2137:
2415:(PRP) test, based on development from Robert Gerbicz in 2017, and a simple way to verify tests developed by Krzysztof Pietrzak in 2018. Due to the low error rate and ease of proof, this nearly halved the computing time to rule out potential primes over the Lucas-Lehmer test (as two users would no longer have to perform the same test to confirm the other's result), although exponents passing the PRP test still require one to confirm their primality.
11798:
1237:
2132:
During the era of manual calculation, all the exponents up to and including 257 were tested with the Lucas–Lehmer test and found to be composite. A notable contribution was made by retired Yale physics professor Horace
Scudder Uhler, who did the calculations for exponents 157, 167, 193, 199, 227, and
5234:
2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667,
3408:
2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667,
1895:
and got the same number, then returned to his seat (to applause) without speaking. He later said that the result had taken him "three years of
Sundays" to find. A correct list of all Mersenne primes in this number range was completed and rigorously verified only about three centuries after Mersenne
944:
5251:
3, 4, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701, 1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321, 986191, 4031399, ..., 13347311, 13372531, ...
1874:, though Mersenne claimed it was composite, and for this reason it is sometimes called Pervushin's number. This was the second-largest known prime number, and it remained so until 1911. Lucas had shown another error in Mersenne's list in 1876 by demonstrating that
3440:
for links to more information. The special number field sieve can factorize numbers with more than one large factor. If a number has only one very large factor then other algorithms can factorize larger numbers by first finding small factors and then running a
2133:
229. Unfortunately for those investigators, the interval they were testing contains the largest known relative gap between
Mersenne primes: the next Mersenne prime exponent, 521, would turn out to be more than four times as large as the previous record of 127.
4756:
3, 2, 2, 5, 2, 3, 2, 3, 5, 5, 2, 3, 2, 3, 3, 7, 2, 17, 2, 3, 3, 11, 2, 3, 11, 0, 3, 7, 2, 109, 2, 5, 3, 11, 31, 5, 2, 3, 53, 17, 2, 5, 2, 103, 7, 5, 2, 7, 1153, 3, 7, 21943, 2, 3, 37, 53, 3, 17, 2, 7, 2, 3, 0, 19, 7, 3, 2, 11, 3, 5, 2, ... (sequence
6732:
2, 2, 2, 3, 2, 2, 7, 2, 2, 3, 2, 17, 3, 2, 2, 5, 3, 2, 5, 2, 2, 229, 2, 3, 3, 2, 3, 3, 2, 2, 5, 3, 2, 3, 2, 2, 3, 3, 2, 7, 2, 3, 37, 2, 3, 5, 58543, 2, 3, 2, 2, 3, 2, 2, 3, 2, 5, 3, 4663, 54517, 17, 3, 2, 5, 2, 3, 3, 2, 2, 47, 61, 19, ... (sequence
4714:
2, 3, 2, 3, 2, 5, 3, 0, 2, 17, 2, 5, 3, 3, 2, 3, 2, 19, 3, 3, 2, 5, 3, 0, 7, 3, 2, 5, 2, 7, 0, 3, 13, 313, 2, 13, 3, 349, 2, 3, 2, 5, 5, 19, 2, 127, 19, 0, 3, 4229, 2, 11, 3, 17, 7, 3, 2, 3, 2, 7, 3, 5, 0, 19, 2, 19, 5, 3, 2, 3, 2, ... (sequence
4285:
2, 3, 5, 7, 11, 19, 29, 47, 73, 79, 113, 151, 157, 163, 167, 239, 241, 283, 353, 367, 379, 457, 997, 1367, 3041, 10141, 14699, 27529, 49207, 77291, 85237, 106693, 160423, 203789, 364289, 991961, 1203793, 1667321, 3704053, 4792057, ... (sequence
4812:
2, 2, 2, 2, 5, 2, 2, 2, 10, 6, 2, 61, 14, 15, 5, 24, 19, 2, 46, 3, 11, 22, 41, 2, 12, 22, 3, 2, 12, 86, 2, 7, 13, 11, 5, 29, 56, 30, 44, 60, 304, 5, 74, 118, 33, 156, 46, 183, 72, 606, 602, 223, 115, 37, 52, 104, 41, 6, 338, 217, ... (sequence
6766:
1, 1, 1, 1, 5, 1, 1, 1, 5, 2, 1, 39, 6, 4, 12, 2, 2, 1, 6, 17, 46, 7, 5, 1, 25, 2, 41, 1, 12, 7, 1, 7, 327, 7, 8, 44, 26, 12, 75, 14, 51, 110, 4, 14, 49, 286, 15, 4, 39, 22, 109, 367, 22, 67, 27, 95, 80, 149, 2, 142, 3, 11, ... (sequence
4837:
3, 2, 2, 2, 2, 2, 2, 2, 2, 7, 2, 16, 61, 2, 6, 10, 6, 2, 5, 46, 18, 2, 49, 16, 70, 2, 5, 6, 12, 92, 2, 48, 89, 30, 16, 147, 19, 19, 2, 16, 11, 289, 2, 12, 52, 2, 66, 9, 22, 5, 489, 69, 137, 16, 36, 96, 76, 117, 26, 3, ... (sequence
1232:{\displaystyle {\begin{aligned}2^{ab}-1&=(2^{a}-1)\cdot \left(1+2^{a}+2^{2a}+2^{3a}+\cdots +2^{(b-1)a}\right)\\&=(2^{b}-1)\cdot \left(1+2^{b}+2^{2b}+2^{3b}+\cdots +2^{(a-1)b}\right).\end{aligned}}}
2345:
for their discovery of a very nearly 13-million-digit
Mersenne prime. The prize, finally confirmed in October 2009, is for the first known prime with at least 10 million digits. The prime was found on a
1320:
that greatly aids this task, making it much easier to test the primality of
Mersenne numbers than that of most other numbers of the same size. The search for the largest known prime has somewhat of a
315:
asserts a one-to-one correspondence between even perfect numbers and
Mersenne primes. Many of the largest known primes are Mersenne primes because Mersenne numbers are easier to check for primality.
1280:
The evidence at hand suggests that a randomly selected
Mersenne number is much more likely to be prime than an arbitrary randomly selected odd integer of similar size. Nonetheless, prime values of
949:
3428:
Since they are prime numbers, Mersenne primes are divisible only by 1 and themselves. However, not all
Mersenne numbers are Mersenne primes. Mersenne numbers are very good test cases for the
1312:
The current lack of any simple test to determine whether a given
Mersenne number is prime makes the search for Mersenne primes a difficult task, since Mersenne numbers grow very rapidly. The
419:
2379:(a number with 22,338,618 digits), as a result of a search executed by a GIMPS server network. This was the fourth Mersenne prime discovered by Cooper and his team in the past ten years.
4915:
2353:
On April 12, 2009, a GIMPS server log reported that a 47th
Mersenne prime had possibly been found. The find was first noticed on June 4, 2009, and verified a week later. The prime is
1754:
His list replicated the known primes of his time with exponents up to 19. His next entry, 31, was correct, but the list then became largely incorrect, as Mersenne mistakenly included
4497:
790:
669:
2397:(a number with 23,249,425 digits), as a result of a search executed by a GIMPS server network. The discovery was made by a computer in the offices of a church in the same town.
8276:
8196:
8330:– status page gives various statistics on search progress, typically updated every week, including progress towards proving the ordering of the largest known Mersenne primes
1859:
719:
2182:
836:
1458:
1507:
450:
2357:. Although it is chronologically the 47th Mersenne prime to be discovered, it is smaller than the largest known at the time, which was the 45th to be discovered.
1343:
of Mersenne number order requires knowing the factorization of that number, so Mersenne primes allow one to find primitive polynomials of very high order. Such
5268:
2, 3, 5, 17, 29, 31, 53, 59, 101, 277, 647, 1061, 2381, 2833, 3613, 3853, 3929, 5297, 7417, 90217, 122219, 173191, 256199, 336353, 485977, 591827, 1059503, ...
2140:
Graph of number of digits in largest known Mersenne prime by year – electronic era. The vertical scale is logarithmic in the number of digits, thus being a
1728:
The first 64 prime exponents with those corresponding to Mersenne primes shaded in cyan and in bold, and those thought to do so by Mersenne in red and bold
9900:
4378:
2, 5, 7, 11, 17, 19, 79, 163, 193, 239, 317, 353, 659, 709, 1049, 1103, 1759, 2029, 5153, 7541, 9049, 10453, 23743, 255361, 534827, 2237561, ... (sequence
3460:
is a 3,829,294-digit probable prime. It was discovered by a GIMPS participant with nickname "Funky Waddle". As of September 2022, the Mersenne number
3254:, so −2 is the product of a residue and a nonresidue and hence it is a nonresidue, which is a contradiction. Hence, the former congruence must be true and
7947:
Zalnezhad, Ali; Zalnezhad, Hossein; Shabani, Ghasem; Zalnezhad, Mehdi (March 2015). "Relationships and Algorithm in order to Achieve the Largest Primes".
5300:
2, 3, 5, 7, 13, 23, 43, 281, 359, 487, 577, 1579, 1663, 1741, 3191, 9209, 11257, 12743, 13093, 17027, 26633, 104243, 134227, 152287, 700897, 1205459, ...
3467:
is the smallest composite Mersenne number with no known factors; it has no prime factors below 2, and is very unlikely to have any factors below 10 (~2).
3436:
is the record-holder, having been factored with a variant of the special number field sieve that allows the factorization of several numbers at once. See
1742:, who compiled what was supposed to be a list of Mersenne primes with exponents up to 257. The exponents listed by Mersenne in 1644 were as follows:
358:
Many fundamental questions about Mersenne primes remain unresolved. It is not even known whether the set of Mersenne primes is finite or infinite.
9387:
5332:
2, 3, 7, 17, 59, 283, 311, 383, 499, 521, 541, 599, 1193, 1993, 2671, 7547, 24019, 46301, 48121, 68597, 91283, 131497, 148663, 184463, 341233, ...
8531:
455:
It is also not known whether infinitely many Mersenne numbers with prime exponents are composite, although this would follow from widely believed
8475:
8463:
8619:
7456:
1344:
8990:
362:
8543:
8119:
1814:
is indeed prime, as Mersenne claimed. This was the largest known prime number for 75 years until 1951, when Ferrier found a larger prime,
12297:
11879:
3395:
334:
project. In December 2020, a major milestone in the project was passed after all exponents below 100 million were checked at least once.
2400:
On December 21, 2018, it was announced that The Great Internet Mersenne Prime Search (GIMPS) discovered the largest known prime number,
1887:
in 1903. Without speaking a word, he went to a blackboard and raised 2 to the 67th power, then subtracted one, resulting in the number
7319:
1324:. Consequently, a large amount of computer power has been expended searching for new Mersenne primes, much of which is now done using
12292:
7055:
6774:
6740:
4845:
4820:
4764:
4722:
4575:
4544:
4532:
4403:
4385:
4318:
4293:
3744:
3477:
3416:
2430:
576:
270:
240:
114:
9893:
9072:
8221:
352:
7571:
11919:
8141:
1336:
8995:
8385:
7914:
7802:
7686:
2338:
2213:
11874:
8909:
8508:
8439:
7632:
7484:
6829:
5390:
3, 43, 59, 191, 223, 349, 563, 709, 743, 1663, 5471, 17707, 19609, 35449, 36697, 45259, 91493, 246497, 265007, 289937, ...
327:
10700:
9886:
8434:
1340:
7511:
3432:
algorithm, so often the largest number factorized with this algorithm has been a Mersenne number. As of June 2019,
11834:
10695:
8612:
7897:
Solinas, Jerome A. (1 January 2011). "Generalized Mersenne Prime". In Tilborg, Henk C. A. van; Jajodia, Sushil (eds.).
7703:
12287:
12193:
10710:
10690:
7879:
372:
11403:
10983:
9246:
8393:
7553:
12013:
12008:
12003:
11998:
11993:
11988:
11983:
11978:
3103:
are coprime. Consequently, a prime number divides at most one prime-exponent Mersenne number. That is, the set of
2048:
1313:
17:
4863:
4300:
Like the sequence of exponents for usual Mersenne primes, this sequence contains only (rational) prime numbers.
10705:
9327:
7378:
2365:
2342:
4304:
12228:
12108:
11489:
8605:
8315:
7403:
3437:
2361:
1348:
205:. Therefore, an equivalent definition of the Mersenne primes is that they are the prime numbers of the form
6803:
5284:
3, 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551, 36913, 43063, 49681, 57917, 483611, 877843, ...
4452:
12113:
12103:
12043:
11155:
10805:
10474:
10267:
9449:
9107:
9020:
3775:
2206:
5438:
3, 7, 11, 13, 47, 127, 149, 181, 619, 929, 3407, 10949, 13241, 13873, 16519, 201359, 396413, 1888279, ...
2287:
was the first with more than a million. In general, the number of digits in the decimal representation of
2188:
The search for Mersenne primes was revolutionized by the introduction of the electronic digital computer.
740:
12058:
11912:
11331:
11190:
11021:
10835:
10825:
10479:
10459:
9474:
8310:
2877:, which asserts the infinitude of primes, distinct from the proof written by Euclid: for every odd prime
2341:
participating in the Great Internet Mersenne Prime Search (GIMPS) won part of a $ 100,000 prize from the
618:
11160:
8488:
sequence A250197 (Numbers n such that the left Aurifeuillian primitive part of 2^n+1 is prime)
12302:
12203:
11280:
10903:
10745:
10660:
10469:
10451:
10345:
10335:
10325:
10161:
9382:
8940:
8528:
6835:
4240:
3998:
3429:
2736:
1905:
1356:
319:
11185:
3281:
With the exception of 1, a Mersenne number cannot be a perfect power. That is, and in accordance with
2136:
12188:
12053:
11408:
10953:
10574:
10360:
10355:
10350:
10340:
10317:
8472:
8460:
8305:
8227:
8147:
2389:
On January 3, 2018, it was announced that Jonathan Pace, a 51-year-old electrical engineer living in
2019:
1871:
11165:
8419:
4974:, the division is necessary for there to be any chance of finding prime numbers.) We can ask which
1817:
682:
12312:
12307:
12266:
10830:
10740:
10393:
9411:
7748:
7231:
2382:
On September 2, 2016, the Great Internet Mersenne Prime Search finished verifying all tests below M
2143:
1405:
1368:
795:
312:
188:
11519:
11484:
11270:
11180:
11054:
11029:
10938:
10928:
10650:
10540:
10522:
10442:
9015:
8540:
8455:
6018:
2, 3, 5, 7, 11, 17, 19, 41, 53, 109, 167, 2207, 3623, 5059, 5471, 7949, 21211, 32993, 60251, ...
3282:
2411:
In late 2020, GIMPS began using a new technique to rule out potential Mersenne primes called the
1418:
7669:
Kleinjung, Thorsten; Bos, Joppe W.; Lenstra, Arjen K. (2014). "Mersenne Factorization Factory".
4392:
The norms (that is, squares of absolute values) of these Eisenstein primes are rational primes:
1463:
12261:
12156:
11905:
11869:
11827:
11779:
11049:
10923:
10554:
10330:
10110:
10037:
9532:
8661:
6870:
6818:
5550:
2, 3, 11, 31, 43, 47, 59, 107, 811, 2819, 4817, 9601, 33581, 38447, 41341, 131891, 196337, ...
3774:
steps, assuming no mistakes are made. The number of rice grains on the whole chessboard in the
2372:(a number with 17,425,170 digits), as a result of a search executed by a GIMPS server network.
2047:
The most efficient method presently known for testing the primality of Mersenne numbers is the
1509:
is a Perfect Number. (Perfect Numbers are Triangular Numbers whose base is a Mersenne Prime.)
326:, is a Mersenne prime. Since 1997, all newly found Mersenne primes have been discovered by the
12223:
12213:
12151:
11743:
11383:
11034:
10888:
10815:
9970:
9869:
9459:
9112:
7930:
7342:
5774:
7, 11, 17, 29, 31, 79, 113, 131, 139, 4357, 44029, 76213, 83663, 173687, 336419, 615997, ...
5422:
2, 5, 7, 13, 19, 37, 59, 67, 79, 307, 331, 599, 1301, 12263, 12589, 18443, 20149, 27983, ...
3986:
2965:
2390:
1904:
Fast algorithms for finding Mersenne primes are available, and as of June 2023, the six
1325:
331:
8450:
7301:
290:. Sometimes, however, Mersenne numbers are defined to have the additional requirement that
11854:
11676:
11570:
11534:
11275:
10998:
10978:
10795:
10464:
10252:
9439:
7327:
6808:
5880:
4, 7, 13, 31, 43, 269, 353, 383, 619, 829, 877, 4957, 5711, 8317, 21739, 24029, 38299, ...
5454:
5, 67, 101, 103, 229, 347, 4013, 23297, 30133, 177337, 193939, 266863, 277183, 335429, ...
5406:
13, 19, 23, 31, 47, 127, 223, 281, 2083, 5281, 7411, 7433, 19051, 27239, 35863, 70327, ...
3470:
The table below shows factorizations for the first 20 composite Mersenne numbers (sequence
3127:
2217:
604:
460:
428:
10755:
10224:
5374:
3, 5, 19, 37, 173, 211, 227, 619, 977, 1237, 2437, 5741, 13463, 23929, 81223, 121271, ...
8:
12098:
11398:
11262:
11257:
11225:
10988:
10963:
10958:
10933:
10863:
10859:
10790:
10680:
10512:
10308:
10277:
9434:
9092:
7408:
7343:"A Brief History of the Investigations on Mersenne Numbers and the Latest Immense Primes"
3445:
on the cofactor. As of September 2022, the largest completely factored number (with
3135:
2874:
2221:
908:
463:
366:
9097:
8093:
8067:
8019:
4537:
corresponding to primes 11, 1111111111111111111, 11111111111111111111111, ... (sequence
2224:. It was the first Mersenne prime to be identified in thirty-eight years; the next one,
12146:
12048:
12038:
11859:
11801:
11555:
11550:
11464:
11438:
11336:
11315:
11087:
10968:
10918:
10840:
10810:
10750:
10517:
10497:
10428:
10141:
9542:
9479:
9469:
9454:
9087:
8945:
8866:
8041:
7997:
7948:
7489:
7152:
6933:
6929:
4204:
4167:
4152:
4136:
3830:
3275:
2480:
1409:
1404:
proved that, conversely, all even perfect numbers have this form. This is known as the
467:
10685:
8513:
8357:
7967:
7844:
12063:
11820:
11797:
11695:
11640:
11494:
11469:
11443:
10898:
10893:
10820:
10800:
10785:
10507:
10489:
10408:
10398:
10383:
10146:
9511:
9486:
9464:
9444:
9067:
9039:
8732:
8580:
8561:
7910:
7798:
7682:
5518:
2, 5, 11, 13, 23, 61, 83, 421, 1039, 1511, 31237, 60413, 113177, 135647, 258413, ...
5486:
2, 3, 5, 7, 17, 19, 109, 509, 661, 709, 1231, 12889, 13043, 26723, 43963, 44789, ...
3104:
2193:
1884:
592:
11220:
8564:
7284:
7267:
4568:
2, 5, 11, 109, 193, 1483, 11353, 21419, 21911, 24071, 106859, 139739, ... (sequence
1799:(which are prime). Mersenne gave little indication of how he came up with his list.
12248:
12068:
12018:
11886:
11731:
11524:
11110:
11082:
11072:
11064:
10948:
10913:
10908:
10875:
10569:
10532:
10423:
10418:
10413:
10403:
10375:
10262:
10209:
10166:
10105:
9421:
9406:
9343:
9190:
9057:
8960:
8583:
7902:
7674:
7585:
7529:
7279:
6813:
5534:
2, 3, 7, 29, 71, 127, 271, 509, 1049, 6389, 6883, 10613, 19889, 79987, 608099, ...
4216:
4200:
3737:
The number of factors for the first 500 Mersenne numbers can be found at (sequence
1883:
was composite without finding a factor. No factor was found until a famous talk by
1352:
907:, all other Mersenne numbers are also congruent to 3 (mod 4). Consequently, in the
198:
10214:
7649:
7430:
7298:
2233:, was found by the computer a little less than two hours later. Three more —
2006:
1802:
12208:
11707:
11596:
11529:
11455:
11378:
11352:
11170:
10883:
10675:
10645:
10635:
10630:
10296:
10204:
10151:
9995:
9935:
9122:
9082:
8965:
8930:
8894:
8849:
8702:
8690:
8547:
8535:
8479:
8467:
7906:
7678:
7636:
6875:
6840:
2978:
2209:
1239:
This rules out primality for Mersenne numbers with a composite exponent, such as
8425:
7629:
7205:
7098:
is a perfect fourth power, it can be shown that there are at most two values of
2350:
745 on August 23, 2008. This was the eighth Mersenne prime discovered at UCLA.
