2137:
2420:(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.
11803:
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
5239:
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,
3413:
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:
5256:
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
3445:
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.
4761:
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
6737:
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
4719:
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
4290:
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
4817:
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
6771:
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
4842:
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:
3433:
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.
4920:
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
4502:
790:
669:
2402:(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.
9905:
8281:
8201:
8335:– 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:
9392:
450:
8624:
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.
362:
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
8995:
5273:
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
12302:
11884:
4383:
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
3465:
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
3400:
3259:, 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
7952:
Zalnezhad, Ali; Zalnezhad, Hossein; Shabani, Ghasem; Zalnezhad, Mehdi (March 2015). "Relationships and Algorithm in order to Achieve the Largest Primes".
5305:
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, ...
3472:
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).
3441:
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
12297:
7060:
6779:
6745:
4850:
4825:
4769:
4727:
4580:
4549:
4537:
4408:
4390:
4323:
4298:
3749:
3482:
3421:
2435:
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:
576:
270:
240:
114:
9898:
9077:
358:
Many fundamental questions about Mersenne primes remain unresolved. It is not even known whether the set of Mersenne primes is finite or infinite.
352:
5337:
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, ...
1336:
11924:
8536:
455:
It is also not known whether infinitely many Mersenne numbers with prime exponents are composite, although this would follow from widely believed
8480:
8468:
9000:
7461:
8914:
7919:
7807:
7691:
2213:
8548:
11879:
6834:
2338:
327:
8124:
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,
334:
project. In December 2020, a major milestone in the project was passed after all exponents below 100 million were checked at least once.
10705:
9891:
2405:
On December 21, 2018, it was announced that The Great Internet Mersenne Prime Search (GIMPS) discovered the largest known prime number,
1344:
1340:
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
11839:
10700:
8617:
7324:
1324:. Consequently, a large amount of computer power has been expended searching for new Mersenne primes, much of which is now done using
10715:
10695:
12292:
12198:
8226:
7576:
11408:
10988:
9251:
8146:
2048:
1313:
8390:
2136:
10710:
9332:
2365:
2342:
11494:
8513:
8444:
7637:
7489:
6808:
5395:
3, 43, 59, 191, 223, 349, 563, 709, 743, 1663, 5471, 17707, 19609, 35449, 36697, 45259, 91493, 246497, 265007, 289937, ...
12233:
12113:
12018:
12013:
12008:
12003:
11998:
11993:
11988:
11983:
8610:
8320:
3442:
2361:
1348:
12118:
12048:
10810:
9454:
9112:
8439:
12108:
11160:
10479:
10272:
7516:
3780:
3437:
algorithm, so often the largest number factorized with this algorithm has been a Mersenne number. As of June 2019,
2206:
11195:
11165:
10840:
10830:
9479:
7902:
Solinas, Jerome A. (1 January 2011). "Generalized Mersenne Prime". In Tilborg, Henk C. A. van; Jajodia, Sushil (eds.).
7708:
11917:
11336:
10750:
10484:
10464:
8945:
8315:
12063:
11026:
11190:
7884:
372:
12307:
12208:
11285:
10908:
10665:
10474:
10456:
10350:
10340:
10330:
9387:
8398:
7558:
6840:
4245:
4003:
3434:
2741:
1905:
1356:
319:
11170:
3108:
are coprime. Consequently, a prime number divides at most one prime-exponent Mersenne number. That is, the set of
12193:
12058:
11413:
10958:
10579:
10365:
10360:
10355:
10345:
10322:
2019:
1871:
4868:
4305:
Like the sequence of exponents for usual Mersenne primes, this sequence contains only (rational) prime numbers.
1405:
1368:
312:
12317:
12312:
12271:
10398:
7383:
188:
10655:
7408:
205:. Therefore, an equivalent definition of the Mersenne primes is that they are the prime numbers of the form
11524:
11489:
11275:
11185:
11059:
11034:
10943:
10933:
10545:
10527:
10447:
9020:
5289:
3, 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551, 36913, 43063, 49681, 57917, 483611, 877843, ...
