633:
629:
614:
613:
553:
550:
547:
531:
527:
524:
519:
515:
507:
502:
498:
465:
464:
450:
446:
442:
438:
434:
430:
426:
422:
412:
404:
400:
385:Main article:
382:
379:
330:
327:
315:
314:
293:
285:
262:
259:
256:
251:
248:
243:
239:
232:
229:
209:
206:
203:
183:
180:
177:
174:
171:
148:
145:
142:
137:
134:
129:
125:
118:
109:of the number
98:
77:repunit primes
33:natural number
15:
9:
6:
4:
3:
2:
3719:
3708:
3705:
3704:
3702:
3687:
3683:
3679:
3678:
3675:
3665:
3662:
3661:
3658:
3653:
3648:
3644:
3634:
3631:
3629:
3626:
3625:
3622:
3617:
3612:
3608:
3598:
3595:
3593:
3590:
3589:
3586:
3581:
3576:
3572:
3562:
3559:
3557:
3554:
3553:
3550:
3546:
3540:
3536:
3526:
3523:
3521:
3518:
3516:
3513:
3512:
3509:
3505:
3500:
3496:
3482:
3479:
3478:
3476:
3472:
3466:
3463:
3461:
3458:
3456:
3455:Polydivisible
3453:
3451:
3448:
3446:
3443:
3441:
3438:
3436:
3433:
3432:
3430:
3426:
3420:
3417:
3415:
3412:
3410:
3407:
3405:
3402:
3400:
3397:
3396:
3394:
3391:
3386:
3380:
3377:
3375:
3372:
3370:
3367:
3365:
3362:
3360:
3357:
3355:
3352:
3350:
3347:
3346:
3344:
3341:
3337:
3329:
3326:
3325:
3324:
3321:
3320:
3318:
3315:
3311:
3299:
3296:
3295:
3294:
3291:
3289:
3286:
3284:
3281:
3279:
3276:
3274:
3271:
3269:
3266:
3264:
3261:
3259:
3256:
3254:
3251:
3250:
3248:
3244:
3238:
3235:
3234:
3232:
3228:
3222:
3219:
3217:
3214:
3213:
3211:
3209:Digit product
3207:
3201:
3198:
3196:
3193:
3191:
3188:
3186:
3183:
3182:
3180:
3178:
3174:
3166:
3163:
3161:
3158:
3157:
3156:
3153:
3152:
3150:
3148:
3143:
3139:
3135:
3130:
3125:
3121:
3111:
3108:
3106:
3103:
3101:
3098:
3096:
3093:
3091:
3088:
3086:
3083:
3081:
3078:
3076:
3073:
3071:
3068:
3066:
3063:
3061:
3058:
3056:
3053:
3051:
3048:
3046:
3045:Erdős–Nicolas
3043:
3041:
3038:
3036:
3033:
3032:
3029:
3024:
3020:
3014:
3010:
2996:
2993:
2991:
2988:
2987:
2985:
2983:
2979:
2973:
2970:
2968:
2965:
2963:
2960:
2958:
2955:
2954:
2952:
2950:
2946:
2940:
2937:
2935:
2932:
2930:
2927:
2925:
2922:
2920:
2917:
2915:
2912:
2911:
2909:
2907:
2903:
2897:
2894:
2892:
2889:
2888:
2886:
2884:
2880:
2874:
2871:
2869:
2866:
2864:
2863:Superabundant
2861:
2859:
2856:
2854:
2851:
2849:
2846:
2844:
2841:
2839:
2836:
2834:
2831:
2829:
2826:
2824:
2821:
2819:
2816:
2814:
2811:
2809:
2806:
2804:
2801:
2799:
2796:
2794:
2791:
2789:
2786:
2784:
2781:
2779:
2776:
2774:
2771:
2769:
2766:
2765:
2763:
2761:
2757:
2753:
2749:
2745:
2740:
2736:
2726:
2723:
2721:
2718:
2716:
2713:
2711:
2708:
2706:
2703:
2701:
2698:
2696:
2693:
2691:
2688:
2686:
2683:
2681:
2678:
2676:
2673:
2671:
2668:
2667:
2664:
2660:
2655:
2651:
2641:
2638:
2636:
2633:
2631:
2628:
2626:
2623:
2622:
2619:
2615:
2610:
2606:
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:
2521:
2518:
2516:
2513:
2511:
2508:
2506:
2503:
2501:
2498:
2497:
2494:
2487:
2483:
2465:
2462:
2460:
2457:
2455:
2452:
2451:
2449:
2445:
2442:
2440:
2439:4-dimensional
2436:
2426:
2423:
2422:
2420:
2418:
2414:
2408:
2405:
2403:
2400:
2398:
2395:
2393:
2390:
2388:
2385:
2383:
2380:
2379:
2377:
2375:
2371:
2365:
2362:
2360:
2357:
2355:
2352:
2350:
2349:Centered cube
2347:
2345:
2342:
2341:
2339:
2337:
2333:
2330:
2328:
2327:3-dimensional
2324:
2314:
2311:
2309:
2306:
2304:
2301:
2299:
2296:
2294:
2291:
2289:
2286:
2284:
2281:
2279:
2276:
2274:
2271:
2269:
2266:
2265:
2263:
2261:
2257:
2251:
2248:
2246:
2243:
2241:
2238:
2236:
2233:
2231:
2228:
2226:
2223:
2221:
2218:
2216:
2213:
2211:
2208:
2207:
2205:
2203:
2199:
2196:
2194:
2193:2-dimensional
2190:
2186:
2182:
2177:
2173:
2163:
2160:
2158:
2155:
2153:
2150:
2148:
2145:
2143:
2140:
2138:
2137:Nonhypotenuse
2135:
2134:
2131:
2124:
2120:
2110:
2107:
2105:
2102:
2100:
2097:
2095:
2092:
2090:
2087:
2086:
2083:
2076:
2072:
2062:
2059:
2057:
2054:
2052:
2049:
2047:
2044:
2042:
2039:
2037:
2034:
2032:
2029:
2027:
2024:
2023:
2020:
2015:
2010:
2006:
1996:
1993:
1991:
1988:
1986:
1983:
1981:
1978:
1976:
1973:
1972:
1969:
1962:
1958:
1948:
1945:
1943:
1940:
1938:
1935:
1933:
1930:
1928:
1925:
1923:
1920:
1918:
1915:
1914:
1911:
1906:
1900:
1896:
1886:
1883:
1881:
1878:
1876:
1875:Perfect power
1873:
1871:
1868:
1866:
1865:Seventh power
1863:
1861:
1858:
1856:
1853:
1851:
1848:
1846:
1843:
1841:
1838:
1836:
1833:
1831:
1828:
1826:
1823:
1821:
1818:
1817:
1814:
1809:
1804:
1800:
1796:
1788:
1783:
1781:
1776:
1774:
1769:
1768:
1765:
1759:
1756:
1751:
1750:
1745:
1742:
1737:
1736:
1716:
1712:
1706:
1690:
1686:
1680:
1665:
1661:
1655:
1647:
1640:
1631:
1627:
1623:
1616:
1607:
1603:
1596:
1587:
1583:
1576:
1569:
1568:Schott (2010)
1564:
1555:
1550:
1543:
1536:
1535:Schott (2010)
1531:
1524:
1523:Schott (2010)
1519:
1512:
1511:Schott (2010)
1507:
1501:
1497:
1491:
1484:
1483:Schott (2010)
1479:
1472:
1471:Schott (2010)
1467:
1459:
1452:
1444:
1440:
1436:
1432:
1425:
1421:
1415:
1407:
1403:
1396:
1381:
1377:
1376:"FAQ on GETs"
1371:
1363:
1359:
1355:
1348:
1341:
1333:
1327:
1323:
1318:
1317:
1308:
1306:
1301:
1291:
1288:
1287:
1281:
1279:
1278:numerological
1265:
1261:
1256:
1251:
1247:
1242:
1238:
1234:
1230:
1226:
1222:
1203:
1198:
1194:
1189:
1185:
1181:
1176:
1164:
1160:
1156:
1152:
1148:
1144:
1133:
1129:
1125:
1121:
1117:
1113:
1112:
1109:
1105:
1101:
1097:
1078:
1075:
1072:
1067:
1064:
1059:
1055:
1048:
1042:
1039:
1036:
1031:
1028:
1023:
1019:
1012:
1009:
1000:
996:
992:
987:
982:
978:
975:
970:
966:. (sequence
963:
959:
950:
946:
942:
938:
934:
925:
921:
918:
913:
908:
904:
901:
897:
892:
888:
883:
879:
878:
874:
869:
863:
858:
850:
846:
842:
838:
834:
830:
826:
825:
821:
817:
813:
809:
803:
799:
795:
785:
781:
777:
771:
767:
766:Trygve Nagell
760:
756:
752:
748:
744:
740:
736:
732:
731:
727:
723:
719:
715:
711:
707:
703:
699:
695:
691:
681:
675:
659:
656:
649:
645:
640:
636:
631:
627:
619:
618:Fermat number
611:
606:
601:
600:
599:
596:
591:
589:
585:
581:
576:
572:
567:
551:
548:
545:
529:
525:
522:
517:
513:
505:
500:
496:
487:
482:
480:
475:
470:
462:
457:
420:
419:
418:
415:
410:
398:
394:
388:
387:Repunit prime
377:
375:
371:
365:
361:
355:
349:
342:
340:
336:
335:Beiler (1966)
326:
324:
320:
312:
307:
302:
301:
300:
296:
291:
282:
277:
260:
257:
254:
249:
246:
241:
237:
230:
227:
207:
204:
201:
181:
178:
175:
172:
169:
146:
143:
140:
135:
132:
127:
123:
116:
96:
89:
84:
82:
78:
74:
70:
66:
62:
58:
54:
50:
46:
42:
38:
34:
30:
27:or sometimes
26:
22:
3419:Transposable
3358:
3283:Narcissistic
3190:Digital root
3110:Super-Poulet
3070:Jordan–Pólya
3019:prime factor
2924:Noncototient
2891:Almost prime
2873:Superperfect
2848:Refactorable
2843:Quasiperfect
2818:Hyperperfect
2659:Pseudoprimes
2630:Wall–Sun–Sun
2565:Ordered Bell
2535:Fuss–Catalan
2447:non-centered
2397:Dodecahedral
2374:non-centered
2260:non-centered
2162:Wolstenholme
1907:× 2 ± 1
1904:
1903:Of the form
1870:Eighth power
1850:Fourth power
1747:
1719:. Retrieved
1715:Cosmopolitan
1714:
1705:
1693:. Retrieved
1688:
1679:
1668:. Retrieved
1666:. 2023-06-29
1663:
1654:
1645:
1639:
1629:
1625:
1615:
1605:
1601:
1595:
1585:
1581:
1575:
1570:, Theorem 1.
1563:
1542:
1530:
1518:
1506:
1490:
1485:, Theorem 4.
1478:
1473:, Theorem 2.
1466:
1457:
1451:
1434:
1430:
1414:
1405:
1395:
1383:. Retrieved
1379:
1370:
1353:
1340:
1315:
1275:
1240:
1228:
1224:
1201:
1187:
1183:
1162:
1158:
1154:
1150:
1146:
1131:
1127:
1123:
1119:
1115:
1107:
1103:
1099:
1095:
980:
961:
957:
948:
944:
940:
936:
932:
923:
906:
899:
895:
881:
872:
853:
848:
844:
840:
836:
832:
828:
815:
810:
801:
797:
793:
783:
779:
775:
769:
763:
758:
754:
750:
746:
742:
738:
734:
725:
721:
717:
713:
709:
705:
704:with 1 <
701:
697:
687:
615:
592:
565:
483:
466:
429:, 31 = 11111
413:
408:
390:
373:
369:
363:
359:
353:
347:
344:
332:
316:
294:
289:
280:
278:
85:
28:
24:
18:
3440:Extravagant
3435:Equidigital
3390:permutation
3349:Palindromic
3323:Automorphic
3221:Sum-product
3200:Sum-product
3155:Persistence
3050:Erdős–Woods
2972:Untouchable
2853:Semiperfect
2803:Hemiperfect
2464:Tesseractic
2402:Icosahedral
2382:Tetrahedral
2313:Dodecagonal
2014:Recursively
1885:Prime power
1860:Sixth power
1855:Fifth power
1835:Power of 10
1793:Classes of
1588:(1): 17–18.