1309:
is prime for only 43 of the first two million prime numbers (up to 32,452,843).
12238:
12129:
12028:
11843:
11712:
11580:
11565:
11429:
11393:
11368:
11244:
11215:
11200:
11077:
10973:
10943:
10670:
10625:
10502:
10100:
10095:
10090:
10062:
9960:
9945:
9923:
9910:
9527:
9501:
9398:
9266:
9117:
9077:
9062:
8934:
8825:
8790:
8745:
8670:
7883:
6896:
6845:
5235:
42643801, 43112609, 57885161, ..., 74207281, ..., 77232917, ..., 82589933, ...
5156:
4258:
4192:
3826:
3756:
3446:
3442:
3322:
2412:
2405:
2317:
2069:
1993:
1980:
1739:
1401:
1385:
1321:
1317:
1263:, this is not the case, and the smallest counterexample is the Mersenne number
225:
184:
39:
8327:
5470:
2, 3, 17, 19, 47, 101, 1709, 2539, 5591, 6037, 8011, 19373, 26489, 27427, ...
3451:
2 − 1 = 1,119,429,257 × 175,573,124,547,437,977 × 8,480,999,878,421,106,991 ×
2968:, every prime modulus in which the number 2 has a square root is congruent to
2375:
On January 19, 2016, Cooper published his discovery of a 49th Mersenne prime,
233:
which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, ... (sequence
12281:
12198:
11973:
11953:
11928:
11635:
11619:
11560:
11514:
11210:
11195:
11105:
10388:
10257:
10219:
10176:
10057:
10042:
10032:
9990:
9980:
9955:
9878:
9537:
9302:
9166:
9139:
8975:
8840:
8778:
8769:
8754:
8717:
8643:
7515:
7461:
7364:
7178:
6865:
6855:
6793:
4591:
3935:
3811:
3799:
2347:
2205:, by this means was achieved at 10:00 pm on January 30, 1952, using the U.S.
1735:
1332:
260:
8405:
7723:
7605:
7219:
6563:
2, 3, 5, 19, 97, 109, 317, 353, 701, 9739, 14951, 37573, 46889, 769543, ...
4307:(that is, squares of absolute values) of these numbers are rational primes:
12233:
11671:
11660:
11575:
11413:
11388:
11305:
11205:
11175:
11150:
11134:
11039:
11006:
10729:
10640:
10579:
10156:
10052:
9985:
9965:
9940:
9858:
9853:
9848:
9843:
9838:
9833:
9828:
9823:
9818:
9813:
9808:
9803:
9798:
9793:
9788:
9783:
9778:
9773:
9768:
9763:
9758:
9753:
9748:
9743:
9738:
9733:
9728:
9723:
9718:
9713:
9708:
9703:
9698:
9693:
9688:
9491:
9214:
8980:
8970:
8955:
8950:
8914:
8628:
7248:
6850:
6798:
6255:
5, 19, 67, 107, 593, 757, 1801, 2243, 2383, 6043, 10181, 11383, 15629, ...
5742:
2, 11, 31, 173, 271, 547, 1823, 2111, 5519, 7793, 22963, 41077, 49739, ...
5502:
4, 5, 7, 19, 29, 61, 137, 883, 1381, 1823, 5227, 25561, 29537, 300893, ...
5043:
4026:
3795:
2404:, having 24,862,048 digits. A computer volunteered by Patrick Laroche from
2041:
1335:, making them popular choices when a prime modulus is desired, such as the
722:
256:
158:
154:
95:
8445:
8333:
6579:
2, 5, 11, 109, 193, 1483, 11353, 21419, 21911, 24071, 106859, 139739, ...
12243:
12183:
11630:
11505:
11310:
10774:
10665:
10620:
10615:
10365:
10272:
10171:
10000:
9975:
9950:
9683:
9678:
9673:
9668:
9663:
9658:
9653:
9648:
9643:
9638:
9633:
9628:
9623:
9618:
9613:
9608:
9603:
9598:
9593:
9588:
9583:
9429:
9102:
9010:
9005:
8985:
8899:
8802:
8678:
8401:
7934:
7654:
7244:
6860:
4196:
4156:
3815:
2189:
365:
claims that there are infinitely many Mersenne primes and predicts their
252:
162:
146:
91:
69:
7768:
6351:
17, 19, 73, 139, 907, 1907, 2029, 4801, 5153, 10867, 20161, 293831, ...
4551:
These primes are called repunit primes. Another example is when we take
4311:
5, 13, 41, 113, 2113, 525313, 536903681, 140737471578113, ... (sequence
3720:
3,391 × 23,279 × 65,993 × 1,868,569 × 1,066,818,132,868,207 (16 digits)
1415:
An alternative form of Perfect Numbers (not affecting the essence): If
921: ) there must be at least one prime factor congruent to 3 (mod 4).
12033:
11767:
11748:
11044:
10655:
9506:
9322:
9230:
9150:
9000:
8904:
8597:
7673:. Lecture Notes in Computer Science. Vol. 8874. pp. 358–377.
4022:
4003:
The simplest generalized Mersenne primes are prime numbers of the form
3850:
456:
341:
248:
244:
87:
83:
6335:
2, 5, 11, 13, 331, 599, 18839, 23747, 24371, 29339, 32141, 67421, ...
3409:
42643801, 43112609, 57885161, 74207281, 77232917, 82589933. (sequence
2386:, thus officially confirming its position as the 45th Mersenne prime.
2196:
in 1949, but the first successful identification of a Mersenne prime,
12139:
12134:
11968:
11373:
11300:
11292:
11097:
11011:
10129:
9547:
9496:
9377:
8588:
8569:
5316:
3, 4, 7, 11, 83, 149, 223, 599, 647, 1373, 8423, 149497, 388897, ...
4184:
does not lead to anything interesting (since it is always −1 for all
3950:
MF(2, 2), MF(2, 3), MF(2, 4), MF(2, 5), MF(3, 2), MF(3, 3), MF(7, 2),
3731:
263 × 10,350,794,431,055,162,386,718,619,237,468,234,569 (38 digits)
3400:
As of 2023, the 51 known Mersenne primes are 2 − 1 for the following
296:
be prime. The smallest composite Mersenne number with prime exponent
8529:
http://www.leyland.vispa.com/numth/factorization/cunningham/main.htm
5080:
is prime. However, this has not been proved for any single value of
3687:
7,432,339,208,719 (13 digits) × 341,117,531,003,194,129 (18 digits)
369:
and frequency: For every number n, there should on average be about
12073:
11474:
8358:
Property of Mersenne numbers with prime exponent that are composite
7953:
5986:
3, 59, 223, 547, 773, 1009, 1823, 3803, 49223, 193247, 703393, ...
5896:
2, 7, 29, 31, 67, 149, 401, 2531, 19913, 30773, 53857, 170099, ...
3838:
3822:
3791:
2260: — were found by the same program in the next several months.
1331:
Arithmetic modulo a Mersenne number is particularly efficient on a
8473:
http://www.leyland.vispa.com/numth/factorization/cunningham/2+.txt
8461:
http://www.leyland.vispa.com/numth/factorization/cunningham/2-.txt
7946:
5726:
2, 3, 5, 19, 41, 47, 8231, 33931, 43781, 50833, 53719, 67211, ...
5662:
5, 13, 131, 149, 1699, 14221, 35201, 126037, 371669, 1264699, ...
4525:
2, 19, 23, 317, 1031, 49081, 86453, 109297, 270343, ... (sequence
11963:
11812:
11479:
11138:
11132:
9049:
8484:
7819:
6823:
6788:
5582:
2, 3, 7, 29, 41, 67, 1327, 1399, 2027, 69371, 86689, 355039, ...
4929:
4421:
3092:
925:
307:
Mersenne primes were studied in antiquity because of their close
175:
11897:
6207:
3, 5, 19, 311, 317, 1129, 4253, 7699, 18199, 35153, 206081, ...
2889:; thus there are always larger primes than any particular prime.
2423:
Mersenne numbers are 0, 1, 3, 7, 15, 31, 63, ... (sequence
12218:
12023:
11958:
7457:"Prime number with 22 million digits is the biggest ever found"
6082:
2, 3, 7, 11, 19, 29, 401, 709, 2531, 15787, 66949, 282493, ...
4440:, it is to simply take out this factor and ask which values of
4330:
One may encounter cases where such a Mersenne prime is also an
3807:
3803:
1389:
10194:
8541:
http://www.leyland.vispa.com/numth/factorization/anbn/main.htm
7701:
7650:"Proof of a result of Euler and Lagrange on Mersenne Divisors"
7379:"UCLA mathematicians discover a 13-million-digit prime number"
4630:
is a perfect power, it can be shown that there is at most one
4508:
can be either positive or negative.) If, for example, we take
4139:, the former is not a prime. This can be remedied by allowing
3423:
3274:
All composite divisors of prime-exponent Mersenne numbers are
9044:
9030:
7485:"New Biggest Prime Number = 2 to the 74 Mil ... Uh, It's Big"
7431:"Mersenne Prime Number discovery – 2 − 1 is Prime!"
6066:
3, 7, 13, 19, 307, 619, 2089, 7297, 75571, 76103, 98897, ...
5694:
2, 5, 23, 73, 101, 401, 419, 457, 811, 1163, 1511, 8011, ...
3845:
then because it is primitive it constrains the odd leg to be
3841:
is always a Mersenne number. For example, if the even leg is
887:, all Mersenne primes are congruent to 3 (mod 4). Other than
7586:"GIMPS Discovers Largest Known Prime Number: 2^82,589,933-1"
6319:
3, 5, 19, 31, 367, 389, 431, 2179, 10667, 13103, 90397, ...
6162:
2, 3, 19, 31, 101, 139, 167, 1097, 43151, 60703, 90499, ...
2563:(which is a contradiction, as neither −1 nor 0 is prime) or
8487:
7299:
Bell, E.T. and Mathematical Association of America (1951).
7059:
6769:
6735:
6627:
6598:
6582:
6566:
6550:
6534:
6518:
6502:
6434:
6418:
6402:
6386:
6370:
6354:
6338:
6322:
6306:
6290:
6274:
6258:
6242:
6226:
6210:
6194:
6165:
6149:
6133:
6117:
6101:
6085:
6069:
6053:
6037:
6021:
6005:
5989:
5960:
5931:
5915:
5899:
5883:
5867:
5851:
5809:
5793:
5777:
5761:
5745:
5729:
5713:
5697:
5681:
5665:
5649:
5633:
5617:
5601:
5585:
5569:
5553:
5537:
5521:
5505:
5489:
5473:
5457:
5441:
5425:
5409:
5393:
5377:
5335:
5319:
5303:
5287:
5271:
5255:
5238:
5220:
4840:
4815:
4759:
4717:
4570:
4539:
4527:
4398:
4396:
7, 271, 2269, 176419, 129159847, 1162320517, ... (sequence
4380:
4313:
4288:
4092:
It is also natural to try to generalize primes of the form
3818:
having been discovered and named during the 19th century).
3750:
3739:
3472:
3411:
2425:
2269:
was the first prime discovered with more than 1000 digits,
1996:
in 1772. The next (in historical, not numerical order) was
571:
265:
235:
8322:
7865:
6431:
7, 11, 181, 421, 2297, 2797, 4129, 4139, 7151, 29033, ...
6178:
2, 3, 5, 11, 19, 1259, 1399, 2539, 2843, 5857, 10589, ...
3709:
745,988,807 × 870,035,986,098,720,987,332,873 (24 digits)
3698:
2,550,183,799 × 3,976,656,429,941,438,590,393 (22 digits)
3321:
The Mersenne number sequence is a member of the family of
2220:, with a computer search program written and run by Prof.
11948:
7153:"GIMPS Project Discovers Largest Known Prime Number: 2-1"
6034:
3, 5, 13, 17, 43, 127, 229, 277, 6043, 11131, 11821, ...
425:
with n decimal digits (i.e. 10 < p < 10) for which
8559:
6146:
5, 7, 19, 31, 53, 67, 293, 641, 2137, 3011, 268207, ...
6130:
2, 19, 23, 317, 1031, 49081, 86453, 109297, 270343, ...
5678:
3, 17, 23, 29, 47, 61, 1619, 18251, 106187, 201653, ...
5566:
3, 4, 5, 17, 397, 409, 643, 1783, 2617, 4583, 8783, ...
3837: ) generates a unique right triangle such that its
3802:
after Marin Mersenne, because 8191 is a Mersenne prime (
3676:
11,447 × 13,842,607,235,828,485,645,766,393 (26 digits)
1961:, was discovered anonymously before 1461; the next two (
8578:
7206:"Heuristics: Deriving the Wagstaff Mersenne Conjecture"
5957:
2, 3, 5, 13, 29, 37, 1021, 1399, 2137, 4493, 5521, ...
1302:(the correct terms on Mersenne's original list), while
7572:"Found: A Special, Mind-Bogglingly Large Prime Number"
1734:
Mersenne primes take their name from the 17th-century
621:
10858:
9578:
9573:
9568:
9563:
8509:
Factorization of completely factored Mersenne numbers
8230:
8150:
7554:"Largest-known prime number found on church computer"
6650:
are not included in the corresponding OEIS sequence.
6383:
3, 5, 17, 67, 83, 101, 1373, 6101, 12119, 61781, ...
5710:
3, 13, 31, 313, 3709, 7933, 14797, 30689, 38333, ...
5630:
3, 7, 19, 109, 131, 607, 863, 2917, 5923, 12421, ...
4866:
4580:
corresponding to primes −11, 19141, 57154490053, ....
4455:
2146:
1820:
1466:
1421:
1291:
increases. For example, eight of the first 11 primes
947:
798:
743:
685:
431:
375:
123:
11243:
6595:
2, 3, 5, 13, 347, 977, 1091, 4861, 4967, 34679, ...
6367:
5, 7, 179, 229, 439, 557, 6113, 223999, 327001, ...
6303:
3, 5, 11, 17, 71, 89, 827, 22307, 45893, 63521, ...
4978:
makes this number prime. It can be shown that such
4852:
4118:
3654:
2,687 × 202,029,703 × 1,113,491,139,767 (13 digits)
838:
cannot be prime. The first four Mersenne primes are
459:
about prime numbers, for example, the infinitude of
8451:
Decimal digits and English names of Mersenne primes
7668:
7530:"Mersenne Prime Discovery - 2^77232917-1 is Prime!"
6547:2, 3, 31, 41, 53, 101, 421, 1259, 4721, 45259, ...
6287:5, 41, 149, 229, 263, 739, 3457, 20269, 98221, ...
941:must also be prime. This follows from the identity
8270:
8190:
7300:
6050:2, 3, 107, 197, 2843, 3571, 4451, ..., 31517, ...
4909:
4491:
4143:to be an algebraic integer instead of an integer:
2418:
2176:
1853:
1501:
1452:
1231:
830:
784:
713:
663:
444:
413:
191:, who studied them in the early 17th century. If
10242:
6611:2, 3, 7, 67, 79, 167, 953, 1493, 3389, 4871, ...
5848:2, 5, 163, 191, 229, 271, 733, 21059, 25237, ...
5614:2, 5, 11, 61, 619, 2879, 2957, 24371, 69247, ...
3665:167 × 57,912,614,113,275,649,087,721 (23 digits)
12279:
9260: = 0, 1, 2, 3, ...
7868:. The On-Line Encyclopedia of Integer Sequences.
5646:3, 7, 19, 79, 431, 1373, 1801, 2897, 46997, ...
5598:3, 5, 7, 113, 397, 577, 7573, 14561, 58543, ...
3643:439 × 2,298,041 × 9,361,973,132,609 (13 digits)
3389:
1899:
1891:. On the other side of the board, he multiplied
286:without the primality requirement may be called
10128:
7340:
5007:is prime. It is a conjecture that for any pair
4191:). Thus, we can regard a ring of "integers" on
4180:are the usual Mersenne primes, and the formula
2892:It follows from this fact that for every prime
2212:at the Institute for Numerical Analysis at the
2051:. Specifically, it can be shown that for prime
878:and because the first Mersenne prime starts at
343:
9922:
9908:
414:{\displaystyle e^{\gamma }\cdot \log _{2}(10)}
318:As of 2023, 51 Mersenne primes are known. The
11913:
11828:
9894:
8613:
8442:– contains factors for small Mersenne numbers
7272:Bulletin of the American Mathematical Society
4982:must be primes themselves or equal to 4, and
4325:
3938:. The only known Mersenne–Fermat primes with
3023:. If the given congruence is satisfied, then
1252:Though the above examples might suggest that
11730:
10080:
7724:"M12720787 Mersenne number exponent details"
7476:
7220:Mersenne Primes: History, Theorems and Lists
6447:2, 5, 7, 107, 383, 17359, 21929, 26393, ...
5928:3, 11, 17, 173, 839, 971, 40867, 45821, ...
5790:2, 19, 1021, 5077, 34031, 46099, 65707, ...
4767:) (notice this OEIS sequence does not allow
1400:) is a perfect number. In the 18th century,
7842:
7243:
6473:2, 3, 13, 31, 59, 131, 223, 227, 1523, ...
4210:
3424:Factorization of composite Mersenne numbers
3396:List of Mersenne primes and perfect numbers
3238:and therefore 2 is a quadratic residue mod
3002:does not hold. By Fermat's little theorem,
2040:) were found early in the 20th century, by
11920:
11906:
11835:
11821:
10195:Possessing a specific set of other numbers
10018:
9901:
9887:
8620:
8606:
7880:"A research of Mersenne and Fermat primes"
7422:
6675:, a difference of two consecutive perfect
5806:2, 3, 7, 19, 31, 67, 89, 9227, 43891, ...
4910:{\displaystyle {\frac {a^{n}-b^{n}}{a-b}}}
4584:It is a conjecture that for every integer
4354:. In these cases, such numbers are called
3621:193,707,721 × 761,838,257,287 (12 digits)
2899:, there is at least one prime of the form
349:Are there infinitely many Mersenne primes?
11658:
10605:
8514:The Cunningham project, factorization of
7952:
7899:Encyclopedia of Cryptography and Security
7365:The Mathematics Department and the Mark 1
7303:Mathematics, queen and servant of science
7283:
7145:
6531:2, 3, 7, 13, 47, 89, 139, 523, 1051, ...
6271:2, 3, 11, 163, 191, 269, 1381, 1493, ...
5864:2, 7, 19, 167, 173, 223, 281, 21647, ...
4426:The other way to deal with the fact that
4368:is an Eisenstein prime for the following
3864:
3833:and has its even leg a power of 2 (
3218:, so −2 would be a quadratic residue mod
2994:is a Mersenne prime, then the congruence
2797:. As a result, for all positive integers
2278:was the first with more than 10,000, and
1748:2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257.
8627:
8422:with hyperlinks to original publications
8334:GIMPS, known factors of Mersenne numbers
7792:
7769:"M1277 Mersenne number exponent details"
7376:
7255:(4th ed.). Oxford University Press.
7253:An Introduction to the Theory of Numbers
6515:2, 3, 7, 89, 101, 293, 4463, 70067, ...
6002:2, 3, 7, 127, 283, 883, 1523, 4001, ...
3751:Mersenne numbers in nature and elsewhere
3610:179,951 × 3,203,431,780,337 (13 digits)
2135:
1247:= 2 − 1 = 15 = 3 × 5 = (2 − 1) × (1 + 2)
337:
243:) and the resulting Mersenne primes are
9138:
7896:
7671:Advances in Cryptology – ASIACRYPT 2014
7197:
4857:Another generalized Mersenne number is
353:(more unsolved problems in mathematics)
14:
12280:
11766:
7795:Famous Puzzles of Great Mathematicians
7428:
6239:2, 7, 11, 17, 37, 521, 877, 2423, ...
5046:, there are infinitely many values of
4594:, there are infinitely many values of
4492:{\displaystyle {\frac {b^{n}-1}{b-1}}}
4275:is a Gaussian prime for the following
3906:natural number, and can be written as
2785:is also the smallest positive integer
1870:was determined to be prime in 1883 by
1408:. It is unknown whether there are any
1287:appear to grow increasingly sparse as
928:about Mersenne numbers states that if
11901:
11816:
11765:
11729:
11693:
11657:
11617:
11242:
11131:
10857:
10772:
10727:
10604:
10294:
10241:
10193:
10127:
10079:
10017:
9921:
9882:
8601:
8579:
8560:
7931:The Prime Glossary: Gaussian Mersenne
7817:
7482:
7454:
7401:
6191:4, 7, 67, 73, 1091, 1483, 10937, ...
3107:Mersenne numbers is pairwise coprime.
2337:In September 2008, mathematicians at
2214:University of California, Los Angeles
1983:in 1588. After nearly two centuries,
363:Lenstra–Pomerance–Wagstaff conjecture
11875:Great Internet Mersenne Prime Search
10295:
8490:– Factorization of Mersenne numbers
7702:Henri Lifchitz and Renaud Lifchitz.