4457:
3287:
12266:
12161:
11910:
11874:
11832:
11784:
11054:
10928:
10559:
10335:
10115:
10042:
9537:
8666:
6875:
6823:
5443:
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:
2882:, which asserts the infinitude of primes, distinct from the proof written by Euclid: for every odd prime
618:
12228:
12218:
12156:
11039:
10893:
10820:
9975:
9874:
9464:
9117:
9025:
8493:
sequence A250197 (Numbers n such that the left Aurifeuillian primitive part of 2^n+1 is prime)
3991:
2970:
2395:
1325:
331:
11748:
11388:
8533:
3286:
With the exception of 1, a Mersenne number cannot be a perfect power. That is, and in accordance with
11859:
11681:
11575:
11539:
11280:
11003:
10983:
10800:
10469:
10257:
10229:
9444:
8477:
8465:
8310:
8232:
8152:
6813:
3132:
2394:
On January 3, 2018, it was announced that Jonathan Pace, a 51-year-old electrical engineer living in
2217:
604:
460:
8424:
4979:, the division is necessary for there to be any chance of finding prime numbers.) We can ask which
1817:
682:
12103:
11403:
11267:
11262:
11230:
10993:
10968:
10963:
10938:
10868:
10864:
10795:
10685:
10517:
10313:
10282:
9439:
9097:
7753:
7413:
7236:
3140:
2879:
2221:
2143:
908:
795:
463:
366:
11802:
2382:
On September 2, 2016, the Great Internet Mersenne Prime Search finished verifying all tests below
12151:
11806:
11560:
11555:
11469:
11443:
11341:
11320:
11092:
10973:
10923:
10845:
10815:
10755:
10522:
10502:
10433:
10146:
9547:
9484:
9474:
9459:
9092:
8950:
8545:
8460:
7953:
7494:
6938:
6934:
6023:
2, 3, 5, 7, 11, 17, 19, 41, 53, 109, 167, 2207, 3623, 5059, 5471, 7949, 21211, 32993, 60251, ...
4209:
4172:
4157:
4141:
3835:
3280:
2485:
2416:
In late 2020, GIMPS began using a new technique to rule out potential Mersenne primes called the
1418:
1409:
467:
12053:
12043:
11864:
10690:
8871:
7674:
Kleinjung, Thorsten; Bos, Joppe W.; Lenstra, Arjen K. (2014). "Mersenne Factorization Factory".
4397:
The norms (that is, squares of absolute values) of these Eisenstein primes are rational primes:
1463:
5555:
2, 3, 11, 31, 43, 47, 59, 107, 811, 2819, 4817, 9601, 33581, 38447, 41341, 131891, 196337, ...
3779:
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
11825:
11700:
11645:
11499:
11474:
11448:
11225:
10903:
10898:
10825:
10805:
10790:
10512:
10494:
10413:
10403:
10388:
10166:
10151:
9516:
9491:
9469:
9449:
9072:
9044:
8737:
8585:
8566:
7915:
7803:
7687:
3109:
2193:
1884:
592:
7935:
7347:
5779:
7, 11, 17, 29, 31, 79, 113, 131, 139, 4357, 44029, 76213, 83663, 173687, 336419, 615997, ...
5427:
2, 5, 7, 13, 19, 37, 59, 67, 79, 307, 331, 599, 1301, 12263, 12589, 18443, 20149, 27983, ...