1500:Prime Pages
1437:: 209–212.
1235:created by
847:> 1 and
449:, 157 = 111
319:imageboards
49:portmanteau
3652:Graphemics
3525:Pernicious
3379:Undulating
3354:Pandigital
3328:Trimorphic
2929:Nontotient
2778:Arithmetic
2392:Octahedral
2293:Heptagonal
2283:Pentagonal
2268:Triangular
2109:Sierpiński
2031:Jacobsthal
1830:Power of 3
1825:Power of 2
1744:"Repdigit"
1721:2023-08-28
1670:2023-08-28
1664:Glamour UK
1632:: 147–168.
1554:1903.04577
1354:Quadrature
1296:References
1272:Numerology
983:(sequence
692:, and the
568:listed in
541:with
441:, 73 = 111
437:, 43 = 111
425:, 13 = 111
356:such that
3409:Parasitic
3258:Factorion
3185:Digit sum
3177:Digit sum
2995:Fortunate
2982:Primorial
2896:Semiprime
2833:Practical
2798:Descartes
2793:Deficient
2783:Betrothed
2625:Wieferich
2454:Pentatope
2417:pyramidal
2308:Decagonal
2303:Nonagonal
2298:Octagonal
2288:Hexagonal
2147:Practical
2094:Congruent
2026:Fibonacci
1990:Loeschian
1749:MathWorld
1695:28 August
1498:" at The
1458:Hypermath
1385:March 14,
1076:−
1065:−
1040:−
1029:−
824:Ljunggren
549:≥
523:−
345:A number
339:Fortaleza
297:− 1
258:−
247:−
231:×
144:−
133:−
29:monodigit
3701:Category
3481:Friedman
3414:Primeval
3359:Repdigit
3316:-related
3263:Kaprekar
3237:Meertens
3160:Additive
3147:dynamics
3055:Friendly
2967:Sociable
2957:Amicable
2768:Abundant
2748:dynamics
2570:Schröder
2560:Narayana
2530:Eulerian
2520:Delannoy
2515:Dedekind
2336:centered
2202:centered
2089:Amenable
2046:Narayana
2036:Leonardo
1932:Mersenne
1880:Powerful
1820:Achilles
1608:: 17–20.
1422:(1974).
1284:See also
761:>= 2.
745:+ ... +
716:in base
372:in base
358:1 <
73:repunits
25:repdigit
3654:related
3618:related
3582:related
3580:Sorting
3465:Vampire
3450:Harshad
3392:related
3364:Repunit
3278:Lychrel
3253:Dudeney
3105:Størmer
3100:Sphenic
3085:Regular
3023:divisor
2962:Perfect
2858:Sublime
2828:Perfect
2555:Motzkin
2510:Catalan
2051:Padovan
1985:Leyland
1980:Idoneal
1975:Hilbert
1947:Woodall
1443:0354535
1264:A279930
1262::
1250:A329383
1248::
1223:In the
1197:A284758
1195::
1175:repunit
1161:,
1157:,
1153:,
1149:,
1130:,
1126:,
1122:,
1118:,
989:in the
986:A290015
972:in the
969:A288783
956:1 <
915:in the
912:A220570
891:A220136
889::
860:in the
857:A208242
851:> 2.