7647:
7265:
7203:
7102:with this property: in these cases,
6830:Great Internet Mersenne Prime Search
6399:3, 103, 271, 523, 23087, 69833, ...
6114:2, 3, 5, 37, 599, 38393, 51431, ...
4410:
2670:is composite. By contraposition, if
2368:, discovered a 48th Mersenne prime,
1951:were known in antiquity. The fifth,
1861:, using a desk calculating machine.
1351:with very large periods such as the
785:{\displaystyle \Phi _{p}(2)=2^{p}-1}
679:or 1. However, it cannot be 1 since
664:{\textstyle {\frac {(2p+1)-1}{2}}=p}
328:Great Internet Mersenne Prime Search
308:
11694:
7268:"On the factoring of large numbers"
6098:2, 31, 103, 617, 10253, 10691, ...
3632:228,479 × 48,544,121 × 212,885,833
3250:, −1 is a quadratic nonresidue mod
2625:is composite, hence can be written
2408:made the find on December 7, 2018.
2393:, had found a 50th Mersenne prime,
2184:function in the value of the prime.
1337:Park–Miller random number generator
591:is congruent to 7 mod 8, so 2 is a
24:
12298:Unsolved problems in number theory
11842:
11618:
8553:
7483:Chang, Kenneth (21 January 2016).
7455:Brook, Robert (January 19, 2016).
7377:Maugh II, Thomas H. (2008-09-27).
6415:2, 7, 53, 67, 71, 443, 26497, ...
4146:
3992:
2927:is an odd prime, then every prime
2687:is an odd prime, then every prime
1772:(which are composite) and omitted
1362:
745:
687:
482:(which is also prime) will divide
25:
12324:
12194:Indefinite and fictitious numbers
11927:
8456:Prime curios: 2305843009213693951
8446:Known factors of Mersenne numbers
8391:
8298:
7901:. Springer US. pp. 509–510.
7845:"JPL Small-Body Database Browser"
7429:Cooper, Curtis (7 January 2016).
7404:"Largest Prime Number Discovered"
6223:5, 31, 271, 929, 2789, 4153, ...
4853:Other generalized Mersenne primes
4415:
3334:(3, 2). That is, Mersenne number
2210:Western Automatic Computer (SWAC)
1355:, generalized shift register and
161:. That is, it is a prime number
12293:Eponymous numbers in mathematics
11796:
11404:Perfect digit-to-digit invariant
10773:
8996:Supersingular (moonshine theory)
4303:As for all Gaussian primes, the
3927:, it is a Mersenne number. When
3160:: 11 and 23 are both prime, and
2044:in 1911 and 1914, respectively.
27:Prime number of the form (2^n)-1
8271:{\displaystyle (a^{n}+b^{n})/c}
8215:
8191:{\displaystyle (a^{n}-b^{n})/c}
8135:
8113:
8087:
8061:
8035:
8013:
7991:
7961:
7940:
7923:
7890:
7872:
7858:
7836:
7811:
7786:
7761:
7741:
7716:
7695:
7662:
7641:
7623:
7598:
7578:
7564:
7546:
7522:
7504:
7448:
7395:
7370:
7357:
7334:
7285:10.1090/S0002-9904-1903-01079-9
7266:Cole, F. N. (1 December 1903).
7066:
6961:
6895:This number is the same as the
6889:
6499:53, 421, 647, 1601, 35527, ...
2595:which is not prime. Therefore,
2419:Theorems about Mersenne numbers
1911:The first four Mersenne primes
344:Unsolved problem in mathematics
8991:Supersingular (elliptic curve)
8440:Will Edgington's Mersenne Page
8257:
8231:
8177:
8151:
7820:"Wheat and Chessboard Problem"
7797:. AMS Bookstore. p. 197.
7630:Will Edgington's Mersenne Page
7312:
7292:
7259:
7237:
7224:
7212:
7171:
7132:can be factored algebraically.
7014:. Thus, in this case the pair
6486:2, 3, 17, 41, 43, 59, 83, ...
3210:. Supposing latter true, then
3186:. By Fermat's little theorem,
2873:This fact leads to a proof of
2447:are natural numbers such that
2366:University of Central Missouri
2343:Electronic Frontier Foundation
2171:
2168:
2162:
2153:
1854:{\displaystyle (2^{148}+1)/17}
1840:
1821:
1488:
1476:
1447:
1422:
1349:pseudorandom number generators
1295:give rise to a Mersenne prime
1209:
1197:
1124:
1105:
1082:
1070:
997:
978:
760:
754:
714:{\displaystyle \Phi _{1}(2)=1}
702:
696:
640:
625:
408:
402:
58:
46:
13:
1:
12109:Conway chained arrow notation
10243:Expressible via specific sums
8772:2 ± 2 ± 1
7606:"GIMPS - The Math - PrimeNet"
7139:
6700:, because it is divisible by
3438:integer factorization records
3390:List of known Mersenne primes
3122:are both prime (meaning that
2977:A Mersenne prime cannot be a
2695:must be 1 plus a multiple of
2177:{\displaystyle \log(\log(y))}
1900:Searching for Mersenne primes
1893:193,707,721 × 761,838,257,287
831:{\displaystyle 2^{p}-1=M_{p}}
309:connection to perfect numbers
8426:report about Mersenne primes
7907:10.1007/978-1-4419-5906-5_32
7679:10.1007/978-3-662-45611-8_19
4693:is prime are (starting with
4261:which will then be called a
4135:, so unless the latter is a
3776:wheat and chessboard problem
3755:In the mathematical problem
3728:272225893536...454145691647
3717:103845937170...992658440191
3706:649037107316...312041152511
3695:101412048018...973625643007
3684:253530120045...993406410751
3673:158456325028...187087900671
3662:967140655691...033397649407
3599:6,361 × 69,431 × 20,394,401
3013:. Therefore, one can write
2755:, for all positive integers
2207:National Bureau of Standards
1992:was verified to be prime by
935:is prime, then the exponent
911:of a Mersenne number (
7:
11332:Multiplicative digital root
8523:= 2, 3, 5, 6, 7, 10, 11, 12
8420:Mersenne prime bibliography
8311:Encyclopedia of Mathematics
7749:"Exponent Status for M1277"
6781:
5155:(some large terms are only
5096:For more information, see
3759:, solving a puzzle with an
3588:2,351 × 4,513 × 13,264,529
2049:Lucas–Lehmer primality test
1906:largest known prime numbers
1889:147,573,952,589,676,412,927
1453:{\displaystyle (M=2^{n}-1)}
1357:Lagged Fibonacci generators
1314:Lucas–Lehmer primality test
128:Mersenne primes (of form 2^
10:
12329:
12204:Largest known prime number
10728:
7793:Petković, Miodrag (2009).
6836:Largest known prime number
6646:is even, then the numbers
6460:7, 1163, 4007, 10159, ...
5912:3, 5, 7, 4703, 30113, ...
4735:, they are (starting with
4419:
4356:Eisenstein Mersenne primes
4326:Eisenstein Mersenne primes
3999:Generalized Mersenne prime
3996:
3430:special number field sieve
3393:
2717:. A composite example is
1512:
1502:{\displaystyle P=M(M+1)/2}
1366:
579:). Since for these primes
320:largest known prime number
12257:
12189:Extended real number line
12169:
12122:
12104:Knuth's up-arrow notation
12091:
12082:
11935:
11850:
11792:
11775:
11761:
11739:
11725:
11703:
11689:
11667:
11653:
11626:
11613:
11589:
11543:
11503:
11454:
11428:
11409:Perfect digital invariant
11361:
11345:
11324:
11291:
11256:
11252:
11238:
11146:
11127:
11096:
11063:
11020:
10997:
10984:Superior highly composite
10874:
10870:
10853:
10781:
10768:
10736:
10723:
10611:
10600:
10562:
10553:
10531:
10488:
10450:
10441:
10374:
10316:
10307:
10303:
10290:
10248:
10237:
10200:
10189:
10137:
10123:
10086:
10075:
10028:
10013:
9931:
9917:
9867:
9556:
9520:
9420:
9397:
9371:
9131:
9029:
8923:
8887:
8636:
7179:"GIMPS Milestones Report"
5758:3, 53, 83, 487, 743, ...
4215:If we regard the ring of
3651:604462909807314587353087
3083:are natural numbers then
2364:, a mathematician at the
2216:, under the direction of
2192:searched for them on the
2020:Ivan Mikheevich Pervushin
1872:Ivan Mikheevich Pervushin
1727:
1388:. In the 4th century BC,
1384:are closely connected to
113:
102:
79:
68:
56:
45:
35:
12288:Classes of prime numbers
12114:Steinhaus–Moser notation
11022:Euler's totient function
10806:Euler–Jacobi pseudoprime
10081:Other polynomial numbers
9378:Mega (1,000,000+ digits)
9247:Arithmetic progression (
8394:"31 and Mersenne Primes"
8222:PRP records, search for
8142:PRP records, search for
7341:Horace S. Uhler (1952).
7320:"h2g2: Mersenne Numbers"
7185:. Mersenne Research, Inc
7049:must be prime. That is,
6882:
6624:47, 401, 509, 8609, ...
4986:can be 4 if and only if
4211:Gaussian Mersenne primes
3577:431 × 9,719 × 2,099,863
1460:is a prime number, then
1274:= 2 − 1 = 2047 = 23 × 89
1259:is prime for all primes
259:, 8191, 131071, 524287,
183:. They are named after
157:that is one less than a
10836:Somer–Lucas pseudoprime
10826:Lucas–Carmichael number
10661:Lazy caterer's sequence
8328:GIMPS Milestones Report
7307:. McGraw-Hill New York.
4964:is always divisible by
4502:be prime. (The integer
4433:is always divisible by
4263:Gaussian Mersenne prime
4128:is always divisible by
3857:and its inradius to be
3640:9444732965739290427391
3629:2361183241434822606847
3292:has no solutions where
2737:Fermat's little theorem
2702:. This holds even when
2661:(2) + (2) + ... + 2 + 1
675:is a prime, it must be
12157:Fast-growing hierarchy
11870:Double Mersenne number
10711:Wedderburn–Etherington
10111:Lucky numbers of Euler
9533:Industrial-grade prime
8910:Newman–Shanks–Williams
8428:– detection in detail
8272:
8192:
7558:christianchronicle.org
7435:Mersenne Research, Inc
7157:Mersenne Research, Inc
6819:Double Mersenne number
6804:Erdős–Borwein constant
4911:
4493:
4176:is either 2 or 0. But
4096:to primes of the form
3871:Mersenne–Fermat number
3865:Mersenne–Fermat primes
3618:147573952589676412927
2907:less than or equal to
2881:, all primes dividing
2546:. In the former case,
2185:
2178:
1855:
1503:
1454:
1316:(LLT) is an efficient
1233:
832:
786:
715:
665:
446:
415:
302:2 − 1 = 2047 = 23 × 89
12214:Long and short scales
12152:Grzegorczyk hierarchy
10999:Prime omega functions
10816:Frobenius pseudoprime
10606:Combinatorial numbers
10475:Centered dodecahedral
10268:Primary pseudoperfect
9870:List of prime numbers
9328:Sophie Germain/Safe (
8273:
8193:
7232:Mersenne's conjecture
5027:are not both perfect
4912:
4494:
4117:). However (see also
4059:; another example is
3987:cyclotomic polynomial
3798:number 8191 is named
3765:-disc tower requires
3566:13,367 × 164,511,353
3050:= 1 + 2 + 2 + ... + 2
2966:quadratic reciprocity
2847:. Furthermore, since
2727:89 = 1 + 4 × (2 × 11)
2391:Germantown, Tennessee
2360:On January 25, 2013,
2179:
2139:
1908:are Mersenne primes.
1856:
1504:
1455:
1326:distributed computing
1234:
833:
787:
716:
666:
461:Sophie Germain primes
447:
445:{\displaystyle M_{p}}
416:
338:About Mersenne primes
332:distributed computing
263:, ... (sequence
11855:Mersenne conjectures
11458:-composition related
11258:Arithmetic functions
10860:Arithmetic functions
10796:Elliptic pseudoprime
10480:Centered icosahedral
10460:Centered tetrahedral
9052:(10 − 1)/9
8228:
8148:
7330:on December 5, 2014.
6809:Mersenne conjectures
4864:
4453:
4334:, being of the form
3544:233 × 1,103 × 2,089
3449:factors allowed) is
3283:Mihăilescu's theorem
3128:Sophie Germain prime
3070:which is impossible.
2957:is a square root of
2765:is also a factor of
2715:31 = 1 + 3 × (2 × 5)
2570:In the latter case,
2144:
1896:published his list.
1818:
1805:proved in 1876 that
1464:
1419:
1406:Euclid–Euler theorem
1369:Euclid–Euler theorem
1345:primitive trinomials
1341:primitive polynomial
945:
796:
741:
683:
619:
605:multiplicative order
470:). For these primes
429:
373:
313:Euclid–Euler theorem
276:Numbers of the form
12229:Orders of magnitude
12099:Scientific notation
11384:Kaprekar's constant
10904:Colossally abundant
10791:Catalan pseudoprime
10691:Schröder–Hipparchus
10470:Centered octahedral
10346:Centered heptagonal
10336:Centered pentagonal
10326:Centered triangular
9926:and related numbers
9361: ± 7, ...
8888:By integer sequence
8673:(2 + 1)/3
7818:Weisstein, Eric W.
7648:Caldwell, Chris K.
7560:. January 12, 2018.
7409:Scientific American
7347:Scripta Mathematica
7218:Chris K. Caldwell,
7084:th powers for some
6871:Gillies' conjecture
5097:
4827:For negative bases
4729:For negative bases
4205:Eisenstein integers
4025:with small integer
3607:576460752303423487
3276:strong pseudoprimes
2914:, for some integer
2859:is odd. Therefore,
2817:. Therefore, since
2777:is not a factor of
2022:in 1883. Two more (
1410:odd perfect numbers
909:prime factorization
32:
12147:Ackermann function
11802:Mathematics portal
11744:Aronson's sequence
11490:Smarandache–Wellin
11247:-dependent numbers
10954:Primitive abundant
10841:Strong pseudoprime
10831:Perrin pseudoprime
10811:Fermat pseudoprime
10751:Wolstenholme prime
10575:Squared triangular
10361:Centered decagonal
10356:Centered nonagonal
10351:Centered octagonal
10341:Centered hexagonal
9543:Formula for primes
9176: + 2 or
9108:Smarandache–Wellin
8581:Weisstein, Eric W.
8562:Weisstein, Eric W.
8546:2016-02-02 at the
8534:2016-03-04 at the
8478:2013-05-02 at the
8466:2014-11-05 at the
8268:
8188:
7847:. Ssd.jpl.nasa.gov
7824:Mathworld. Wolfram
7635:2014-10-14 at the
7574:. January 5, 2018.
7490:The New York Times
7159:. 21 December 2018
6934:quadratic equation
6679:th powers, and if
5165:are checked up to
5095:
5031:th powers for any
4907:
4489:
4219:, we get the case
3555:223 × 616,318,177
3304:are integers with
3212:2 = (2) ≡ −2 (mod
2593:0 − 1 = 0 − 1 = −1
2186:
2174:
1851:
1499:
1450:
1229:
1227:
828:
782:
711:
661:
442:
411:
109:(December 7, 2018)
103:Largest known term
30:
12303:Integer sequences
12275:
12274:
12165:
12164:
11895:
11894:
11810:
11809:
11788:
11787:
11757:
11756:
11721:
11720:
11685:
11684:
11649:
11648:
11609:
11608:
11605:
11604:
11424:
11423:
11234:
11233:
11123:
11122:
11119:
11118:
11065:Aliquot sequences
10876:Divisor functions
10849:
10848:
10821:Lucas pseudoprime
10801:Euler pseudoprime
10786:Carmichael number
10764:
10763:
10719:
10718:
10596:
10595:
10592:
10591:
10588:
10587:
10549:
10548:
10437:
10436:
10394:Square triangular
10286:
10285:
10233:
10232:
10185:
10184:
10119:
10118:
10071:
10070:
10009:
10008:
9876:
9875:
9487:Carmichael number
9422:Composite numbers
9357: ± 3, 8
9353: ± 1, 4
9316: ± 1, …
9312: ± 1, 4
9308: ± 1, 2
9298:
9297:
8843:3·2 − 1
8748:2·3 + 1
8662:Double Mersenne (
8565:"Mersenne number"
8306:"Mersenne number"
7916:978-1-4419-5905-8
7843:Alan Chamberlin.
7804:978-0-8218-4814-2
7704:"PRP Top Records"
7688:978-3-662-45607-1
7383:Los Angeles Times
7230:The Prime Pages,
7204:Caldwell, Chris.