1904:
Fast algorithms for finding Mersenne primes are available, and as of June 2023, the six
12073:
12023:
11891:
11736:
11529:
11115:
11087:
11077:
11069:
10953:
10918:
10913:
10880:
10574:
10537:
10428:
10423:
10418:
10408:
10380:
10267:
10219:
10214:
10171:
10110:
9426:
9416:
9411:
9348:
9195:
9062:
8965:
8455:
7907:
7679:
7306:
7284:
6818:
4309:
4221:
4205:
2006:
1802:
1352:
290:. Sometimes, however, Mersenne numbers are defined to have the additional requirement that
198:
7332:
5885:
4, 7, 13, 31, 43, 269, 353, 383, 619, 829, 877, 4957, 5711, 8317, 21739, 24029, 38299, ...
5459:
5, 67, 101, 103, 229, 347, 4013, 23297, 30133, 177337, 193939, 266863, 277183, 335429, ...
5411:
13, 19, 23, 31, 47, 127, 223, 281, 2083, 5281, 7411, 7433, 19051, 27239, 35863, 70327, ...
3475:
The table below shows factorizations for the first 20 composite Mersenne numbers (sequence
428:
12213:
11712:
11601:
11534:
11460:
11383:
11357:
11175:
10888:
10745:
10680:
10650:
10640:
10635:
10301:
10209:
10156:
10000:
9940:
9127:
9087:
8970:
8935:
8899:
8854:
8707:
8695:
8552:
8540:
8484:
8472:
7641:
6880:
6845:
5379:
3, 5, 19, 37, 173, 211, 227, 619, 977, 1237, 2437, 5741, 13463, 23929, 81223, 121271, ...
2983:
2209:
7348:"A Brief History of the Investigations on Mersenne Numbers and the Latest Immense Primes"
3450:
on the cofactor. As of September 2022, the largest completely factored number (with
8098:
8072:
8024:
4542:
corresponding to primes 11, 1111111111111111111, 11111111111111111111111, ... (sequence
2224:. It was the first Mersenne prime to be identified in thirty-eight years; the next one,
12243:
12134:
12068:
12033:
11848:
11717:
11585:
11570:
11434:
11398:
11373:
11249:
11220:
11205:
11082:
10978:
10948:
10675:
10630:
10507:
10105:
10100:
10095:
10067:
9965:
9950:
9928:
9915:
9532:
9506:
9403:
9271:
9122:
9082:
9067:
8939:
8830:
8795:
8750:
8675:
8046:
8002:
7157:
6901:
6850:
5161:
4263:
4197:
3831:
3761:
3451:
3447:
3327:
2417:
2410:
2317:
2069:
1993:
1980:
1739:
1404:
proved that, conversely, all even perfect numbers have this form. This is known as the
1401:
1385:
1321:
1317:
225:
184:
39:
8518:
8362:
7972:
7849:
12286:
12203:
11978:
11958:
11933:
11640:
11624:
11565:
11519:
11215:
11200:
11110:
10835:
10393:
10262:
10224:
10181:
10062:
10047:
10037:
9995:
9985:
9960:
9542:
9307:
9171:
9144:
8980:
8845:
8783:
8774:
8759:
8722:
8648:
7466:
6870:
6860:
6798:
5523:
2, 5, 11, 13, 23, 61, 83, 421, 1039, 1511, 31237, 60413, 113177, 135647, 258413, ...
5491:
2, 3, 5, 7, 17, 19, 109, 509, 661, 709, 1231, 12889, 13043, 26723, 43963, 44789, ...
4596:
3940:
3816:
3804:
2347:
1735:
1332:
260:
8569:
7289:
7272:
4573:
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.
12238:
11676:
11665:
11580:
11418:
11393:
11310:
11210:
11180:
11155:
11139:
11044:
11011:
10760:
10734:
10645:
10584:
10161:
10057:
9990:
9970:
9945:
9863:
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:
9496:
9219:
9102:
8985:
8975:
8960:
8955:
8919:
8633:
8588:
7590:
7534:
7253:
6855:
6803:
5539:
2, 3, 7, 29, 71, 127, 271, 509, 1049, 6389, 6883, 10613, 19889, 79987, 608099, ...