845:b, n, t
694:squares
608:in the
605:A220627
575:A004023
573::
486:decimal
477:in the
474:A306759
459:in the
456:A085104
421:7 = 111
397:repunit
329:History
309:in the
306:A125134
45:decimal
3520:Odious
3445:Frugal
3399:Cyclic
3388:Digit-
3095:Smooth
3080:Pronic
3040:Cyclic
3017:Other
2990:Euclid
2640:Wilson
2614:Primes
2273:Square
2142:Polite
2104:Riesel
2099:Knödel
2061:Perrin
1942:Thabit
1927:Fermat
1917:Cullen
1840:Square
1808:Powers
1689:Allure
1441:
1328:
1142:, and,
839:+...+
831:= 1 +
737:= 1 +
690:primes
580:binary
536:
467:While
350:> 0
162:where
65:999999
63:, and
3561:Prime
3556:Lucky
3545:sieve
3474:Other
3460:Smith
3340:Digit
3298:Happy
3273:Keith
3246:Other
3090:Rough
3060:Giuga
2525:Euler
2387:Cubic
2041:Lucas
1937:Proth
1549:arXiv
1427:(PDF)
1380:4chan
1350:(PDF)
1169:= 111
1138:= 111
1094:with
979:The 2
960:<
954:with
843:with
749:with
708:<
616:If a
433:= 111
393:prime
362:<
323:4chan
321:like
39:in a
37:digit
31:is a
3515:Evil
3195:Self
3145:and
3035:Blum
2746:and
2550:Lobb
2505:Cake
2500:Bell
2250:Star
2157:Ulam
2056:Pell
1845:Cube
1697:2023
1387:2007
1326:ISBN
1260:OEIS
1246:OEIS
1216:= 22
1212:= 44
1208:= 55
1193:OEIS
991:OEIS
974:OEIS
917:OEIS
887:OEIS
862:OEIS
700:and
610:OEIS
571:OEIS
484:The
479:OEIS
461:OEIS
311:OEIS
205:<
179:<
173:<
88:base
61:4444
23:, a
3633:Ban
3021:or
2540:Lah
1358:doi
964:– 1
928:≥ 8
366:– 1
57:666
19:In
3703::
1746:.
1713:.
1687:.
1662:.
1630:48
1628:.
1624:.
1606:25
1584:.
1439:MR
1435:12
1433:.
1429:.
1404:.
1378:.
1352:.
1324:.
1322:83
1304:^
1218:19
1171:90
1106:,
1098:,
1001::
976:).
951:–1
945:aa
943:=
939:Ă—
935:=
919:).
868:.
866:18
835:+
800:,
796:,
782:,
778:,
753:,
741:+
714:xx
680:.
678:73
590:.
514:10
451:12
405:10
313:).
276:.
255:10
238:10
59:,
55:,
53:11
1905:a
1786:e
1779:t
1772:v
1752:.
1724:.
1699:.
1673:.
1634:.
1610:.
1590:.
1586:3
1557:.
1551::
1445:.
1408:.
1389:.
1364:.
1360::
1334:.
1267:.
1220:.
1214:9
1210:7
1206:3
1188:k
1167:2
1163:n
1159:m
1155:y
1151:x
1147:p
1145:(
1140:5
1136:2
1132:n
1128:m
1124:y
1120:x
1116:p
1114:(
1108:m
1104:n
1100:y
1096:x
1079:1
1073:y
1068:1
1060:n
1056:y
1049:=
1043:1
1037:x
1032:1
1024:m
1020:x
1013:=
1010:p
962:b
958:a
949:b
941:b
937:a
933:n
902:.
896:k
882:n
873:k
849:q
841:b
837:b
833:b
829:n
816:b
806:7
802:q
798:b
794:p
790:3
784:q
780:b
776:p
774:(
770:p
759:b
755:q
751:p
747:b
743:b
739:b
735:p
726:p
722:p
718:y
710:y
706:x
702:y
698:x
660:1
657:+
650:n
646:2
641:2
637:=
632:n
628:F
612:)
566:n
552:3
546:n
530:9
526:1
518:n
506:=
501:n
497:R
463:)
447:2
443:8
439:6
435:5
431:2
427:3
423:2
414:b
409:b
401:4
374:b
370:n
364:n
360:b
354:b
348:n
295:n
290:n
286:8
261:1
250:1
242:5
228:7
208:y
202:1
182:B
176:x
170:0
147:1
141:B
136:1
128:y
124:B
117:x
97:B
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