7080:are both perfect
6633:
6632:
5361:2, 3 (no others)
5042:is not a perfect
4905:
4487:
4411:Divide an integer
4217:Gaussian integers
4201:Gaussian integers
3735:
3734:
3596:9007199254740991
3503:Factorization of
3222:. However, since
2723:23 = 1 + (2 × 11)
2320:(or equivalently
2194:Manchester Mark 1
1885:Frank Nelson Cole
1732:
1731:
653:
593:quadratic residue
143:
142:
16:(Redirected from
12320:
12089:
12088:
12019:Eddington number
11964:Hundred thousand
11922:
11915:
11908:
11899:
11898:
11887:Mersenne Twister
11837:
11830:
11823:
11814:
11813:
11800:
11763:
11762:
11732:Natural language
11727:
11726:
11691:
11690:
11659:Generated via a
11655:
11654:
11615:
11614:
11520:Digit-reassembly
11485:Self-descriptive
11289:
11288:
11254:
11253:
11240:
11239:
11191:Lucas–Carmichael
11181:Harmonic divisor
11129:
11128:
11055:Sparsely totient
11030:Highly cototient
10939:Multiply perfect
10929:Highly composite
10872:
10871:
10855:
10854:
10770:
10769:
10725:
10724:
10706:Telephone number
10602:
10601:
10560:
10559:
10541:Square pyramidal
10523:Stella octangula
10448:
10447:
10314:
10313:
10305:
10304:
10297:Figurate numbers
10292:
10291:
10239:
10238:
10191:
10190:
10125:
10124:
10077:
10076:
10015:
10014:
9919:
9918:
9903:
9896:
9889:
9880:
9879:
9407:Eisenstein prime
9362:
9338:
9317:
9289:
9261:
9241:
9225:
9209:
9204: + 6,
9200: + 2,
9185:
9180: + 4,
9161:
9136:
9135:
9053:
9016:Highly cototient
8878:
8877:
8871:
8861:
8844:
8835:
8820:
8797:
8796:·2 − 1
8785:
8784:·2 + 1
8773:
8764:
8749:
8740:
8727:
8712:
8697:
8685:
8684:·2 + 1
8674:
8665:
8656:
8647:
8622:
8615:
8608:
8599:
8598:
8594:
8593:
8584:"Mersenne prime"
8575:
8574:
8524:
8504:
8498:
8486:
8431:
8416:
8414:
8413:
8404:. Archived from
8384:
8356:
8319:
8293:
8291:
8279:
8277:
8275:
8274:
8269:
8264:
8256:
8255:
8243:
8242:
8219:
8213:
8211:
8199:
8197:
8195:
8194:
8189:
8184:
8176:
8175:
8163:
8162:
8139:
8133:
8130:
8126:
8117:
8111:
8108:
8104:
8091:
8085:
8082:
8078:
8065:
8059:
8056:
8052:
8039:
8033:
8030:
8026:
8017:
8011:
8008:
8004:
7995:
7989:
7986:
7982:
7974:
7965:
7959:
7958:
7956:
7944:
7938:
7929:Chris Caldwell:
7927:
7921:
7920:
7894:
7888:
7887:
7882:. Archived from
7876:
7870:
7869:
7862:
7856:
7855:
7853:
7852:
7840:
7834:
7833:
7831:
7830:
7815:
7809:
7808:
7790:
7784:
7783:
7781:
7779:
7765:
7759:
7758:
7756:
7755:
7745:
7739:
7738:
7736:
7734:
7720:
7714:
7713:
7711:
7710:
7699:
7693:
7692:
7666:
7660:
7659:
7645:
7639:
7627:
7621:
7620:
7618:
7616:
7610:www.mersenne.org
7602:
7596:
7595:
7593:
7592:
7582:
7576:
7575:
7568:
7562:
7561:
7550:
7544:
7543:
7541:
7540:
7534:www.mersenne.org
7526:
7520:
7519:
7514:. Archived from
7508:
7502:
7501:
7499:
7497:
7480:
7474:
7473:
7471:
7469:
7452:
7446:
7445:
7443:
7441:
7426:
7420:
7419:
7417:
7416:
7399:
7393:
7392:
7390:
7389:
7374:
7368:
7361:
7355:
7354:
7338:
7332:
7331:
7326:. Archived from
7316:
7310:
7308:
7306:
7296:
7290:
7289:
7287:
7263:
7257:
7256:
7241:
7235:
7228:
7222:
7216:
7210:
7209:
7201:
7195:
7194:
7192:
7190:
7175:
7169:
7168:
7166:
7164:
7149:
7133:
7131:
7130:
7128:
7127:
7118:
7115:
7101:
7097:
7090:
7083:
7079:
7075:
7070:
7064:
7062:
7052:
7048:
7037:
7025:
7013:
6995:
6993:
6992:
6983:
6980:
6965:
6959:
6957:
6927:
6923:
6919:
6893:
6814:Mersenne twister
6772:
6761:
6750:
6738:
6727:
6716:
6709:
6699:
6692:
6688:
6678:
6674:
6663:
6649:
6645:
6641:
5216:
5215:
5209:
5200:
5196:
5195:
5189:
5180:
5179:
5177:
5168:
5164:
5163:
5152:
5151:
5150:
5148:
5147:
5138:
5135:
5120:
5119:
5112:
5111:
5105:
5104:
5098:
5094:
5091:
5079:
5078:
5076:
5075:
5066:
5063:
5049:
5041:
5034:
5030:
5026:
5022:
5018:
5006:
4996:
4985:
4981:
4977:
4973:
4963:
4953:
4938:
4927:
4923:
4916:
4914:
4913:
4908:
4906:
4904:
4893:
4892:
4891:
4879:
4878:
4868:
4843:
4832:
4818:
4807:
4806:
4804:
4803:
4797:
4794:
4783:
4773:
4762:
4751:
4745:
4741:
4734:
4720:
4709:
4703:
4699:
4692:
4691:
4689:
4688:
4682:
4679:
4668:
4659:
4658:
4656:
4655:
4649:
4646:
4636:value such that
4635:
4629:
4624:is prime. (When
4623:
4622:
4620:
4619:
4613:
4610:
4599:
4589:
4573:
4563:
4557:
4542:
4530:
4520:
4514:
4507:
4498:
4496:
4495:
4490:
4488:
4486:
4475:
4468:
4467:
4457:
4445:
4439:
4432:
4401:
4383:
4373:
4367:
4353:
4343:
4332:Eisenstein prime
4316:
4291:
4280:
4274:
4256:
4248:
4238:
4228:
4190:
4183:
4179:
4175:
4165:
4155:of integers (on
4134:
4127:
4116:
4109:
4102:
4095:
4088:
4069:
4063:, in this case,
4062:
4058:
4039:
4033:, in this case,
4032:
4029:. An example is
4021:is a low-degree
4020:
4009:
3984:
3978:
3955:
3951:
3944:
3933:
3926:
3918:
3905:
3899:
3893:
3892:
3890:
3889:
3886:
3883:
3860:
3856:
3848:
3844:
3836:
3786:
3773:
3764:
3742:
3585:140737488355327
3511:
3500:
3490:
3483:
3482:
3475:
3459:
3455:
3454:
3435:
3414:
3384:
3374:
3364:
3333:
3317:
3310:
3303:
3299:
3295:
3291:
3268:
3261:
3253:
3249:
3246:is congruent to
3245:
3241:
3237:
3234:is congruent to
3233:
3229:
3226:is congruent to
3225:
3221:
3217:
3209:
3201:
3193:
3185:
3177:
3167:
3164:, so 23 divides
3163:
3153:
3149:
3141:
3133:
3125:
3121:
3113:
3102:
3098:
3090:
3086:
3082:
3078:
3069:
3066:, and therefore
3065:
3058:
3051:
3048:
3047:
3045:
3044:
3041:
3038:
3029:
3022:
3012:
3001:
2993:
2971:
2963:
2956:
2952:
2938:
2935:is congruent to
2934:
2930:
2926:
2917:
2913:
2906:
2898:
2888:
2885:are larger than
2884:
2880:
2875:Euclid's theorem
2869:
2858:
2855:, which is odd,
2854:
2850:
2846:
2835:
2828:
2824:
2820:
2816:
2812:
2808:
2804:
2800:
2796:
2792:
2788:
2784:
2780:
2776:
2772:
2768:
2764:
2760:
2754:
2750:
2746:
2742:
2728:
2724:
2720:
2716:
2712:
2705:
2701:
2694:
2690:
2686:
2673:
2669:
2665:
2655:
2652:
2649:
2645:
2638:
2634:
2624:
2614:
2610:
2601:
2594:
2590:
2583:
2576:
2569:
2562:
2555:
2545:
2538:
2527:
2520:
2509:
2498:
2487:
2467:
2460:
2453:
2446:
2442:
2428:
2403:
2396:
2378:
2371:
2356:
2333:
2315:
2307:
2295:
2286:
2277:
2268:
2259:
2250:
2241:
2232:
2204:
2183:
2181:
2180:
2175:
2128:
2121:
2101:
2091:
2079:
2067:
2057:
2039:
2030:
2017:
2004:
1991:
1979:) were found by
1978:
1969:
1960:
1950:
1940:
1930:
1920:
1894:
1890:
1882:
1869:
1860:
1858:
1857:
1852:
1847:
1833:
1832:
1813:
1798:
1789:
1780:
1771:
1762:
1517:
1516:
1508:
1506:
1505:
1500:
1495:
1459:
1457:
1456:
1451:
1440:
1439:
1399:
1395:
1383:
1373:Mersenne primes
1353:Mersenne twister
1308:
1301:
1294:
1290:
1286:
1275:
1262:
1258:
1248:
1238:
1236:
1235:
1230:
1228:
1221:
1217:
1216:
1215:
1182:
1181:
1166:
1165:
1150:
1149:
1117:
1116:
1098:
1094:
1090:
1089:
1088:
1055:
1054:
1039:
1038:
1023:
1022:
990:
989:
964:
963:
940:
934:
920:
906:
896:
886:
877:
867:
857:
847:
837:
835:
834:
829:
827:
826:
808:
807:
791:
789:
788:
783:
775:
774:
753:
752:
736:
728:
725:, so it must be
720:
718:
717:
712:
695:
694:
678:
674:
670:
668:
667:
662:
654:
649:
623:
614:
602:
590:
582:
574:
568:
558:
548:
538:
528:
518:
508:
498:
488:
481:
473:
451:
449:
448:
443:
441:
440:
420:
418:
417:
412:
398:
397:
385:
384:
345:
325:
303:
295:
288:Mersenne numbers
285:
268:
238:
232:
220:
214:
204:
199:composite number
196:
182:
173:
108:
75:Mersenne numbers
60:
48:
33:
29:
21:
12328:
12327:
12323:
12322:
12321:
12319:
12318:
12317:
12313:Perfect numbers
12308:Mersenne primes
12278:
12277:
12276:
12271:
12253:
12209:List of numbers
12177:
12175:
12173:
12171:
12161:
12118:
12084:
12078:
12049:Graham's number
12039:Skewes's number
11941:
11939:
11937:
11931:
11926:
11896:
11891:
11860:Mersenne's laws
11846:
11841:
11811:
11806:
11784:
11780:Strobogrammatic
11771:
11753:
11735:
11717:
11699:
11681:
11663:
11645:
11622:
11601:
11585:
11544:Divisor-related
11539:
11499:
11450:
11420:
11357:
11341:
11320:
11287:
11260:
11248:
11230:
11142:
11141:related numbers
11115:
11092:
11059:
11050:Perfect totient
11016:
10993:
10924:Highly abundant
10866:
10845:
10777:
10760:
10732:
10715:
10701:Stirling second
10607:
10584:
10545:
10527:
10484:
10433:
10370:
10331:Centered square
10299:
10282:
10244:
10229:
10196:
10181:
10133:
10132:defined numbers
10115:
10082:
10067:
10038:Double Mersenne
10024:
10005:
9927:
9913:
9911:natural numbers
9907:
9877:
9872:
9863:
9557:First 60 primes
9552:
9516:
9416:
9399:Complex numbers
9393:
9367:
9345:
9329:
9304:
9303:Bi-twin chain (
9294:
9268:
9248:
9232:
9216:
9192:
9168:
9152:
9127:
9113:Strobogrammatic
9051:
9025:
8919:
8883:
8875:
8869:
8868:
8851:
8842:
8827:
8804:
8792:
8780:
8771:
8756:
8747:
8734:
8726:# + 1
8724:
8719:
8711:# ± 1
8709:
8704:
8696:! ± 1
8692:
8680:
8672:
8664:2 − 1
8663:
8655:2 − 1
8654:
8646:2 + 1
8645:
8632:
8626:
8556:
8554:MathWorld links
8548:Wayback Machine
8536:Wayback Machine
8515:
8500:
8496:
8491:
8480:Wayback Machine
8468:Wayback Machine
8429:
8411:
8409:
8371:
8363:
8346:
8338:
8323:GIMPS home page
8304:
8301:
8296:
8281:
8260:
8251:
8247:
8238:
8234:
8229:
8226:
8225:
8223:
8220:
8216:
8201:
8180:
8171:
8167:
8158:
8154:
8149:
8146:
8145:
8143:
8140:
8136:
8128:
8120:
8118:
8114:
8106:
8094:
8092:
8088:
8080:
8068:
8066:
8062:
8054:
8042:
8040:
8036:
8028:
8020:
8018:
8014:
8006:
7998:
7996:
7992:
7984:
7976:
7968:
7966:
7962:
7945:
7941:
7928:
7924:
7917:
7895:
7891:
7878:
7877:
7873:
7864:
7863:
7859:
7850:
7848:
7841:
7837:
7828:
7826:
7816:
7812:
7805:
7791:
7787:
7777:
7775:
7773:www.mersenne.ca
7767:
7766:
7762:
7753:
7751:
7747:
7746:
7742:
7732:
7730:
7728:www.mersenne.ca
7722:
7721:
7717:
7708:
7706:
7700:
7696:
7689:
7667:
7663:
7646:
7642:
7637:Wayback Machine
7628:
7624:
7614:
7612:
7604:
7603:
7599:
7590:
7588:
7584:
7583:
7579:
7570:
7569:
7565:
7552:
7551:
7547:
7538:
7536:
7528:
7527:
7523:
7510:
7509:
7505:
7495:
7493:
7481:
7477:
7467:
7465:
7453:
7449:
7439:
7437:
7427:
7423:
7414:
7412:
7400:
7396:
7387:
7385:
7375:
7371:
7362:
7358:
7339:
7335:
7318:
7317:
7313:
7297:
7293:
7264:
7260:
7242:
7238:
7229:
7225:
7217:
7213:
7202:
7198:
7188:
7186:
7177:
7176:
7172:
7162:
7160:
7151:
7150:
7146:
7142:
7137:
7136:
7119:
7116:
7107:
7106:
7104:
7103:
7099:
7092:
7085:
7081:
7077:
7073:
7071:
7067:
7054:
7050:
7039:
7027:
7015:
6984:
6981:
6972:
6971:
6969:
6968:
6966:
6962:
6936:
6925:
6921:
6904:
6899:
6894:
6890:
6885:
6880:
6876:Williams number
6841:Wieferich prime
6784:
6768:
6752:
6748:
6734:
6718:
6714:
6701:
6694:
6690:
6689:is prime, then
6680:
6676:
6665:
6654:
6647:
6643:
6636:
5205:
5203:
5202:
5198:
5185:
5183:
5182:
5173:
5171:
5170:
5166:
5161:
5160:
5157:probable primes
5154:
5139:
5136:
5127:
5126:
5124:
5123:
5122:
5117:
5116:
5109:
5108:
5102:
5101:
5081:
5067:
5064:
5055:
5054:
5052:
5051:
5047:
5036:
5032:
5028:
5024:
5020:
5008:
4998:
4987:
4983:
4979:
4975:
4965:
4955:
4940:
4933:
4925:
4921:
4894:
4887:
4883:
4874:
4870:
4869:
4867:
4865:
4862:
4861:
4855:
4839:
4828:
4814:
4798:
4795:
4789:
4788:
4786:
4785:
4779:
4768:
4758:
4747:
4743:
4736:
4730:
4716:
4705:
4701:
4694:
4683:
4680:
4674:
4673:
4671:
4670:
4664:
4650:
4647:
4641:
4640:
4638:
4637:
4631:
4625:
4614:
4611:
4605:
4604:
4602:
4601:
4595:
4590:which is not a
4585:
4579:
4569:
4559:
4552:
4538:
4536:
4526:
4516:
4509:
4503:
4476:
4463:
4459:
4458:
4456:
4454:
4451:
4450:
4441:
4434:
4427:
4424:
4418:
4413:
4397:
4379:
4369:
4361:
4345:
4335:
4328:
4312:
4287:
4276:
4268:
4250:
4244:
4239:, and can ask (
4230:
4220:
4213:
4193:complex numbers
4185:
4181:
4177:
4171:
4160:
4149:
4147:Complex numbers
4129:
4122:
4111:
4104:
4097:
4093:
4071:
4064:
4060:
4041:
4034:
4030:
4011:
4004:
4001:
3995:
3993:Generalizations
3980:
3975:
3961:
3953:
3949:
3939:
3928:
3921:
3908:
3901:
3895:
3887:
3884:
3878:
3877:
3875:
3874:
3867:
3859:2 − 1
3858:
3855:4 + 1
3854:
3847:4 − 1
3846:
3842:
3834:
3784:
3779:
3771:
3766:
3760:
3753:
3738:
3509:
3504:
3498:
3493:
3486:
3471:
3466:
3457:
3452:
3450:
3433:
3426:
3410:
3398:
3392:
3382:
3376:
3372:
3366:
3363:
3353:
3343:
3335:
3332:
3326:
3323:Lucas sequences
3312:
3305:
3301:
3297:
3293:
3286:
3285:, the equation
3267:
3263:
3255:
3251:
3247:
3243:
3239:
3235:
3231:
3227:
3223:
3219:
3211:
3203:
3195:
3187:
3179:
3175:
3165:
3161:
3151:
3143:
3139:
3131:
3123:
3115:
3111:
3100:
3096:
3095:if and only if
3088:
3084:
3080:
3076:
3067:
3060:
3052:
3049:
3042:
3039:
3036:
3035:
3033:
3031:
3024:
3014:
3003:
2995:
2988:
2979:Wieferich prime
2969:
2958:
2954:
2946:
2936:
2932:
2928:
2924:
2915:
2912:
2908:
2900:
2893:
2886:
2882:
2878:
2860:
2856:
2852:
2851:is a factor of
2848:
2837:
2830:
2829:is a factor of
2826:
2822:
2821:is a factor of
2818:
2814:
2813:is a factor of
2810:
2809:if and only if
2806:
2805:is a factor of
2802:
2798:
2794:
2793:is a factor of
2790:
2786:
2782:
2778:
2774:
2770:
2766:
2762:
2756:
2752:
2751:is a factor of
2748:
2744:
2743:is a factor of
2740:
2726:
2722:
2719:2 − 1 = 23 × 89
2718:
2714:
2710:
2703:
2696:
2692:
2688:
2684:
2671:
2667:
2664:
2660:
2656:
2653:
2650:
2647:
2640:
2636:
2626:
2622:
2621:: Suppose that
2612:
2611:is prime, then
2608:
2596:
2592:
2585:
2578:
2571:
2564:
2557:
2547:
2540:
2529:
2522:
2511:
2500:
2489:
2475:
2462:
2455:
2454:is prime, then
2448:
2444:
2438:
2424:
2421:
2401:
2394:
2385:
2376:
2369:
2354:
2330:
2325:
2321:
2309:
2305:
2297:
2293:
2288:
2285:
2279:
2276:
2270:
2267:
2261:
2258:
2252:
2249:
2243:
2240:
2234:
2231:
2225:
2203:
2197:
2145:
2142:
2141:
2123:
2119:
2108:
2103:
2099:
2093:
2090:
2081:
2077:
2072:
2064:
2059:
2052:
2038:
2032:
2029:
2023:
2016:
2010:
2003:
1997:
1990:
1984:
1977:
1971:
1968:
1962:
1958:
1952:
1948:
1942:
1938:
1932:
1928:
1922:
1918:
1912:
1902:
1892:
1888:
1881:
1875:
1868:
1862:
1843:
1828:
1824:
1819:
1816:
1815:
1812:
1806:
1797:
1791:
1788:
1782:
1779:
1773:
1770:
1764:
1761:
1755:
1515:
1491:
1465:
1462:
1461:
1435:
1431:
1420:
1417:
1416:
1397:
1396:is prime, then
1393:
1392:proved that if
1386:perfect numbers
1382:
1374:
1371:
1365:
1363:Perfect numbers
1333:binary computer
1307:
1303:
1300:
1296:
1292:
1288:
1285:
1281:
1273:
1267:
1260:
1257:
1253:
1246:
1240:
1226:
1225:
1196:
1192:
1174:
1170:
1158:
1154:
1145:
1141:
1134:
1130:
1112:
1108:
1096:
1095:
1069:
1065:
1047:
1043:
1031:
1027:
1018:
1014:
1007:
1003:
985:
981:
971:
956:
952:
948:
946:
943:
942:
936:
933:
929:
919:
912:
904:
898:
894:
888:
885:
879:
875:
869:
865:
859:
855:
849:
845:
839:
822:
818:
803:
799:
797:
794:
793:
770:
766:
748:
744:
742:
739:
738:
730:
726:
690:
686:
684:
681:
680:
676:
672:
624:
622:
620:
617:
616:
608:
596:
584:
580:
570:
567:
560:
557:
550:
547:
540:
537:
530:
527:
520:
517:
510:
507:
500:
497:
490:
489:, for example,
487:
483:
475:
471:
436:
432:
430:
427:
426:
393:
389:
380:
376:
374:
371:
370:
367:order of growth
356:
355:
350:
347:
340:
323:
301:
291:
282:
277:
264:
234:
228:
216:
215:for some prime
211:
206:
202:
192:
178:
170:
165:
139:
106:
28:
23:
22:
18:Mersenne number
15:
12:
11:
5:
12326:
12316:
12315:
12310:
12305:
12300:
12295:
12290:
12273:
12272:
12270:
12269:
12264:
12258:
12255:
12254:
12252:
12251:
12246:
12241:
12239:Power of three
12236:
12231:
12226:
12221:
12219:Number systems
12216:
12211:
12206:
12201:
12196:
12191:
12186:
12180:
12178:
12174:(alphabetical
12167:
12166:
12163:
12162:
12160:
12159:
12154:
12149:
12144:
12143:
12142:
12137:
12130:Hyperoperation
12126:
12124:
12120:
12119:
12117:
12116:
12111:
12106:
12101:
12095:
12093:
12086:
12080:
12079:
12077:
12076:
12071:
12066:
12061:
12056:
12051:
12046:
12044:Moser's number
12041:
12036:
12031:
12029:Shannon number
12026:
12021:
12016:
12011:
12006:
12001:
11996:
11991:
11986:
11981:
11976:
11971:
11966:
11961:
11956:
11951:
11945:
11943:
11933:
11932:
11925:
11924:
11917:
11910:
11902:
11893:
11892:
11890:
11889:
11884:
11883:
11882:
11877:
11872:
11865:Mersenne prime
11862:
11857:
11851:
11848:
11847:
11844:Marin Mersenne
11840:
11839:
11832:
11825:
11817:
11808:
11807:
11805:
11804:
11793:
11790:
11789:
11786:
11785:
11783:
11782:
11776:
11773:
11772:
11759:
11758:
11755:
11754:
11752:
11751:
11746:
11740:
11737:
11736:
11723:
11722:
11719:
11718:
11716:
11715:
11713:Sorting number
11710:
11708:Pancake number
11704:
11701:
11700:
11687:
11686:
11683:
11682:
11680:
11679:
11674:
11668:
11665:
11664:
11651:
11650:
11647:
11646:
11644:
11643:
11638:
11633:
11627:
11624:
11623:
11620:Binary numbers
11611:
11610:
11607:
11606:
11603:
11602:
11600:
11599:
11593:
11591:
11587:
11586:
11584:
11583:
11578:
11573:
11568:
11563:
11558:
11553:
11547:
11545:
11541:
11540:
11538:
11537:
11532:
11527:
11522:
11517:
11511:
11509:
11501:
11500:
11498:
11497:
11492:
11487:
11482:
11477:
11472:
11467:
11461:
11459:
11452:
11451:
11449:
11448:
11447:
11446:
11435:
11433:
11430:P-adic numbers
11426:
11425:
11422:
11421:
11419:
11418:
11417:
11416:
11406:
11401:
11396:
11391:
11386:
11381:
11376:
11371:
11365:
11363:
11359:
11358:
11356:
11355:
11349:
11347:
11346:Coding-related
11343:
11342:
11340:
11339:
11334:
11328:
11326:
11322:
11321:
11319:
11318:
11313:
11308:
11303:
11297:
11295:
11286:
11285:
11284:
11283:
11281:Multiplicative
11278:
11267:
11265:
11250:
11249:
11245:Numeral system
11236:
11235:
11232:
11231:
11229:
11228:
11223:
11218:
11213:
11208:
11203:
11198:
11193:
11188:
11183:
11178:
11173:
11168:
11163:
11158:
11153:
11147:
11144:
11143:
11125:
11124:
11121:
11120:
11117:
11116:
11114:
11113:
11108:
11102:
11100:
11094:
11093:
11091:
11090:
11085:
11080:
11075:
11069:
11067:
11061:
11060:
11058:
11057:
11052:
11047:
11042:
11037:
11035:Highly totient
11032:
11026:
11024:
11018:
11017:
11015:
11014:
11009:
11003:
11001:
10995:
10994:
10992:
10991:
10986:
10981:
10976:
10971:
10966:
10961:
10956:
10951:
10946:
10941:
10936:
10931:
10926:
10921:
10916:
10911:
10906:
10901:
10896:
10891:
10889:Almost perfect
10886:
10880:
10878:
10868:
10867:
10851:
10850:
10847:
10846:
10844:
10843:
10838:
10833:
10828:
10823:
10818:
10813:
10808:
10803:
10798:
10793:
10788:
10782:
10779:
10778:
10766:
10765:
10762:
10761:
10759:
10758:
10753:
10748:
10743:
10737:
10734:
10733:
10721:
10720:
10717:
10716:
10714:
10713:
10708:
10703:
10698:
10696:Stirling first
10693:
10688:
10683:
10678:
10673:
10668:
10663:
10658:
10653:
10648:
10643:
10638:
10633:
10628:
10623:
10618:
10612:
10609:
10608:
10598:
10597:
10594:
10593:
10590:
10589:
10586:
10585:
10583:
10582:
10577:
10572:
10566:
10564:
10557:
10551:
10550:
10547:
10546:
10544:
10543:
10537:
10535:
10529:
10528:
10526:
10525:
10520:
10515:
10510:
10505:
10500:
10494:
10492:
10486:
10485:
10483:
10482:
10477:
10472:
10467:
10462:
10456:
10454:
10445:
10439:
10438:
10435:
10434:
10432:
10431:
10426:
10421:
10416:
10411:
10406:
10401:
10396:
10391:
10386:
10380:
10378:
10372:
10371:
10369:
10368:
10363:
10358:
10353:
10348:
10343:
10338:
10333:
10328:
10322:
10320:
10311:
10301:
10300:
10288:
10287:
10284:
10283:
10281:
10280:
10275:
10270:
10265:
10260:
10255:
10249:
10246:
10245:
10235:
10234:
10231:
10230:
10228:
10227:
10222:
10217:
10212:
10207:
10201:
10198:
10197:
10187:
10186:
10183:
10182:
10180:
10179:
10174:
10169:
10164:
10159:
10154:
10149:
10144:
10138:
10135:
10134:
10121:
10120:
10117:
10116:
10114:
10113:
10108:
10103:
10098:
10093:
10087:
10084:
10083:
10073:
10072:
10069:
10068:
10066:
10065:
10060:
10055:
10050:
10045:
10040:
10035:
10029:
10026:
10025:
10011:
10010:
10007:
10006:
10004:
10003:
9998:
9993:
9988:
9983:
9978:
9973:
9968:
9963:
9958:
9953:
9948:
9943:
9938:
9932:
9929:
9928:
9915:
9914:
9906:
9905:
9898:
9891:
9883:
9874:
9873:
9868:
9865:
9864:
9862:
9861:
9856:
9851:
9846:
9841:
9836:
9831:
9826:
9821:
9816:
9811:
9806:
9801:
9796:
9791:
9786:
9781:
9776:
9771:
9766:
9761:
9756:
9751:
9746:
9741:
9736:
9731:
9726:
9721:
9716:
9711:
9706:
9701:
9696:
9691:
9686:
9681:
9676:
9671:
9666:
9661:
9656:
9651:
9646:
9641:
9636:
9631:
9626:
9621:
9616:
9611:
9606:
9601:
9596:
9591:
9586:
9581:
9576:
9571:
9566:
9560:
9558:
9554:
9553:
9551:
9550:
9545:
9540:
9535:
9530:
9528:Probable prime
9524:
9522:
9521:Related topics
9518:
9517:
9515:
9514:
9509:
9504:
9502:Sphenic number
9499:
9494:
9489:
9484:
9483:
9482:
9477:
9472:
9467:
9462:
9457:
9452:
9447:
9442:
9437:
9426:
9424:
9418:
9417:
9415:
9414:
9412:Gaussian prime
9409:
9403:
9401:
9395:
9394:
9392:
9391:
9390:
9380:
9375:
9373:
9369:
9368:
9366:
9365:
9341:
9337: + 1
9325:
9320:
9299:
9296:
9295:
9293:
9292:
9264:
9244:
9240: + 6
9228:
9224: + 4
9212:
9208: + 8
9188:
9184: + 6
9164:
9160: + 2
9147:
9145:
9133:
9129:
9128:
9126:
9125:
9120:
9115:
9110:
9105:
9100:
9095:
9090:
9085:
9080:
9075:
9070:
9065:
9060:
9055:
9047:
9042:
9036:
9034:
9027:
9026:
9024:
9023:
9018:
9013:
9008:
9003:
8998:
8993:
8988:
8983:
8978:
8973:
8968:
8963:
8958:
8953:
8948:
8943:
8938:
8927:
8925:
8921:
8920:
8918:
8917:
8912:
8907:
8902:
8897:
8891:
8889:
8885:
8884:
8882:
8881:
8864:
8860: − 1
8847:
8838:
8823:
8800:
8788:
8776:
8767:
8752:
8743:
8739: + 1
8730:
8722:
8715:
8707:
8700:
8688:
8676:
8668:
8659:
8650:
8640:
8638:
8634:
8633:
8625:
8624:
8617:
8610:
8602:
8596:
8595:
8576:
8555:
8552:
8551:
8550:
8538:
8526:
8511:
8506:
8494:
8482:
8470:
8458:
8453:
8448:
8443:
8437:
8432:
8423:
8417:
8392:Grime, James.