5048:
4031:
3800:
3742:
The number of factors for the first 500 Mersenne numbers can be found at (sequence
2041:
1883:
was composite without finding a factor. No factor was found until a famous talk by
907:, all other Mersenne numbers are also congruent to 3 (mod 4). Consequently, in the
722:
256:
158:
154:
95:
7654:
7435:
2233:, was found by the computer a little less than two hours later. Three more —
7911:
7683:
1239:
This rules out primality for Mersenne numbers with a composite exponent, such as
12248:
12188:
11635:
11510:
11315:
10779:
10670:
10625:
10620:
10370:
10277:
10176:
10005:
9980:
9955:
9688:
9683:
9678:
9673:
9668:
9663:
9658:
9653:
9648:
9643:
9638:
9633:
9628:
9623:
9618:
9613:
9608:
9603:
9598:
9593:
9588:
9434:
9107:
9015:
9010:
8990:
8904:
8807:
8683:
8430:
8406:
7939:
7659:
7634:
7249:
7210:
7103:
is a perfect fourth power, it can be shown that there are at most two values of
6865:
4201:
4161:
3820:
2350:
745 on August 23, 2008. This was the eighth Mersenne prime discovered at UCLA.
2189:
1309:
is prime for only 43 of the first two million prime numbers (up to 32,452,843).
252:
162:
146:
91:
69:
7888:
5240:
42643801, 43112609, 57885161, ..., 74207281, ..., 77232917, ..., 82589933, ...
1263:, this is not the case, and the smallest counterexample is the Mersenne number
17:
12253:
12038:
11772:
11753:
11049:
10660:
9511:
9327:
9235:
9155:
9005:
8909:
8332:
5475:
2, 3, 17, 19, 47, 101, 1709, 2539, 5591, 6037, 8011, 19373, 26489, 27427, ...
4027:
3855:
3456:
2 − 1 = 1,119,429,257 × 175,573,124,547,437,977 × 8,480,999,878,421,106,991 ×
2973:, 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,
456:
248:
244:
233:
which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, ... (sequence
87:
83:
9883:
7520:
7369:
7183:
2205:, by this means was achieved at 10:00 pm on January 30, 1952, using the U.S.
12144:
12139:
11973:
11378:
11305:
11297:
11102:
11016:
10134:
9552:
9501:
9382:
8593:
8574:
8410:
7728:
7610:
7224:
6568:
2, 3, 5, 19, 97, 109, 317, 353, 701, 9739, 14951, 37573, 46889, 769543, ...
4312:(that is, squares of absolute values) of these numbers are rational primes:
6260:
5, 19, 67, 107, 593, 757, 1801, 2243, 2383, 6043, 10181, 11383, 15629, ...
5747:
2, 11, 31, 173, 271, 547, 1823, 2111, 5519, 7793, 22963, 41077, 49739, ...
5507:
4, 5, 7, 19, 29, 61, 137, 883, 1381, 1823, 5227, 25561, 29537, 300893, ...
2409:, having 24,862,048 digits. A computer volunteered by Patrick Laroche from
1335:, making them popular choices when a prime modulus is desired, such as the
8450:
8338:
6584:
2, 5, 11, 109, 193, 1483, 11353, 21419, 21911, 24071, 106859, 139739, ...
12078:
11479:
3843:
3827:
3796:
365:
claims that there are infinitely many Mersenne primes and predicts their
7773:
6356:
17, 19, 73, 139, 907, 1907, 2029, 4801, 5153, 10867, 20161, 293831, ...
4556:
These primes are called repunit primes. Another example is when we take
4316:
5, 13, 41, 113, 2113, 525313, 536903681, 140737471578113, ... (sequence
3725:
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).
11968:
11484:
11143:
9054:
8602:
7678:. Lecture Notes in Computer Science. Vol. 8874. pp. 358–377.
6828:
6793:
4934:
4426:
4008:
The simplest generalized Mersenne primes are prime numbers of the form
3097:
925:
341:
175:
6340:
2, 5, 11, 13, 331, 599, 18839, 23747, 24371, 29339, 32141, 67421, ...
3414:
42643801, 43112609, 57885161, 74207281, 77232917, 82589933. (sequence
2391:, thus officially confirming its position as the 45th Mersenne prime.
2196:
in 1949, but the first successful identification of a Mersenne prime,
12223:
12028:
11963:
5321:
3, 4, 7, 11, 83, 149, 223, 599, 647, 1373, 8423, 149497, 388897, ...
4189:
does not lead to anything interesting (since it is always −1 for all
3955:
MF(2, 2), MF(2, 3), MF(2, 4), MF(2, 5), MF(3, 2), MF(3, 3), MF(7, 2),
3812:
3808:
3736:
263 × 10,350,794,431,055,162,386,718,619,237,468,234,569 (38 digits)
3405:
As of 2023, the 51 known Mersenne primes are 2 − 1 for the following
1389:
296:
be prime. The smallest composite Mersenne number with prime exponent
8534:
http://www.leyland.vispa.com/numth/factorization/cunningham/main.htm
5085:
is prime. However, this has not been proved for any single value of
3692:
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
8363:
Property of Mersenne numbers with prime exponent that are composite
7958:
5991:
3, 59, 223, 547, 773, 1009, 1823, 3803, 49223, 193247, 703393, ...
5901:
2, 7, 29, 31, 67, 149, 401, 2531, 19913, 30773, 53857, 170099, ...
2260: — were found by the same program in the next several months.
1331:
Arithmetic modulo a Mersenne number is particularly efficient on a
9049:
9035:
8478:
http://www.leyland.vispa.com/numth/factorization/cunningham/2+.txt
8466:
http://www.leyland.vispa.com/numth/factorization/cunningham/2-.txt
5731:
2, 3, 5, 19, 41, 47, 8231, 33931, 43781, 50833, 53719, 67211, ...
5667:
5, 13, 131, 149, 1699, 14221, 35201, 126037, 371669, 1264699, ...
4530:
2, 19, 23, 317, 1031, 49081, 86453, 109297, 270343, ... (sequence
2135:
11817:
8489:
7824:
5587:
2, 3, 7, 29, 41, 67, 1327, 1399, 2027, 69371, 86689, 355039, ...
307:
Mersenne primes were studied in antiquity because of their close
11902:
6212:
3, 5, 19, 311, 317, 1129, 4253, 7699, 18199, 35153, 206081, ...
5225:
2894:; thus there are always larger primes than any particular prime.
2428:
Mersenne numbers are 0, 1, 3, 7, 15, 31, 63, ... (sequence
11906:
11821:
11770:
11734:
11698:
11662:
11622:
11247:
11136:
10862:
10777:
10732:
10609:
10299:
10246:
10198:
10132:
10084:
10022:
9926:
9887:
8606:
7462:"Prime number with 22 million digits is the biggest ever found"
6087:
2, 3, 7, 11, 19, 29, 401, 709, 2531, 15787, 66949, 282493, ...
4445:, it is to simply take out this factor and ask which values of
4335:
One may encounter cases where such a Mersenne prime is also an
2337:
In September 2008, mathematicians at UCLA participating in the
11953:
8546:
http://www.leyland.vispa.com/numth/factorization/anbn/main.htm
7655:"Proof of a result of Euler and Lagrange on Mersenne Divisors"
7384:"UCLA mathematicians discover a 13-million-digit prime number"
4635:
is a perfect power, it can be shown that there is at most one
4513:
can be either positive or negative.) If, for example, we take
4144:, the former is not a prime. This can be remedied by allowing
3279:
All composite divisors of prime-exponent Mersenne numbers are
7490:"New Biggest Prime Number = 2 to the 74 Mil ... Uh, It's Big"
7436:"Mersenne Prime Number discovery – 2 − 1 is Prime!"