8389:
8367:
8361:
8342:
8336:
8331:
8325:
8320:
8300:
8299:External links
8297:
8295:
8294:
8267:
8263:
8259:
8254:
8250:
8246:
8241:
8237:
8233:
8214:
8187:
8183:
8179:
8174:
8170:
8166:
8161:
8157:
8153:
8134:
8112:
8086:
8060:
8034:
8012:
7990:
7960:
7939:
7922:
7915:
7889:
7886:on 2012-05-29.
7871:
7866:"OEIS A016131"
7857:
7835:
7810:
7803:
7785:
7760:
7740:
7715:
7694:
7687:
7661:
7640:
7622:
7597:
7577:
7563:
7545:
7521:
7518:on 2016-09-03.
7503:
7475:
7447:
7421:
7394:
7369:
7363:Brian Napper,
7356:
7333:
7311:
7291:
7278:(3): 134–138.
7258:
7236:
7223:
7211:
7196:
7170:
7143:
7141:
7138:
7135:
7134:
7065:
6960:
6902:
6887:
6886:
6884:
6881:
6879:
6878:
6873:
6868:
6863:
6858:
6853:
6848:
6846:Wagstaff prime
6843:
6838:
6833:
6827:
6821:
6816:
6811:
6806:
6801:
6796:
6791:
6785:
6783:
6780:
6779:
6778:
6745:
6744:
6631:
6630:
6625:
6622:
6619:
6615:
6614:
6612:
6609:
6606:
6602:
6601:
6596:
6593:
6590:
6586:
6585:
6580:
6577:
6574:
6570:
6569:
6564:
6561:
6558:
6554:
6553:
6548:
6545:
6542:
6538:
6537:
6532:
6529:
6526:
6522:
6521:
6516:
6513:
6510:
6506:
6505:
6500:
6497:
6494:
6490:
6489:
6487:
6484:
6481:
6477:
6476:
6474:
6471:
6468:
6464:
6463:
6461:
6458:
6455:
6451:
6450:
6448:
6445:
6442:
6438:
6437:
6432:
6429:
6426:
6422:
6421:
6416:
6413:
6410:
6406:
6405:
6400:
6397:
6394:
6390:
6389:
6384:
6381:
6378:
6374:
6373:
6368:
6365:
6362:
6358:
6357:
6352:
6349:
6346:
6342:
6341:
6336:
6333:
6330:
6326:
6325:
6320:
6317:
6314:
6310:
6309:
6304:
6301:
6298:
6294:
6293:
6288:
6285:
6282:
6278:
6277:
6272:
6269:
6266:
6262:
6261:
6256:
6253:
6250:
6246:
6245:
6240:
6237:
6234:
6230:
6229:
6224:
6221:
6218:
6214:
6213:
6208:
6205:
6202:
6198:
6197:
6192:
6189:
6186:
6182:
6181:
6179:
6176:
6173:
6169:
6168:
6163:
6160:
6157:
6153:
6152:
6147:
6144:
6141:
6137:
6136:
6131:
6128:
6125:
6121:
6120:
6115:
6112:
6109:
6105:
6104:
6099:
6096:
6093:
6089:
6088:
6083:
6080:
6077:
6073:
6072:
6067:
6064:
6061:
6057:
6056:
6051:
6048:
6045:
6041:
6040:
6035:
6032:
6029:
6025:
6024:
6019:
6016:
6013:
6009:
6008:
6003:
6000:
5997:
5993:
5992:
5987:
5984:
5981:
5977:
5976:
5974:
5971:
5968:
5964:
5963:
5958:
5955:
5952:
5948:
5947:
5945:
5944:2 (no others)
5942:
5939:
5935:
5934:
5929:
5926:
5923:
5919:
5918:
5913:
5910:
5907:
5903:
5902:
5897:
5894:
5891:
5887:
5886:
5881:
5878:
5875:
5871:
5870:
5865:
5862:
5859:
5855:
5854:
5849:
5846:
5843:
5839:
5838:
5836:
5835:2 (no others)
5833:
5830:
5826:
5825:
5823:
5822:3 (no others)
5820:
5817:
5813:
5812:
5807:
5804:
5801:
5797:
5796:
5791:
5788:
5785:
5781:
5780:
5775:
5772:
5769:
5765:
5764:
5759:
5756:
5753:
5749:
5748:
5743:
5740:
5737:
5733:
5732:
5727:
5724:
5721:
5717:
5716:
5711:
5708:
5705:
5701:
5700:
5695:
5692:
5689:
5685:
5684:
5679:
5676:
5673:
5669:
5668:
5663:
5660:
5657:
5653:
5652:
5647:
5644:
5641:
5637:
5636:
5631:
5628:
5625:
5621:
5620:
5615:
5612:
5609:
5605:
5604:
5599:
5596:
5593:
5589:
5588:
5583:
5580:
5577:
5573:
5572:
5567:
5564:
5561:
5557:
5556:
5551:
5548:
5545:
5541:
5540:
5535:
5532:
5529:
5525:
5524:
5519:
5516:
5513:
5509:
5508:
5503:
5500:
5497:
5493:
5492:
5487:
5484:
5481:
5477:
5476:
5471:
5468:
5465:
5461:
5460:
5455:
5452:
5449:
5445:
5444:
5439:
5436:
5433:
5429:
5428:
5423:
5420:
5417:
5413:
5412:
5407:
5404:
5401:
5397:
5396:
5391:
5388:
5385:
5381:
5380:
5375:
5372:
5369:
5365:
5364:
5362:
5359:
5356:
5352:
5351:
5349:
5348:2 (no others)
5346:
5343:
5339:
5338:
5333:
5330:
5327:
5323:
5322:
5317:
5314:
5311:
5307:
5306:
5301:
5298:
5295:
5291:
5290:
5285:
5282:
5279:
5275:
5274:
5269:
5266:
5263:
5259:
5258:
5253:
5249:
5246:
5242:
5241:
5236:
5232:
5229:
5225:
5224:
5218:
5113:
5106:
4918:
4917:
4903:
4900:
4897:
4890:
4886:
4882:
4877:
4873:
4854:
4851:
4850:
4849:
4825:
4824:
4776:
4775:
4727:
4726:
4582:
4581:
4549:
4548:
4500:
4499:
4485:
4482:
4479:
4474:
4471:
4466:
4462:
4420:Main article:
4417:
4416:Repunit primes
4414:
4412:
4409:
4408:
4407:
4390:
4389:
4327:
4324:
4323:
4322:
4298:
4297:
4259:Gaussian prime
4212:
4209:
4148:
4145:
4119:theorems above
3997:Main article:
3994:
3991:
3973:
3958:
3957:
3873:is defined as
3866:
3863:
3827:right triangle
3782:
3769:
3757:Tower of Hanoi
3752:
3749:
3733:
3732:
3729:
3726:
3722:
3721:
3718:
3715:
3711:
3710:
3707:
3704:
3700:
3699:
3696:
3693:
3689:
3688:
3685:
3682:
3678:
3677:
3674:
3671:
3667:
3666:
3663:
3660:
3656:
3655:
3652:
3649:
3645:
3644:
3641:
3638:
3634:
3633:
3630:
3627:
3623:
3622:
3619:
3616:
3612:
3611:
3608:
3605:
3601:
3600:
3597:
3594:
3590:
3589:
3586:
3583:
3579:
3578:
3575:
3574:8796093022207
3572:
3568:
3567:
3564:
3563:2199023255551
3561:
3557:
3556:
3553:
3550:
3546:
3545:
3542:
3539:
3535:
3534:
3531:
3528:
3524:
3523:
3520:
3517:
3513:
3512:
3507:
3501:
3496:
3491:
3464:
3447:probable prime
3443:primality test
3425:
3422:
3421:
3420:
3394:Main article:
3391:
3388:
3387:
3386:
3380:
3370:
3358:
3348:
3339:
3328:
3319:
3279:
3278:to the base 2.
3272:
3271:
3270:
3265:
3169:
3162:11 = 2 × 4 + 3
3108:
3073:
3072:
3071:
3068:−1 = 0 (mod p)
2975:
2974:
2973:
2921:
2920:
2919:
2910:
2890:
2871:
2730:
2713:is prime, and
2681:
2680:
2679:
2674:is prime then
2662:
2658:
2605:
2604:
2603:
2420:
2417:
2413:Probable prime
2406:Ocala, Florida
2383:
2328:
2323:
2318:floor function
2303:
2291:
2283:
2274:
2265:
2256:
2247:
2238:
2229:
2222:R. M. Robinson
2201:
2173:
2170:
2167:
2164:
2161:
2158:
2155:
2152:
2149:
2114:
2106:
2097:
2085:
2075:
2070:if and only if
2062:
2036:
2027:
2014:
2009:in 1876, then
2001:
1994:Leonhard Euler
1988:
1981:Pietro Cataldi
1975:
1966:
1956:
1946:
1936:
1926:
1916:
1901:
1898:
1879:
1866:
1850:
1846:
1842:
1839:
1836:
1831:
1827:
1823:
1810:
1795:
1786:
1777:
1768:
1759:
1752:
1751:
1750:
1749:
1740:Marin Mersenne
1730:
1729:
1725:
1724:
1721:
1718:
1715:
1712:
1709:
1706:
1703:
1699:
1698:
1695:
1692:
1689:
1686:
1683:
1680:
1677:
1673:
1672:
1669:
1666:
1663:
1660:
1657:
1654:
1651:
1647:
1646:
1643:
1640:
1637:
1634:
1631:
1628:
1625:
1621:
1620:
1617:
1614:
1611:
1608:
1605:
1602:
1599:
1595:
1594:
1591:
1588:
1585:
1582:
1579:
1576:
1573:
1569:
1568:
1565:
1562:
1559:
1556:
1553:
1550:
1547:
1543:
1542:
1539:
1536:
1533:
1530:
1527:
1524:
1521:
1514:
1511:
1498:
1494:
1490:
1487:
1484:
1481:
1478:
1475:
1472:
1469:
1449:
1446:
1443:
1438:
1434:
1430:
1427:
1424:
1402:Leonhard Euler
1378:
1367:Main article:
1364:
1361:
1322:cult following
1318:primality test
1305:
1298:
1283:
1278:
1277:
1271:
1255:
1244:
1224:
1220:
1214:
1211:
1208:
1205:
1202:
1199:
1195:
1191:
1188:
1185:
1180:
1177:
1173:
1169:
1164:
1161:
1157:
1153:
1148:
1144:
1140:
1137:
1133:
1129:
1126:
1123:
1120:
1115:
1111:
1107:
1104:
1101:
1099:
1097:
1093:
1087:
1084:
1081:
1078:
1075:
1072:
1068:
1064:
1061:
1058:
1053:
1050:
1046:
1042:
1037:
1034:
1030:
1026:
1021:
1017:
1013:
1010:
1006:
1002:
999:
996:
993:
988:
984:
980:
977:
974:
972:
970:
967:
962:
959:
955:
951:
950:
931:
917:
902:
892:
883:
873:
863:
853:
843:
825:
821:
817:
814:
811:
806:
802:
781:
778:
773:
769:
765:
762:
759:
756:
751:
747:
710:
707:
704:
701:
698:
693:
689:
660:
657:
652:
648:
645:
642:
639:
636:
633:
630:
627:
565:
555:
545:
535:
525:
515:
505:
495:
485:
439:
435:
421:≈ 5.92 primes
410:
407:
404:
401:
396:
392:
388:
383:
379:
351:
348:
342:
339:
336:
280:
209:
185:Marin Mersenne
168:
151:Mersenne prime
141:
140:
138:
137:
126:
120:
118:
111:
110:
104:
100:
99:
81:
77:
76:
73:
66:
65:
62:
54:
53:
50:
49:of known terms
43:
42:
40:Marin Mersenne
37:
31:Mersenne prime
26:
9:
6:
4:
3:
2:
12325:
12314:
12311:
12309:
12306:
12304:
12301:
12299:
12296:
12294:
12291:
12289:
12286:
12285:
12283:
12268:
12265:
12263:
12260:
12259:
12256:
12250:
12247:
12245:
12242:
12240:
12237:
12235:
12232:
12230:
12227:
12225:
12222:
12220:
12217:
12215:
12212:
12210:
12207:
12205:
12202:
12200:
12199:Infinitesimal
12197:
12195:
12192:
12190:
12187:
12185:
12182:
12181:
12179:
12168:
12158:
12155:
12153:
12150:
12148:
12145:
12141:
12138:
12136:
12133:
12132:
12131:
12128:
12127:
12125:
12121:
12115:
12112:
12110:
12107:
12105:
12102:
12100:
12097:
12096:
12094:
12090:
12087:
12081:
12075:
12072:
12070:
12069:Rayo's number
12067:
12065:
12062:
12060:
12057:
12055:
12052:
12050:
12047:
12045:
12042:
12040:
12037:
12035:
12032:
12030:
12027:
12025:
12022:
12020:
12017:
12015:
12012:
12010:
12007:
12005:
12002:
12000:
11997:
11995:
11992:
11990:
11987:
11985:
11982:
11980:
11977:
11975:
11972:
11970:
11967:
11965:
11962:
11960:
11957:
11955:
11952:
11950:
11947:
11946:
11944:
11934:
11930:
11929:Large numbers
11923:
11918:
11916:
11911:
11909:
11904:
11903:
11900:
11888:
11885:
11881:
11878:
11876:
11873:
11871:
11868:
11867:
11866:
11863:
11861:
11858:
11856:
11853:
11852:
11849:
11845:
11838:
11833:
11831:
11826:
11824:
11819:
11818:
11815:
11803:
11799:
11795:
11794:
11791:
11781:
11778:
11777:
11774:
11769:
11764:
11760:
11750:
11747:
11745:
11742:
11741:
11738:
11733:
11728:
11724:
11714:
11711:
11709:
11706:
11705:
11702:
11697:
11692:
11688:
11678:
11675:
11673:
11670:
11669:
11666:
11662:
11656:
11652:
11642:
11639:
11637:
11634:
11632:
11629:
11628:
11625:
11621:
11616:
11612:
11598:
11595:
11594:
11592:
11588:
11582:
11579:
11577:
11574:
11572:
11571:Polydivisible
11569:
11567:
11564:
11562:
11559:
11557:
11554:
11552:
11549:
11548:
11546:
11542:
11536:
11533:
11531:
11528:
11526:
11523:
11521:
11518:
11516:
11513:
11512:
11510:
11507:
11502:
11496:
11493:
11491:
11488:
11486:
11483:
11481:
11478:
11476:
11473:
11471:
11468:
11466:
11463:
11462:
11460:
11457:
11453:
11445:
11442:
11441:
11440:
11437:
11436:
11434:
11431:
11427:
11415:
11412:
11411:
11410:
11407:
11405:
11402:
11400:
11397:
11395:
11392:
11390:
11387:
11385:
11382:
11380:
11377:
11375:
11372:
11370:
11367:
11366:
11364:
11360:
11354:
11351:
11350:
11348:
11344:
11338:
11335:
11333:
11330:
11329:
11327:
11325:Digit product
11323:
11317:
11314:
11312:
11309:
11307:
11304:
11302:
11299:
11298:
11296:
11294:
11290:
11282:
11279:
11277:
11274:
11273:
11272:
11269:
11268:
11266:
11264:
11259:
11255:
11251:
11246:
11241:
11237:
11227:
11224:
11222:
11219:
11217:
11214:
11212:
11209:
11207:
11204:
11202:
11199:
11197:
11194:
11192:
11189:
11187:
11184:
11182:
11179:
11177:
11174:
11172:
11169:
11167:
11164:
11162:
11161:Erdős–Nicolas
11159:
11157:
11154:
11152:
11149:
11148:
11145:
11140:
11136:
11130:
11126:
11112:
11109:
11107:
11104:
11103:
11101:
11099:
11095:
11089:
11086:
11084:
11081:
11079:
11076:
11074:
11071:
11070:
11068:
11066:
11062:
11056:
11053:
11051:
11048:
11046:
11043:
11041:
11038:
11036:
11033:
11031:
11028:
11027:
11025:
11023:
11019:
11013:
11010:
11008:
11005:
11004:
11002:
11000:
10996:
10990:
10987:
10985:
10982:
10980:
10979:Superabundant
10977:
10975:
10972:
10970:
10967:
10965:
10962:
10960:
10957:
10955:
10952:
10950:
10947:
10945:
10942:
10940:
10937:
10935:
10932:
10930:
10927:
10925:
10922:
10920:
10917:
10915:
10912:
10910:
10907:
10905:
10902:
10900:
10897:
10895:
10892:
10890:
10887:
10885:
10882:
10881:
10879:
10877:
10873:
10869:
10865:
10861:
10856:
10852:
10842:
10839:
10837:
10834:
10832:
10829:
10827:
10824:
10822:
10819:
10817:
10814:
10812:
10809:
10807:
10804:
10802:
10799:
10797:
10794:
10792:
10789:
10787:
10784:
10783:
10780:
10776:
10771:
10767:
10757:
10754:
10752:
10749:
10747:
10744:
10742:
10739:
10738:
10735:
10731:
10726:
10722:
10712:
10709:
10707:
10704:
10702:
10699:
10697:
10694:
10692:
10689:
10687:
10684:
10682:
10679:
10677:
10674:
10672:
10669:
10667:
10664:
10662:
10659:
10657:
10654:
10652:
10649:
10647:
10644:
10642:
10639:
10637:
10634:
10632:
10629:
10627:
10624:
10622:
10619:
10617:
10614:
10613:
10610:
10603:
10599:
10581:
10578:
10576:
10573:
10571:
10568:
10567:
10565:
10561:
10558:
10556:
10555:4-dimensional
10552:
10542:
10539:
10538:
10536:
10534:
10530:
10524:
10521:
10519:
10516:
10514:
10511:
10509:
10506:
10504:
10501:
10499:
10496:
10495:
10493:
10491:
10487:
10481:
10478:
10476:
10473:
10471:
10468:
10466:
10465:Centered cube
10463:
10461:
10458:
10457:
10455:
10453:
10449:
10446:
10444:
10443:3-dimensional
10440:
10430:
10427:
10425:
10422:
10420:
10417:
10415:
10412:
10410:
10407:
10405:
10402:
10400:
10397:
10395:
10392:
10390:
10387:
10385:
10382:
10381:
10379:
10377:
10373:
10367:
10364:
10362:
10359:
10357:
10354:
10352:
10349:
10347:
10344:
10342:
10339:
10337:
10334:
10332:
10329:
10327:
10324:
10323:
10321:
10319:
10315:
10312:
10310:
10309:2-dimensional
10306:
10302:
10298:
10293:
10289:
10279:
10276:
10274:
10271:
10269:
10266:
10264:
10261:
10259:
10256:
10254:
10253:Nonhypotenuse
10251:
10250:
10247:
10240:
10236:
10226:
10223:
10221:
10218:
10216:
10213:
10211:
10208:
10206:
10203:
10202:
10199:
10192:
10188:
10178:
10175:
10173:
10170:
10168:
10165:
10163:
10160:
10158:
10155:
10153:
10150:
10148:
10145:
10143:
10140:
10139:
10136:
10131:
10126:
10122:
10112:
10109:
10107:
10104:
10102:
10099:
10097:
10094:
10092:
10089:
10088:
10085:
10078:
10074:
10064:
10061:
10059:
10056:
10054:
10051:
10049:
10046:
10044:
10041:
10039:
10036:
10034:
10031:
10030:
10027:
10022:
10016:
10012:
10002:
9999:
9997:
9994:
9992:
9991:Perfect power
9989:
9987:
9984:
9982:
9981:Seventh power
9979:
9977:
9974:
9972:
9969:
9967:
9964:
9962:
9959:
9957:
9954:
9952:
9949:
9947:
9944:
9942:
9939:
9937:
9934:
9933:
9930:
9925:
9920:
9916:
9912:
9904:
9899:
9897:
9892:
9890:
9885:
9884:
9881:
9871:
9866:
9860:
9857:
9855:
9852:
9850:
9847:
9845:
9842:
9840:
9837:
9835:
9832:
9830:
9827:
9825:
9822:
9820:
9817:
9815:
9812:
9810:
9807:
9805:
9802:
9800:
9797:
9795:
9792:
9790:
9787:
9785:
9782:
9780:
9777:
9775:
9772:
9770:
9767:
9765:
9762:
9760:
9757:
9755:
9752:
9750:
9747:
9745:
9742:
9740:
9737:
9735:
9732:
9730:
9727:
9725:
9722:
9720:
9717:
9715:
9712:
9710:
9707:
9705:
9702:
9700:
9697:
9695:
9692:
9690:
9687:
9685:
9682:
9680:
9677:
9675:
9672:
9670:
9667:
9665:
9662:
9660:
9657:
9655:
9652:
9650:
9647:
9645:
9642:
9640:
9637:
9635:
9632:
9630:
9627:
9625:
9622:
9620:
9617:
9615:
9612:
9610:
9607:
9605:
9602:
9600:
9597:
9595:
9592:
9590:
9587:
9585:
9582:
9580:
9577:
9575:
9572:
9570:
9567:
9565:
9562:
9561:
9559:
9555:
9549:
9546:
9544:
9541:
9539:
9538:Illegal prime
9536:
9534:
9531:
9529:
9526:
9525:
9523:
9519:
9513:
9510:
9508:
9505:
9503:
9500:
9498:
9495:
9493:
9490:
9488:
9485:
9481:
9478:
9476:
9473:
9471:
9468:
9466:
9463:
9461:
9458:
9456:
9453:
9451:
9448:
9446:
9443:
9441:
9438:
9436:
9433:
9432:
9431:
9428:
9427:
9425:
9423:
9419:
9413:
9410:
9408:
9405:
9404:
9402:
9400:
9396:
9389:
9386:
9385:
9384:
9383:Largest known
9381:
9379:
9376:
9374:
9370:
9364:
9360:
9356:
9352:
9348:
9342:
9340:
9336:
9332:
9326:
9324:
9321:
9319:
9315:
9311:
9307:
9301:
9300:
9291:
9288:
9285: +
9284:
9280:
9276:
9273: −
9272:
9265:
9263:
9259:
9255:
9252: +
9251:
9245:
9243:
9239:
9235:
9229:
9227:
9223:
9219:
9213:
9211:
9207:
9203:
9199:
9195:
9189:
9187:
9183:
9179:
9175:
9171:
9165:
9163:
9159:
9155:
9149:
9148:
9146:
9144:
9142:
9137:
9134:
9130:
9124:
9121:
9119:
9116:
9114:
9111:
9109:
9106:
9104:
9101:
9099:
9096:
9094:
9091:
9089:
9086:
9084:
9081:
9079:
9076:
9074:
9071:
9069:
9066:
9064:
9061:
9059:
9056:
9054:
9048:
9046:
9043:
9041:
9038:
9037:
9035:
9032:
9028:
9022:
9019:
9017:
9014:
9012:
9009:
9007:
9004:
9002:
8999:
8997:
8994:
8992:
8989:
8987:
8984:
8982:
8979:
8977:
8974:
8972:
8969:
8967:
8964:
8962:
8959:
8957:
8954:
8952:
8949:
8947:
8944:
8942:
8939:
8936:
8932:
8929:
8928:
8926:
8922:
8916:
8913:
8911:
8908:
8906:
8903:
8901:
8898:
8896:
8893:
8892:
8890:
8886:
8880:
8874:
8865:
8863:
8859:
8855:
8848:
8846:
8839:
8837:
8834:
8831: +
8830:
8824:
8822:
8819:
8816: −
8815:
8811:
8808: −
8807:
8801:
8799:
8795:
8789:
8787:
8783:
8777:
8775:
8768:
8766:
8763:
8760: +
8759:
8753:
8751:
8744:
8742:
8738:
8733:Pythagorean (
8731:
8729:
8725:
8716:
8714:
8710:
8701:
8699:
8695:
8689:
8687:
8683:
8677:
8675:
8669:
8667:
8660:
8658:
8651:
8649:
8642:
8641:
8639:
8635:
8630:
8623:
8618:
8616:
8611:
8609:
8604:
8603:
8600:
8591:
8590:
8585:
8582:
8577:
8572:
8571:
8566:
8563:
8558:
8557:
8549:
8545:
8542:
8539:
8537:
8533:
8530:
8527:
8525:
8522:
8518:
8512:
8510:
8507:
8503:
8497:
8489:
8483:
8481:
8477:
8474:
8471:
8469:
8465:
8462:
8459:
8457:
8454:
8452:
8449:
8447:
8444:
8441:
8438:
8436:
8433:
8427:
8424:
8421:
8418:
8408:on 2013-05-31
8407:
8403:
8399:
8395:
8390:
8387:
8383:
8379:
8375:
8370:
8366:
8362:
8359:
8354:
8350:
8345:
8341:
8337:
8335:
8332:
8329:
8326:
8324:
8321:
8317:
8313:
8312:
8307:
8303:
8302:
8292:
8289:
8285:
8265:
8261:
8252:
8248:
8244:
8239:
8235:
8218:
8212:
8209:
8205:
8185:
8181:
8172:
8168:
8164:
8159:
8155:
8138:
8132:
8124:
8116:
8110:
8102:
8098:
8090:
8084:
8076:
8072:
8064:
8058:
8050:
8046:
8038:
8032:
8024:
8016:
8010:
8002:
7994:
7988:
7980:
7972:
7964:
7955:
7950:
7943:
7936:
7933:(part of the
7932:
7926:
7918:
7912:
7908:
7904:
7900:
7893:
7885:
7881:
7875:
7867:
7861:
7846:
7839:
7825:
7821:
7814:
7806:
7800:
7796:
7789:
7774:
7770:
7764:
7750:
7744:
7729:
7725:
7719:
7705:
7698:
7690:
7684:
7680:
7676:
7672:
7665:
7657:
7656:
7651:
7644:
7638:
7634:
7631:
7626:
7611:
7607:
7601:
7587:
7581:
7573:
7567:
7559:
7555:
7549:
7535:
7531:
7525:
7517:
7513:
7507:
7492:
7491:
7486:
7479:
7464:
7463:
7462:New Scientist
7458:
7451:
7436:
7432:
7425:
7411:
7410:
7405:
7398:
7384:
7380:
7373:
7366:
7360:
7352:
7348:
7344:
7337:
7329:
7325:
7321:
7315:
7305:
7304:
7295:
7286:
7281:
7277:
7273:
7269:
7262:
7254:
7250:
7249:Wright, E. M.
7246:
7240:
7233:
7227:
7221:
7215:
7207:
7200:
7184:
7180:
7174:
7158:
7154:
7148:
7144:
7126:
7122:
7114:
7110:
7096:
7088:
7069:
7061:
7057:
7046:
7042:
7035:
7031:
7023:
7019:
7011:
7007:
7003:
6999:
6991:
6987:
6979:
6975:
6964:
6955:
6951:
6947:
6943:
6939:
6935:
6931:
6917:
6913:
6909:
6905:
6898:
6892:
6888:
6877:
6874:
6872:
6869:
6867:
6866:Solinas prime
6864:
6862:
6859:
6857:
6856:Woodall prime
6854:
6852:
6849:
6847:
6844:
6842:
6839:
6837:
6834:
6831:
6828:
6825:
6822:
6820:
6817:
6815:
6812:
6810:
6807:
6805:
6802:
6800:
6797:
6795:
6794:Fermat number
6792:
6790:
6787:
6786:
6776:
6771:
6765:
6764:
6763:
6762:is prime are
6760:
6756:
6742:
6737:
6731:
6730:
6729:
6728:is prime are
6726:
6722:
6711:
6708:
6704:
6697:
6687:
6683:
6673:
6669:
6661:
6657:
6651:
6639:
6629:
6626:
6623:
6620:
6617:
6616:
6613:
6610:
6607:
6604:
6603:
6600:
6597:
6594:
6591:
6588:
6587:
6584:
6581:
6578:
6575:
6572:
6571:
6568:
6565:
6562:
6559:
6556:
6555:
6552:
6549:
6546:
6543:
6540:
6539:
6536:
6533:
6530:
6527:
6524:
6523:
6520:
6517:
6514:
6511:
6508:
6507:
6504:
6501:
6498:
6495:
6492:
6491:
6488:
6485:
6482:
6479:
6478:
6475:
6472:
6469:
6466:
6465:
6462:
6459:
6456:
6453:
6452:
6449:
6446:
6443:
6440:
6439:
6436:
6433:
6430:
6427:
6424:
6423:
6420:
6417:
6414:
6411:
6408:
6407:
6404:
6401:
6398:
6395:
6392:
6391:
6388:
6385:
6382:
6379:
6376:
6375:
6372:
6369:
6366:
6363:
6360:
6359:
6356:
6353:
6350:
6347:
6344:
6343:
6340:
6337:
6334:
6331:
6328:
6327:
6324:
6321:
6318:
6315:
6312:
6311:
6308:
6305:
6302:
6299:
6296:
6295:
6292:
6289:
6286:
6283:
6280:
6279:
6276:
6273:
6270:
6267:
6264:
6263:
6260:
6257:
6254:
6251:
6248:
6247:
6244:
6241:
6238:
6235:
6232:
6231:
6228:
6225:
6222:
6219:
6216:
6215:
6212:
6209:
6206:
6203:
6200:
6199:
6196:
6193:
6190:
6187:
6184:
6183:
6180:
6177:
6174:
6171:
6170:
6167:
6164:
6161:
6158:
6155:
6154:
6151:
6148:
6145:
6142:
6139:
6138:
6135:
6132:
6129:
6126:
6123:
6122:
6119:
6116:
6113:
6110:
6107:
6106:
6103:
6100:
6097:
6094:
6091:
6090:
6087:
6084:
6081:
6078:
6075:
6074:
6071:
6068:
6065:
6062:
6059:
6058:
6055:
6052:
6049:
6046:
6043:
6042:
6039:
6036:
6033:
6030:
6027:
6026:
6023:
6020:
6017:
6014:
6011:
6010:
6007:
6004:
6001:
5998:
5995:
5994:
5991:
5988:
5985:
5982:
5979:
5978:
5975:
5972:
5969:
5966:
5965:
5962:
5959:
5956:
5953:
5950:
5949:
5946:
5943:
5940:
5937:
5936:
5933:
5930:
5927:
5924:
5921:
5920:
5917:
5914:
5911:
5908:
5905:
5904:
5901:
5898:
5895:
5892:
5889:
5888:
5885:
5882:
5879:
5876:
5873:
5872:
5869:
5866:
5863:
5860:
5857:
5856:
5853:
5850:
5847:
5844:
5841:
5840:
5837:
5834:
5831:
5828:
5827:
5824:
5821:
5818:
5815:
5814:
5811:
5808:
5805:
5802:
5799:
5798:
5795:
5792:
5789:
5786:
5783:
5782:
5779:
5776:
5773:
5770:
5767:
5766:
5763:
5760:
5757:
5754:
5751:
5750:
5747:
5744:
5741:
5738:
5735:
5734:
5731:
5728:
5725:
5722:
5719:
5718:
5715:
5712:
5709:
5706:
5703:
5702:
5699:
5696:
5693:
5690:
5687:
5686:
5683:
5680:
5677:
5674:
5671:
5670:
5667:
5664:
5661:
5658:
5655:
5654:
5651:
5648:
5645:
5642:
5639:
5638:
5635:
5632:
5629:
5626:
5623:
5622:
5619:
5616:
5613:
5610:
5607:
5606:
5603:
5600:
5597:
5594:
5591:
5590:
5587:
5584:
5581:
5578:
5575:
5574:
5571:
5568:
5565:
5562:
5559:
5558:
5555:
5552:
5549:
5546:
5543:
5542:
5539:
5536:
5533:
5530:
5527:
5526:
5523:
5520:
5517:
5514:
5511:
5510:
5507:
5504:
5501:
5498:
5495:
5494:
5491:
5488:
5485:
5482:
5479:
5478:
5475:
5472:
5469:
5466:
5463:
5462:
5459:
5456:
5453:
5450:
5447:
5446:
5443:
5440:
5437:
5434:
5431:
5430:
5427:
5424:
5421:
5418:
5415:
5414:
5411:
5408:
5405:
5402:
5399:
5398:
5395:
5392:
5389:
5386:
5383:
5382:
5379:
5376:
5373:
5370:
5367:
5366:
5363:
5360:
5357:
5354:
5353:
5350:
5347:
5344:
5341:
5340:
5337:
5334:
5331:
5328:
5325:
5324:
5321:
5318:
5315:
5312:
5309:
5308:
5305:
5302:
5299:
5296:
5293:
5292:
5289:
5286:
5283:
5280:
5277:
5276:
5273:
5270:
5267:
5264:
5261:
5260:
5257:
5254:
5250:
5247:
5244:
5243:
5240:
5237:
5233:
5230:
5227:
5226:
5222:
5219:
5213:
5208:
5204:5 < |
5193:
5188:
5176:
5158:
5146:
5142:
5134:
5130:
5114:
5107:
5100:
5099:
5093:
5089:
5085:
5074:
5070:
5062:
5058:
5045:
5040:
5016:
5012:
5005:
5001:
4994:
4990:
4972:
4968:
4962:
4958:
4952:
4948:
4944:
4936:
4931:
4901:
4898:
4895:
4888:
4884:
4880:
4875:
4871:
4860:
4859:
4858:
4847:
4842:
4836:
4835:
4834:
4831:
4822:
4817:
4811:
4810:
4809:
4808:is prime are
4801:
4792:
4782:
4771:
4766:
4761:
4755:
4754:
4753:
4750:
4739:
4733:
4724:
4719:
4713:
4712:
4711:
4708:
4697:
4686:
4677:
4667:
4661:
4653:
4644:
4634:
4628:
4617:
4608:
4598:
4593:
4592:perfect power
4588:
4577:
4572:
4567:
4566:
4565:
4562:
4555:
4546:
4541:
4534:
4529:
4524:
4523:
4522:
4519:
4512:
4506:
4483:
4480:
4477:
4472:
4469:
4464:
4460:
4449:
4448:
4447:
4444:
4437:
4430:
4423:
4405:
4400:
4395:
4394:
4393:
4387:
4382:
4377:
4376:
4375:
4372:
4365:
4359:
4357:
4352:
4348:
4342:
4338:
4333:
4320:
4315:
4310:
4309:
4308:
4306:
4301:
4295:
4290:
4284:
4283:
4282:
4279:
4272:
4266:
4264:
4260:
4254:
4247:
4242:
4237:
4233:
4227:
4223:
4218:
4208:
4206:
4202:
4198:
4194:
4188:
4174:
4169:
4163:
4158:
4154:
4144:
4142:
4138:
4132:
4125:
4120:
4114:
4107:
4100:
4090:
4086:
4082:
4078:
4074:
4067:
4056:
4052:
4048:
4044:
4037:
4028:
4024:
4018:
4014:
4007:
4000:
3990:
3988:
3983:
3976:
3969:
3965:
3948:
3947:
3946:
3942:
3937:
3936:Fermat number
3931:
3924:
3919:
3916:
3912:
3904:
3898:
3881:
3872:
3862:
3852:
3840:
3832:
3828:
3825:, an integer
3824:
3819:
3817:
3813:
3812:31 Euphrosyne
3809:
3805:
3801:
3800:8191 Mersenne
3797:
3793:
3788:
3785:
3777:
3772:
3763:
3758:
3748:
3746:
3741:
3730:
3727:
3724:
3723:
3719:
3716:
3713:
3712:
3708:
3705:
3702:
3701:
3697:
3694:
3691:
3690:
3686:
3683:
3680:
3679:
3675:
3672:
3669:
3668:
3664:
3661:
3658:
3657:
3653:
3650:
3647:
3646:
3642:
3639:
3636:
3635:
3631:
3628:
3625:
3624:
3620:
3617:
3614:
3613:
3609:
3606:
3603:
3602:
3598:
3595:
3592:
3591:
3587:
3584:
3581:
3580:
3576:
3573:
3570:
3569:
3565:
3562:
3559:
3558:
3554:
3552:137438953471
3551:
3548:
3547:
3543:
3540:
3537:
3536:
3533:47 × 178,481
3532:
3529:
3526:
3525:
3521:
3518:
3515:
3514:
3510:
3502:
3499:
3492:
3489:
3485:
3484:
3481:
3479:
3474:
3468:
3463:
3448:
3444:
3439:
3431:
3418:
3413:
3407:
3406:
3405:
3403:
3397:
3379:
3369:
3361:
3357:
3351:
3347:
3342:
3338:
3331:
3324:
3320:
3315:
3308:
3290:
3284:
3280:
3277:
3273:
3259:
3242:. Also since
3215:
3207:
3199:
3191:
3183:
3173:
3170:
3159:
3156:
3155:
3147:
3137:
3129:
3119:
3109:
3106:
3094:
3074:
3064:
3056:
3027:
3021:
3017:
3010:
3006:
2999:
2991:
2987:: We show if
2986:
2983:
2982:
2980:
2976:
2967:
2962:
2950:
2944:
2941:
2940:
2931:that divides
2922:
2904:
2896:
2891:
2876:
2872:
2867:
2863:
2844:
2840:
2833:
2773:is prime and
2759:
2738:
2734:
2731:
2709:For example,
2708:
2707:
2700:
2691:that divides
2682:
2677:
2643:
2633:
2629:
2620:
2617:
2616:
2606:
2599:
2588:
2581:
2574:
2567:
2560:
2554:
2550:
2543:
2536:
2532:
2528:is prime, so
2525:
2518:
2514:
2507:
2504:− 1 ≡ 0 (mod
2503:
2496:
2492:
2485:
2482:
2478:
2473:
2470:
2469:
2465:
2458:
2451:
2441:
2436:
2435:
2434:
2432:
2427:
2416:
2414:
2409:
2407:
2398:
2392:
2387:
2380:
2373:
2367:
2363:
2362:Curtis Cooper
2358:
2351:
2349:
2348:Dell OptiPlex
2344:
2340:
2335:
2331:
2319:
2313:
2301:
2294:
2282:
2273:
2264:
2255:
2246:
2237:
2228:
2223:
2219:
2215:
2211:
2208:
2200:
2195:
2191:
2165:
2159:
2156:
2150:
2147:
2138:
2134:
2130:
2126:
2117:
2113:
2109:
2096:
2088:
2084:
2078:
2071:
2065:
2055:
2050:
2045:
2043:
2035:
2026:
2021:
2013:
2008:
2007:Édouard Lucas
2000:
1995:
1987:
1982:
1974:
1965:
1955:
1945:
1935:
1925:
1915:
1909:
1907:
1897:
1886:
1878:
1873:
1865:
1848:
1844:
1837:
1834:
1829:
1825:
1809:
1804:
1803:Édouard Lucas
1800:
1794:
1785:
1776:
1767:
1758:
1747:
1746:
1745:
1744:
1743:
1741:
1737:
1726:
1722:
1719:
1716:
1713:
1710:
1707:
1704:
1701:
1700:
1696:
1693:
1690:
1687:
1684:
1681:
1678:
1675:
1674:
1670:
1667:
1664:
1661:
1658:
1655:
1652:
1649:
1648:
1644:
1641:
1638:
1635:
1632:
1629:
1626:
1623:
1622:
1618:
1615:
1612:
1609:
1606:
1603:
1600:
1597:
1596:
1592:
1589:
1586:
1583:
1580:
1577:
1574:
1571:
1570:
1566:
1563:
1560:
1557:
1554:
1551:
1548:
1545:
1544:
1540:
1537:
1534:
1531:
1528:
1525:
1522:
1519:
1518:
1510:
1496:
1492:
1485:
1482:
1479:
1473:
1470:
1467:
1444:
1441:
1436:
1432:
1428:
1425:
1413:
1411:
1407:
1403:
1391:
1387:
1381:
1377:
1370:
1360:
1358:
1354:
1350:
1346:
1342:
1339:. To find a
1338:
1334:
1329:
1327:
1323:
1319:
1315:
1310:
1270:
1266:
1265:
1264:
1250:
1243:
1222:
1218:
1212:
1206:
1203:
1200:
1193:
1189:
1186:
1183:
1178:
1175:
1171:
1167:
1162:
1159:
1155:
1151:
1146:
1142:
1138:
1135:
1131:
1127:
1121:
1118:
1113:
1109:
1102:
1100:
1091:
1085:
1079:
1076:
1073:
1066:
1062:
1059:
1056:
1051:
1048:
1044:
1040:
1035:
1032:
1028:
1024:
1019:
1015:
1011:
1008:
1004:
1000:
994:
991:
986:
982:
975:
973:
968:
965:
960:
957:
953:
939:
927:
922:
916:
910:
901:
891:
882:
872:
862:
852:
842:
823:
819:
815:
812:
809:
804:
800:
779:
776:
771:
767:
763:
757:
749:
734:
724:
723:prime factors
721:and 1 has no
708:
705:
699:
691:
658:
655:
650:
646:
643:
637:
634:
631:
628:
612:
606:
600:
594:
588:
578:
573:
564:
554:
544:
534:
524:
514:
504:
494:
479:
469:
465:
462:
458:
453:
437:
433:
424:
405:
399:
394:
390:
386:
381:
377:
368:
364:
359:
354:
335:
333:
329:
321:
316:
314:
310:
305:
299:
294:
289:
283:
274:
272:
267:
262:
258:
254:
250:
246:
242:
237:
231:
227:
222:
219:
212:
200:
195:
190:
186:
181:
177:
171:
164:
160:
156:
152:
148:
135:
131:
127:
125:
122:
121:
119:
116:
112:
105:
101:
97:
93:
89:
85:
82:
78:
74:
71:
67:
63:
55:
51:
44:
41:
38:
34:
19:
12234:Power of two
12224:Number names
11959:Ten thousand
11864:
11535:Transposable
11399:Narcissistic
11306:Digital root
11226:Super-Poulet
11186:Jordan–Pólya
11135:prime factor
11040:Noncototient
11007:Almost prime
10989:Superperfect
10964:Refactorable
10959:Quasiperfect
10934:Hyperperfect
10775:Pseudoprimes
10746:Wall–Sun–Sun
10681:Ordered Bell
10651:Fuss–Catalan
10563:non-centered
10513:Dodecahedral
10490:non-centered
10376:non-centered
10278:Wolstenholme
10047:
10023:× 2 ± 1
10020:
10019:Of the form
9986:Eighth power
9966:Fourth power
9492:Almost prime
9450:Euler–Jacobi
9358:
9354:
9350:
9346:
9344:Cunningham (
9334:
9330:
9313:
9309:
9305:
9286:
9282:
9278:
9274:
9270:
9269:consecutive
9257:
9253:
9249:
9237:
9233:
9221:
9217:
9205:
9201:
9197:
9193:
9191:Quadruplet (
9181:
9177:
9173:
9169:
9157:
9153:
9140:
9088:Full reptend
8946:Wolstenholme
8941:Wall–Sun–Sun
8872:
8857:
8853:
8832:
8828:
8817:
8813:
8809:
8805:
8793:
8781:
8761:
8757:
8736:
8720:
8705:
8693:
8681:
8652:
8629:Prime number
8587:
8568:
8520:
8516:
8501:
8492:
8410:. Retrieved
8406:the original
8397:
8381:
8377:
8373:
8368:
8364:
8352:
8348:
8343:
8339:
8309:
8287:
8283:
8217:
8207:
8203:
8137:
8122:
8115:
8100:
8096:
8089:
8074:
8070:
8063:
8048:
8044:
8037:
8022:
8015:
8000:
7993:
7978:
7970:
7963:
7942:
7925:
7898:
7892:
7884:the original
7874:
7860:
7849:. Retrieved
7838:
7827:. Retrieved
7823:
7813:
7794:
7788:
7776:. Retrieved
7772:
7763:
7752:. Retrieved
7743:
7731:. Retrieved
7727:
7718:
7707:. Retrieved
7697:
7670:
7664:
7653:
7643:
7625:
7613:. Retrieved
7609:
7600:
7589:. Retrieved
7580:
7566:
7557:
7548:
7537:. Retrieved
7533:
7524:
7516:the original
7512:"Milestones"
7506:
7494:. Retrieved
7488:
7478:
7466:. Retrieved
7460:
7450:
7438:. Retrieved
7434:
7424:
7413:. Retrieved
7407:
7397:
7386:. Retrieved
7382:
7372:
7359:
7350:
7346:
7336:
7328:the original
7323:
7314:
7302:
7294:
7275:
7271:
7261:
7252:
7245:Hardy, G. H.
7239:
7226:
7214:
7199:
7187:. Retrieved
7183:Mersenne.org
7182:
7173:
7161:. Retrieved
7156:
7147:
7124:
7120:
7112:
7108:
7094:
7086:
7068:
7044:
7040:
7033:
7029:
7021:
7017:
7009:
7005:
7001:
6997:
6989:
6985:
6977:
6973:
6963:
6953:
6949:
6945:
6941:
6937:
6915:
6911:
6907:
6900:
6897:Lucas number
6891:
6851:Cullen prime
6799:Power of two
6758:
6754:
6746:
6724:
6720:
6712:
6706:
6702:
6695:
6685:
6681:
6671:
6667:
6659:
6655:
6652:
6637:
6634:
5211:
5210:| <
5206:
5191:
5186:
5174:
5144:
5140:
5132:
5128:
5087:
5083:
5072:
5068:
5060:
5056:
5044:fourth power
5038:
5014:
5010:
5003:
4999:
4992:
4988:
4970:
4966:
4960:
4956:
4950:
4946:
4942:
4934:
4919:
4856:
4829:
4826:
4799:
4790:
4780:
4777:
4769:
4748:
4737:
4731:
4728:
4706:
4695:
4684:
4675:
4665:
4662:
4651:
4642:
4632:
4626:
4615:
4606:
4596:
4586:
4583:
4560:
4553:
4550:
4517:
4510:
4504:
4501:
4442:
4435:
4428:
4425:
4391:
4370:
4363:
4360:
4355:
4350:
4346:
4340:
4336:
4331:
4329:
4302:
4299:
4277:
4270:
4267:
4262:
4252:
4245:
4243:) for which
4235:
4231:
4225:
4221:
4214:
4197:real numbers
4186:
4172:
4161:
4157:real numbers
4150:
4140:
4130:
4123:
4112:
4105:
4098:
4091:
4084:
4080:
4076:
4072:
4065:
4054:
4050:
4046:
4042:
4035:
4027:coefficients
4016:
4012:
4005:
4002:
3981:
3971:
3967:
3963:
3959:
3940:
3929:
3922:
3914:
3910:
3907:
3902:
3896:
3880:
3870:
3868:
3820:
3796:minor planet
3789:
3780:
3767:
3761:
3754:
3736:
3505:
3494:
3487:
3469:
3461:
3427:
3401:
3399:
3377:
3367:
3359:
3355:
3349:
3345:
3340:
3336:
3329:
3313:
3306:
3288:
3257:
3213:
3205:
3204:2 ≡ −1 (mod
3197:
3194:, so either
3189:
3181:
3171:
3157:
3145:
3117:
3062:
3054:
3030:, therefore
3025:
3019:
3015:
3008:
3004:
2997:
2989:
2984:
2960:
2948:
2942:
2902:
2894:
2865:
2861:
2842:
2838:
2831:
2757:
2732:
2698:
2675:
2641:
2631:
2627:
2618:
2597:
2586:
2579:
2572:
2565:
2558:
2552:
2548:
2541:
2534:
2530:
2523:
2516:
2512:
2505:
2501:
2494:
2490:
2483:
2476:
2471:
2463:
2456:
2449:
2439:
2422:
2410:
2399:
2388:
2381:
2374:
2359:
2352:
2336:
2326:
2316:denotes the
2311:
2299:
2289:
2280:
2271:
2262:
2253:
2244:
2235:
2226:
2218:D. H. Lehmer
2198:
2187:
2131:
2124:
2115:
2111:
2104:
2094:
2086:
2082:
2073:
2060:
2053:
2046:
2042:R. E. Powers
2033:
2024:
2011:
1998:
1985:
1972:
1963:
1953:
1943:
1933:
1923:
1913:
1910:
1903:
1876:
1863:
1807:
1801:
1792:
1783:
1774:
1765:
1756:
1753:
1733:
1414:
1379:
1375:
1372:
1347:are used in
1330:
1311:
1279:
1268:
1251:
1241:
937:
923:
914:
899:
889:
880:
870:
860:
850:
840:
732:
615:must divide
610:
598:
586:
562:
552:
542:
532:
522:
512:
502:
492:
477:
454:
422:
360:
357:
317:
306:
297:
292:
287:
278:
275:
229:
223:
217:
207:
193:
179:
166:
159:power of two
155:prime number
150:
144:
133:
129:
57:Conjectured
12244:Power of 10
12184:Busy beaver
11989:Quintillion
11984:Quadrillion
11556:Extravagant
11551:Equidigital
11506:permutation
11465:Palindromic
11439:Automorphic
11337:Sum-product
11316:Sum-product
11271:Persistence
11166:Erdős–Woods
11088:Untouchable
10969:Semiperfect
10919:Hemiperfect
10580:Tesseractic
10518:Icosahedral
10498:Tetrahedral
10429:Dodecagonal
10130:Recursively
10001:Prime power
9976:Sixth power
9971:Fifth power
9951:Power of 10
9909:Classes of
9475:Somer–Lucas
9430:Pseudoprime
9068:Truncatable
9040:Palindromic
8924:By property
8703:Primorial (
8691:Factorial (
8505:up to 1280)
8430:(in German)
8402:Brady Haran
8398:Numberphile
8386:math thesis
8280:, that is,
8200:, that is,
7935:Prime Pages
7733:5 September
7655:Prime Pages
7402:Tia Ghose.
7163:21 December
7053:must be in
6861:Proth prime
4833:, they are
4778:Least base
4746:if no such
4704:if no such
4564:values of:
4521:values of:
4249:the number
4195:instead of
3816:127 Johanna
3196:2 ≡ 1 (mod
3188:2 ≡ 1 (mod
3057:mod (2 − 1)
2996:2 ≡ 1 (mod
2947:2 ≡ 2 (mod
2591:, however,
2521:. However,
2190:Alan Turing
2005:, found by
457:conjectures
452:is prime.
201:then so is
189:Minim friar
187:, a French
163:of the form
147:mathematics
136:is a prime)
80:First terms
70:Subsequence
36:Named after
12282:Categories
12249:Sagan Unit
12083:Expression
12034:Googolplex
11999:Septillion
11994:Sextillion
11940:numerical
11768:Graphemics
11641:Pernicious
11495:Undulating
11470:Pandigital
11444:Trimorphic
11045:Nontotient
10894:Arithmetic
10508:Octahedral
10409:Heptagonal
10399:Pentagonal
10384:Triangular
10225:Sierpiński
10147:Jacobsthal
9946:Power of 3
9941:Power of 2
9512:Pernicious
9507:Interprime
9267:Balanced (
9058:Permutable
9033:-dependent
8850:Williams (
8746:Pierpont (
8671:Wagstaff
8653:Mersenne (
8637:By formula
8435:GIMPS wiki
8412:2013-04-06
8131:= 2 to 200
8109:= 1 to 107
8057:= 1 to 160
8031:= 2 to 160
8009:= 2 to 160
7954:1503.07688
7851:2011-05-21
7829:2023-02-11
7754:2021-07-21
7709:2022-09-05
7591:2019-01-01
7539:2018-01-03
7496:22 January
7468:19 January
7440:22 January
7415:2013-02-07
7388:2011-05-21
7353:: 122–131.
7189:5 December
7140:References
6751:such that
6717:such that
5178:| ≤ 5
5121:such that
5050:such that
5019:such that
4932:integers,
4784:such that
4669:such that
4660:is prime)
4600:such that
4023:polynomial
3934:, it is a
3851:hypotenuse
3541:536870911
3105:pernicious
2970:±1 (mod 8)
2937:±1 (mod 8)
2864:≡ 1 (mod 2
2789:such that
2711:2 − 1 = 31
2706:is prime.
2615:is prime.
2384:37,156,667
603:, and the
569:(sequence
261:2147483647
132:− 1 where
12140:Pentation
12135:Tetration
12123:Operators
12092:Notations
12014:Decillion
12009:Nonillion
12004:Octillion
11936:Examples
11525:Parasitic
11374:Factorion
11301:Digit sum
11293:Digit sum
11111:Fortunate
11098:Primorial
11012:Semiprime
10949:Practical
10914:Descartes
10909:Deficient
10899:Betrothed
10741:Wieferich
10570:Pentatope
10533:pyramidal
10424:Decagonal
10419:Nonagonal
10414:Octagonal
10404:Hexagonal
10263:Practical
10210:Congruent
10142:Fibonacci
10106:Loeschian
9548:Prime gap
9497:Semiprime
9460:Frobenius
9167:Triplet (
8966:Ramanujan
8961:Fortunate
8931:Wieferich
8895:Fibonacci
8826:Leyland (
8791:Woodall (
8770:Solinas (
8755:Quartan (
8589:MathWorld
8570:MathWorld
8316:EMS Press
8165:−
8083:= 1 to 40
7987:= 2 to 50
6635:Note: if
5223:sequence
5190:| =
4954:. (Since
4899:−
4881:−
4558:, we get
4515:, we get
4481:−
4470:−
4061:2 − 2 − 1
4031:2 − 2 + 1
3960:In fact,
3954:MF(59, 2)
3835:≥ 4
3831:primitive
3248:3 (mod 4)
3236:7 (mod 8)
3228:3 (mod 4)
3140:3 (mod 4)
3136:congruent
2841:≡ 1 (mod
2678:is prime.
2657:= (2 − 1)
2654:= (2) − 1
2493:≡ 1 (mod
2284:6,972,593
2160:
2151:
2068:is prime
1442:−
1204:−
1187:⋯
1128:⋅
1119:−
1077:−
1060:⋯
1001:⋅
992:−
966:−
810:−
777:−
746:Φ
729:. Hence,
688:Φ
644:−
607:of 2 mod
464:congruent
400:
387:⋅
382:γ
226:exponents
174:for some
12172:articles
12170:Related
12074:Infinity
11979:Trillion
11954:Thousand
11597:Friedman
11530:Primeval
11475:Repdigit
11432:-related
11379:Kaprekar
11353:Meertens
11276:Additive
11263:dynamics
11171:Friendly
11083:Sociable
11073:Amicable
10884:Abundant
10864:dynamics
10686:Schröder
10676:Narayana
10646:Eulerian
10636:Delannoy
10631:Dedekind
10452:centered
10318:centered
10205:Amenable
10162:Narayana
10152:Leonardo
10048:Mersenne
9996:Powerful
9936:Achilles
9440:Elliptic
9215:Cousin (
9132:Patterns
9123:Tetradic
9118:Dihedral
9083:Primeval
9078:Delicate
9063:Circular
9050:Repunit
8841:Thabit (
8779:Cullen (
8718:Euclid (
8644:Fermat (
8544:Archived
8532:Archived
8476:Archived
8464:Archived
8105:for odd
7633:Archived
7324:BBC News
7251:(1959).
7091:or when
7026:must be
6928:are the
6920:, since
6826:/ MPrime
6782:See also
6693:must be
6664:, it is
5159:, these
5153:is prime
5115:numbers
4752:exists)
4710:exists)
4010:, where
3979:, where
3839:inradius
3829:that is
3823:geometry
3792:asteroid
3530:8388607
3522:23 × 89
3456:, where
3325:. It is
3287:2 − 1 =
3262:divides
3150:divides
3059:. Hence
2769:. Since
2747:. Since
2721:, where
2556:, hence
2544:− 1 = ±1
2308:, where
2092:, where
2080:divides
1738:scholar
924:A basic
913:≥
737:divides
671:. Since
64:Infinite
61:of terms
12267:History
12085:methods
12059:SSCG(3)
12054:TREE(3)
11974:Billion
11969:Million
11949:Hundred
11770:related
11734:related
11698:related
11696:Sorting
11581:Vampire
11566:Harshad
11508:related
11480:Repunit
11394:Lychrel
11369:Dudeney
11221:Størmer
11216:Sphenic
11201:Regular
11139:divisor
11078:Perfect
10974:Sublime
10944:Perfect
10671:Motzkin
10626:Catalan
10167:Padovan
10101:Leyland
10096:Idoneal
10091:Hilbert
10063:Woodall
9435:Catalan
9372:By size
9143:-tuples
9073:Minimal
8976:Regular
8867:Mills (
8803:Cuban (
8679:Proth (
8631:classes
8318:, 2001
8278:
8224:
8198:
8144:
7778:24 June
7615:29 June
7309:p. 228.