6071:
3, 7, 13, 19, 307, 619, 2089, 7297, 75571, 76103, 98897, ...
5699:
2, 5, 23, 73, 101, 401, 419, 457, 811, 1163, 1511, 8011, ...
3850:
then because it is primitive it constrains the odd leg to be
3846:
is always a Mersenne number. For example, if the even leg is
887:, all Mersenne primes are congruent to 3 (mod 4). Other than
7591:"GIMPS Discovers Largest Known Prime Number: 2^82,589,933-1"
6324:
3, 5, 19, 31, 367, 389, 431, 2179, 10667, 13103, 90397, ...
6167:
2, 3, 19, 31, 101, 139, 167, 1097, 43151, 60703, 90499, ...
2568:(which is a contradiction, as neither −1 nor 0 is prime) or
8492:
7304:
Bell, E.T. and Mathematical Association of America (1951).
7064:
6774:
6740:
6632:
6603:
6587:
6571:
6555:
6539:
6523:
6507:
6439:
6423:
6407:
6391:
6375:
6359:
6343:
6327:
6311:
6295:
6279:
6263:
6247:
6231:
6215:
6199:
6170:
6154:
6138:
6122:
6106:
6090:
6074:
6058:
6042:
6026:
6010:
5994:
5965:
5936:
5920:
5904:
5888:
5872:
5856:
5814:
5798:
5782:
5766:
5750:
5734:
5718:
5702:
5686:
5670:
5654:
5638:
5622:
5606:
5590:
5574:
5558:
5542:
5526:
5510:
5494:
5478:
5462:
5446:
5430:
5414:
5398:
5382:
5340:
5324:
5308:
5292:
5276:
5260:
5243:
4845:
4820:
4764:
4722:
4575:
4544:
4532:
4403:
4401:
7, 271, 2269, 176419, 129159847, 1162320517, ... (sequence
4385:
4318:
4293:
4097:
It is also natural to try to generalize primes of the form
3823:
having been discovered and named during the 19th century).
3744:
3477:
3416:
2430:
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:
8327:
7870:
6436:
7, 11, 181, 421, 2297, 2797, 4129, 4139, 7151, 29033, ...
6183:
2, 3, 5, 11, 19, 1259, 1399, 2539, 2843, 5857, 10589, ...
3714:
745,988,807 × 870,035,986,098,720,987,332,873 (24 digits)
3703:
2,550,183,799 × 3,976,656,429,941,438,590,393 (22 digits)
3326:
The Mersenne number sequence is a member of the family of
2220:, with a computer search program written and run by Prof.
9583:
9578:
9573:
9568:
7158:"GIMPS Project Discovers Largest Known Prime Number: 2-1"
6039:
3, 5, 13, 17, 43, 127, 229, 277, 6043, 11131, 11821, ...
425:
with n decimal digits (i.e. 10 < p < 10) for which
6151:
5, 7, 19, 31, 53, 67, 293, 641, 2137, 3011, 268207, ...
6135:
2, 19, 23, 317, 1031, 49081, 86453, 109297, 270343, ...
5683:
3, 17, 23, 29, 47, 61, 1619, 18251, 106187, 201653, ...
5571:
3, 4, 5, 17, 397, 409, 643, 1783, 2617, 4583, 8783, ...
3842: ) generates a unique right triangle such that its
3807:
after Marin Mersenne, because 8191 is a Mersenne prime (
3681:
11,447 × 13,842,607,235,828,485,645,766,393 (26 digits)
1961:, was discovered anonymously before 1461; the next two (
7211:"Heuristics: Deriving the Wagstaff Mersenne Conjecture"
5962:
2, 3, 5, 13, 29, 37, 1021, 1399, 2137, 4493, 5521, ...
1302:(the correct terms on Mersenne's original list), while
7577:"Found: A Special, Mind-Bogglingly Large Prime Number"
1734:
Mersenne primes take their name from the 17th-century
621:
8514:
Factorization of completely factored Mersenne numbers
8235:
8155:
7559:"Largest-known prime number found on church computer"
6655:
are not included in the corresponding OEIS sequence.
6388:
3, 5, 17, 67, 83, 101, 1373, 6101, 12119, 61781, ...
5715:
3, 13, 31, 313, 3709, 7933, 14797, 30689, 38333, ...
5635:
3, 7, 19, 109, 131, 607, 863, 2917, 5923, 12421, ...
4871:
4585:
corresponding to primes −11, 19141, 57154490053, ....
4460:
2146:
1820:
1466:
1421:
1291:
increases. For example, eight of the first 11 primes
947:
798:
743:
685:
431:
375:
123:
6600:
2, 3, 5, 13, 347, 977, 1091, 4861, 4967, 34679, ...
6372:
5, 7, 179, 229, 439, 557, 6113, 223999, 327001, ...
6308:
3, 5, 11, 17, 71, 89, 827, 22307, 45893, 63521, ...
4983:
makes this number prime. It can be shown that such
4123:
3659:
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
12174:
12127:
12096:
12087:
11940:
11594:
11548:
11508:
11459:
11433:
11366:
11350:
11329:
11296:
11261:
11101:
11068:
11025:
11002:
10879:
10567:
10558:
10536:
10493:
10455:
10446:
10379:
10321:
10312:
9561:
9525:
9425:
9402:
9376:
9143:
9136:
9034:
8928:
8892:
8641:
8456:
Decimal digits and English names of Mersenne primes
7535:"Mersenne Prime Discovery - 2^77232917-1 is Prime!"
6552:2, 3, 31, 41, 53, 101, 421, 1259, 4721, 45259, ...
6292:5, 41, 149, 229, 263, 739, 3457, 20269, 98221, ...
941:must also be prime. This follows from the identity
113:
102:
79:
68:
56:
45:
35:
8275:
8195:
7305:
6055:2, 3, 107, 197, 2843, 3571, 4451, ..., 31517, ...
4914:
4496:
4148:to be an algebraic integer instead of an integer:
2176:
1853:
1501:
1452:
1231:
830:
784:
713:
663:
444:
413:
191:, who studied them in the early 17th century. If
6616:2, 3, 7, 67, 79, 167, 953, 1493, 3389, 4871, ...
5853:2, 5, 163, 191, 229, 271, 733, 21059, 25237, ...
5619:2, 5, 11, 61, 619, 2879, 2957, 24371, 69247, ...
3670:167 × 57,912,614,113,275,649,087,721 (23 digits)
9265: = 0, 1, 2, 3, ...
7873:. The On-Line Encyclopedia of Integer Sequences.
5651:3, 7, 19, 79, 431, 1373, 1801, 2897, 46997, ...
5603:3, 5, 7, 113, 397, 577, 7573, 14561, 58543, ...
3648:439 × 2,298,041 × 9,361,973,132,609 (13 digits)
1891:. On the other side of the board, he multiplied
286:without the primality requirement may be called
5012:is prime. It is a conjecture that for any pair
4196:). Thus, we can regard a ring of "integers" on
4185:are the usual Mersenne primes, and the formula
2897:It follows from this fact that for every prime
2341:(GIMPS) won part of a $ 100,000 prize from the
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
414:{\displaystyle e^{\gamma }\cdot \log _{2}(10)}
318:As of 2023, 51 Mersenne primes are known. The
11918:
11833:
9899:
8618:
8447:– contains factors for small Mersenne numbers
7277:Bulletin of the American Mathematical Society
4987:must be primes themselves or equal to 4, and
3943:. The only known Mersenne–Fermat primes with
3028:. If the given congruence is satisfied, then
1252:Though the above examples might suggest that
8:
7729:"M12720787 Mersenne number exponent details"
7225:