7129:
7105:
7060:A027861
7058::
6994:
6970:
6932:of the
6832:(GIMPS)
6824:Prime95
6789:Repunit
6773:in the
6770:A222119
6757:+ 1) −
6739:in the
6736:A058013
6723:+ 1) −
6670:+ 1) −
6628:A213216
6599:A128341
6583:A057178
6567:A004064
6551:A128348
6535:A273814
6519:A062578
6503:A185239
6435:A128340
6419:A224501
6403:A128070
6387:A125957
6371:A057177
6355:A005808
6339:A210506
6323:A128027
6307:A216181
6291:A128347
6275:A273598
6259:A273599
6243:A273600
6227:A273601
6211:A062577
6195:A217095
6166:A128069
6150:A001562
6134:A004023
6118:A128026
6102:A273403
6086:A062576
6070:A187819
6054:A301369
6038:A128339
6022:A211409
6006:A125956
5990:A057175
5973:(none)
5961:A173718
5932:A128346
5916:A273010
5900:A059803
5884:A181141
5868:A128338
5852:A128068
5810:A128025
5794:A128345
5778:A062574
5762:A187805
5746:A128337
5730:A218373
5714:A128067
5698:A125955
5682:A057173
5666:A004063
5650:A215487
5634:A128024
5618:A213073
5602:A128344
5586:A062573
5570:A128336
5554:A057172
5538:A004062
5522:A062572
5506:A128335
5490:A122853
5474:A082387
5458:A057171
5442:A004061
5426:A082182
5410:A121877
5394:A059802
5378:A128066
5336:A059801
5320:A057469
5304:A007658
5288:A028491
5272:A057468
5256:A000978
5239:A000043
5149:
5125:
5077:
5053:
4930:coprime
4844:in the
4841:A103795
4819:in the
4816:A066180
4805:
4787:
4763:in the
4760:A084742
4721:in the
4718:A084740
4690:
4672:
4657:
4639:
4621:
4603:
4574:in the
4571:A057178
4543:in the
4540:A004022
4531:in the
4528:A004023
4422:Repunit
4402:in the
4399:A066413
4384:in the
4381:A066408
4317:in the
4314:A182300
4292:in the
4289:A057429
4199:, like
4170:, then
4151:In the
3985:is the
3920:. When
3900:prime,
3891:
3876:
3743:in the
3740:A046800
3476:in the
3473:A244453
3415:in the
3412:A000043
3158:Example
3142:, then
3130:), and
3093:coprime
3046:
3034:
3028:| 2 − 1
2992:= 2 − 1
2651:= 2 − 1
2646:. Then
2510:. Thus
2488:. Then
2429:in the
2426:A000225
2296:equals
2066:= 2 − 1
1513:History
1398:2(2 − 1
926:theorem
575:in the
572:A002515
284:= 2 − 1
269:in the
266:A000668
239:in the
236:A000043
213:= 2 − 1
176:integer
172:= 2 − 1
124:A000668
12176:order)
12024:Googol
11636:Odious
11561:Frugal
11515:Cyclic
11504:Digit-
11211:Smooth
11196:Pronic
11156:Cyclic
11133:Other
11106:Euclid
10756:Wilson
10730:Primes
10389:Square
10258:Polite
10220:Riesel
10215:Knödel
10177:Perrin
10058:Thabit
10043:Fermat
10033:Cullen
9956:Square
9924:Powers
9480:Strong
9470:Perrin
9455:Fermat
9231:Sexy (
9151:Twin (
9093:Unique
9021:Unique
8981:Strong
8971:Pillai
8951:Wilson
8915:Perrin
8351:) − (3
8073:+ 1, −
7913:
7801:
7685:
7089:> 1
7032:+ 1, −
6967:Since
6747:Least
6713:Least
6640:< 0
5184:|
5172:|
5167:100000
4937:> 1
4663:Least
4349:= 1 −
4339:= 1 +
4234:= 1 −
4224:= 1 +
4189:> 0
4159:), if
4115:> 1
4070:, and
4040:, and
3943:> 1
3853:to be
3849:, the
3808:7 Iris
3804:3 Juno
3316:> 1
3309:> 1
3300:, and
3174:: Let
3018:− 1 =
2964:. By
2959:2 mod
2897:> 2
2644:> 1
2561:= 0, 1
2533:− 1 =
2515:− 1 |
2306:2⌋ + 1
2275:44,497
2251:, and
2127:> 0
2056:> 2
1959:= 8191
1790:, and
1736:French
1390:Euclid
561:503 |
559:, and
551:479 |
541:383 |
531:359 |
521:263 |
511:167 |
466:to 3 (
311:: the
98:, 8191
12262:Names
12064:BH(3)
11942:order
11677:Prime
11672:Lucky
11661:sieve
11590:Other
11576:Smith
11456:Digit
11414:Happy
11389:Keith
11362:Other
11206:Rough
11176:Giuga
10641:Euler
10503:Cubic
10157:Lucas
10053:Proth
9465:Lucas
9445:Euler
9098:Happy
9045:Emirp
9011:Higgs
9006:Super
8986:Stern
8956:Lucky
8900:Lucas
8519:± 1,
8360:(PDF)
8125:, −1)
8099:+ 2,
8047:+ 1,
8025:, −1)
7981:, −1)
7949:arXiv
7072:When
6930:roots
6883:Notes
6653:When
5199:20000
4949:<
4945:<
4920:with
4556:= −12
4446:make
4366:) − 1
4362:(1 +
4305:norms
4273:) − 1
4269:(1 +
4257:is a
4255:) − 1
4251:(1 +
4182:0 − 1
4178:2 − 1
4166:is a
4103:(for
4094:2 − 1
3894:with
3888:2 − 1
3794:with
3519:2047
3434:2 − 1
3365:with
3172:Proof
3166:2 − 1
3152:2 − 1
3126:is a
3101:2 − 1
3097:2 − 1
3043:2 − 1
3037:2 − 1
2985:Proof
2953:, so
2943:Proof
2933:2 − 1
2883:2 − 1
2853:2 − 1
2823:2 − 1
2807:2 − 1
2795:2 − 1
2779:2 − 1
2767:2 − 1
2753:2 − 1
2745:2 − 1
2735:: By
2733:Proof
2704:2 − 1
2693:2 − 1
2672:2 − 1
2668:2 − 1
2648:2 − 1
2635:with
2619:Proof
2609:2 − 1
2584:. If
2499:, so
2479:≡ 1 (
2472:Proof
2402:2 − 1
2395:2 − 1
2377:2 − 1
2370:2 − 1
2355:2 − 1
2332:⌋ + 1
2302:× log
2266:4,423
2120:) − 2
1949:= 127
1394:2 − 1
876:= 127
501:47 |
491:23 |
468:mod 4
324:2 − 1
203:2 − 1
197:is a
153:is a
117:index
107:2 − 1
11880:List
11631:Evil
11311:Self
11261:and
11151:Blum
10862:and
10666:Lobb
10621:Cake
10616:Bell
10366:Star
10273:Ulam
10172:Pell
9961:Cube
9388:list
9323:Chen
9103:Self
9031:Base
9001:Good
8935:pair
8905:Pell
8856:−1)·
8485:OEIS
8388:(PS)
8347:= (8
8127:for
8079:for
8053:for
8027:for
8005:for
8003:, 1)
7983:for
7975:and
7973:, 1)
7911:ISBN
7799:ISBN
7780:2022
7735:2022
7683:ISBN
7617:2021
7498:2016
7470:2016
7442:2016
7191:2020
7165:2018
7076:and
7056:OEIS
7047:+ 1)
7038:and
6924:and
6775:OEIS
6741:OEIS
6642:and
6621:−11
6496:−10
5221:OEIS
5201:for
5169:for
5035:and
5023:and
4997:and
4939:and
4928:any
4846:OEIS
4821:OEIS
4765:OEIS
4740:= −2
4723:OEIS
4576:OEIS
4545:OEIS
4533:OEIS
4513:= 10
4404:OEIS
4386:OEIS
4344:and
4319:OEIS
4294:OEIS
4241:WLOG
4229:and
4203:and
4168:unit
4153:ring
4137:unit
4110:and
4079:) =
4068:= 64
4049:) =
4038:= 32
3970:) =
3952:and
3945:are
3814:and
3790:The
3745:OEIS
3725:131
3714:113
3703:109
3692:103
3681:101
3478:OEIS
3465:1277
3417:OEIS
3375:and
3311:and
3114:and
3099:and
3091:are
3087:and
3079:and
3061:p |
3032:0 ≡
2725:and
2639:and
2568:= 1.
2508:− 1)
2497:− 1)
2486:− 1)
2443:and
2431:OEIS
2339:UCLA
2322:⌊log
2257:2281
2248:2203
2239:1279
2122:for
2102:and
2031:and
1970:and
1941:and
1939:= 31
1763:and
1723:311
1720:307
1717:293
1714:283
1711:281
1708:277
1705:271
1702:269
1697:263
1694:257
1691:251
1688:241
1685:239
1682:233
1679:229
1676:227
1671:223
1668:211
1665:199
1662:197
1659:193
1656:191
1653:181
1650:179
1645:173
1642:167
1639:163
1636:157
1633:151
1630:149
1627:139
1624:137
1619:131
1616:127
1613:113
1610:109
1607:107
1604:103
1601:101
897:and
868:and
866:= 31
792:and
595:mod
577:OEIS
361:The
330:, a
271:OEIS
241:OEIS
224:The
149:, a
115:OEIS
11749:Ban
11137:or
10656:Lah
9859:281
9854:277
9849:271
9844:269
9839:263
9834:257
9829:251
9824:241
9819:239
9814:233
9809:229
9804:227
9799:223
9794:211
9789:199
9784:197
9779:193
9774:191
9769:181
9764:179
9759:173
9754:167
9749:163
9744:157
9739:151
9734:149
9729:139
9724:137
9719:131
9714:127
9709:113
9704:109
9699:107
9694:103
9689:101
9349:, 2
9333:, 2
9254:a·n
8812:)/(
8286:, −
7903:doi
7675:doi
7280:doi
7043:+ (
6996:= (
6956:= 0
6940:− (
6698:+ 1
6662:+ 1
6618:12
6608:−7
6605:12
6592:−5
6589:12
6576:−1
6573:12
6557:12
6541:12
6525:12
6512:11
6509:12
6493:11
6483:−9
6480:11
6470:−8
6467:11
6457:−7
6454:11
6444:−6
6441:11
6428:−5
6425:11
6412:−4
6409:11
6396:−3
6393:11
6380:−2
6377:11
6364:−1
6361:11
6345:11
6329:11
6313:11
6297:11
6281:11
6265:11
6249:11
6233:11
6217:11
6204:10
6201:11
6188:−9
6185:10
6175:−7
6172:10
6159:−3
6156:10
6143:−1
6140:10
6124:10
6108:10
6092:10
6076:10
6063:−8
6047:−7
6031:−5
6015:−4
5999:−2
5983:−1
5877:−7
5861:−5
5845:−3
5832:−1
5755:−6
5739:−5
5723:−4
5707:−3
5691:−2
5675:−1
5563:−5
5547:−1
5499:−4
5483:−3
5467:−2
5451:−1
5371:−3
5358:−1
5313:−2
5297:−1
5248:−1
5214:− 1
5194:− 1
5181:or
4995:= 1
4802:− 1
4793:− 1
4772:= 2
4698:= 2
4687:− 1
4678:− 1
4654:− 1
4645:− 1
4618:− 1
4609:− 1
4438:− 1
4431:− 1
4164:− 1
4133:− 1
4126:− 1
4121:),
4108:≠ 2
4101:− 1
4087:− 1
4057:+ 1
4008:(2)
3977:(2)
3962:MF(
3932:= 2
3925:= 1
3909:MF(
3882:− 1
3821:In
3778:is
3747:).
3670:97
3659:83
3648:79
3637:73
3626:71
3615:67
3604:59
3593:53
3582:47
3571:43
3560:41
3549:37
3538:29
3527:23
3516:11
3480:).
3383:= 1
3373:= 0
3354:- 2
3344:= 3
3260:+ 1
3202:or
3184:+ 1
3178:be
3148:+ 1
3138:to
3134:is
3120:+ 1
3110:If
3075:If
3011:− 1
2923:If
2836:so
2834:− 1
2683:If
2666:so
2607:If
2600:= 2
2589:= 0
2582:= 0
2577:or
2575:= 2
2539:or
2537:− 1
2526:− 1
2519:− 1
2481:mod
2466:= 1
2461:or
2459:= 2
2452:− 1
2437:If
2433:).
2334:).
2230:607
2202:521
2157:log
2148:log
2118:− 1
2110:= (
2100:= 4
2089:− 2
2037:107
2018:by
2002:127
1929:= 7
1919:= 3
1830:148
1811:127
1796:107
1769:257
1598:97
1593:89
1590:83
1587:79
1584:73
1581:71
1578:67
1575:61
1572:59
1567:53
1564:47
1561:43
1558:41
1555:37
1552:31
1549:29
1546:23
1541:19
1538:17
1535:13
1532:11
905:= 1
895:= 0
856:= 7
846:= 3
735:+ 1
613:+ 1
601:+ 1
589:+ 1
566:251
556:239
546:191
536:179
526:131
480:+ 1
391:log
300:is
273:).
257:127
145:In
96:127
59:no.
47:No.
12284::
11938:in
9684:97
9679:89
9674:83
9669:79
9664:73
9659:71
9654:67
9649:61
9644:59
9639:53
9634:47
9629:43
9624:41
9619:37
9614:31
9609:29
9604:23
9599:19
9594:17
9589:13
9584:11
9281:,
9277:,
9256:,
9236:,
9220:,
9196:,
9172:,
9156:,
8586:.
8567:.
8400:.
8396:.
8376:+
8372:=
8353:qy
8314:,
8308:,
8206:,
7909:.
7822:.
7771:.
7726:.
7681:.
7652:.
7608:.
7556:.
7532:.
7487:.
7459:.
7433:.
7406:.
7381:.
7351:18
7349:.
7345:.
7322:.
7276:10
7274:.
7270:.
7247:;
7181:.
7155:.
7123:−
7111:−
7095:ab
7093:−4
7020:,
7008:+
7004:)(
7000:+
6988:−
6976:−
6954:ab
6952:+
6944:+
6916:ab
6914:,
6910:+
6710:.
6705:−
6684:−
6658:=
6560:1
6544:5
6528:7
6348:1
6332:2
6316:3
6300:4
6284:5
6268:6
6252:7
6236:8
6220:9
6127:1
6111:3
6095:7
6079:9
6060:9
6044:9
6028:9
6012:9
5996:9
5980:9
5970:1
5967:9
5954:2
5951:9
5941:4
5938:9
5925:5
5922:9
5909:7
5906:9
5893:8
5890:9
5874:8
5858:8
5842:8
5829:8
5819:1
5816:8
5803:3
5800:8
5787:5
5784:8
5771:7
5768:8
5752:7
5736:7
5720:7
5704:7
5688:7
5672:7
5659:1
5656:7
5643:2
5640:7
5627:3
5624:7
5611:4
5608:7
5595:5
5592:7
5579:6
5576:7
5560:6
5544:6
5531:1
5528:6
5515:5
5512:6
5496:5
5480:5
5464:5
5448:5
5435:1
5432:5
5419:2
5416:5
5403:3
5400:5
5387:4
5384:5
5368:4
5355:4
5345:1
5342:4
5329:3
5326:4
5310:3
5294:3
5281:1
5278:3
5265:2
5262:3
5245:2
5231:1
5228:2
5217:)
5197:,
5143:−
5131:−
5092:.
5086:,
5071:−
5059:−
5039:ab
5037:−4
5013:,
5002:+
4991:+
4969:−
4959:−
4924:,
4742:,
4700:,
4578:),
4547:).
4535:),
4374::
4358:.
4321:).
4281::
4265:.
4207:.
4089:.
4083:−
4053:−
3989:.
3966:,
3913:,
3869:A
3861:.
3810:,
3806:,
3787:.
3783:64
3404::
3362:-2
3352:-1
3296:,
3230:,
3154:.
3053:≡
3020:mλ
3007:|
2981:.
2945::
2939:.
2905:+1
2903:kp
2825:,
2801:,
2781:,
2761:,
2739:,
2632:ab
2630:=
2551:=
2474::
2468:.
2324:10
2304:10
2242:,
2129:.
2058:,
2028:89
2015:61
1989:31
1976:19
1967:17
1957:13
1931:,
1921:,
1880:67
1867:61
1849:17
1787:89
1781:,
1778:61
1760:67
1529:7
1526:5
1523:3
1520:2
1412:.
1359:.
1328:.
1272:11
1249:.
858:,
848:,
583:,
549:,
539:,
529:,
519:,
516:83
509:,
506:23
499:,
496:11
474:,
406:10
322:,
304:.
255:,
253:31
251:,
247:,
221:.
94:,
92:31
90:,
86:,
72:of
52:51
11921:e
11914:t
11907:v
11836:e
11829:t
11822:v
10021:a
9902:e
9895:t
9888:v
9579:7
9574:5
9569:3
9564:2
9363:)
9359:p
9355:p
9351:p
9347:p
9339:)
9335:p
9331:p
9318:)
9314:n
9310:n
9306:n
9290:)
9287:n
9283:p
9279:p
9275:n
9271:p
9262:)
9258:n
9250:p
9242:)
9238:p
9234:p
9226:)
9222:p
9218:p
9210:)
9206:p
9202:p
9198:p
9194:p
9186:)
9182:p
9178:p
9174:p
9170:p
9162:)
9158:p
9154:p
9141:k
8937:)
8933:(
8879:)
8876:⌋
8873:A
8870:⌊
8862:)
8858:b
8854:b
8852:(
8845:)
8836:)
8833:y
8829:x
8821:)
8818:y
8814:x
8810:y
8806:x
8798:)
8794:n
8786:)
8782:n
8774:)
8765:)
8762:y
8758:x
8750:)
8741:)
8737:n
8735:4
8728:)
8723:n
8721:p
8713:)
8708:n
8706:p
8698:)
8694:n
8686:)
8682:k
8666:)
8657:)
8648:)
8621:e
8614:t
8607:v
8592:.
8573:.
8521:b
8517:b
8502:n
8499:(
8495:n
8493:M
8415:.
8382:y
8380:·
8378:d
8374:x
8369:q
8365:M
8355:)
8349:x
8344:q
8340:M
8290:)
8288:b
8284:a
8282:(
8266:c
8262:/
8258:)
8253:n
8249:b
8245:+
8240:n
8236:a
8232:(
8210:)
8208:b
8204:a
8202:(
8186:c
8182:/
8178:)
8173:n
8169:b
8160:n
8156:a
8152:(
8129:x
8123:x
8121:(
8107:x
8103:)
8101:x
8097:x
8095:(
8081:x
8077:)
8075:x
8071:x
8069:(
8055:x
8051:)
8049:x
8045:x
8043:(
8029:x
8023:x
8021:(
8007:x
8001:x
7999:(
7985:x
7979:x
7977:(
7971:x
7969:(
7957:.
7951::
7937:)
7919:.
7905::
7854:.
7832:.
7807:.
7782:.
7757:.
7737:.
7712:.
7691:.
7677::
7658:.
7619:.
7594:.
7542:.
7500:.
7472:.
7444:.
7418:.
7391:.
7367:.
7288:.
7282::
7234:.
7208:.
7193:.
7167:.
7125:b
7121:a
7117:/
7113:b
7109:a
7100:n
7087:r
7082:r
7078:b
7074:a
7063:.
7051:x
7045:x
7041:x
7036:)
7034:x
7030:x
7028:(
7024:)
7022:b
7018:a
7016:(
7012:)
7010:b
7006:a
7002:b
6998:a
6990:b
6986:a
6982:/
6978:b
6974:a
6958:.
6950:x
6948:)
6946:b
6942:a
6938:x
6926:b
6922:a
6918:)
6912:b
6908:a
6906:(
6903:n
6901:U
6777:)
6759:b
6755:b
6753:(
6749:b
6743:)
6725:b
6721:b
6719:(
6715:n
6707:b
6703:a
6696:b
6691:a
6686:b
6682:a
6677:n
6672:b
6668:b
6666:(
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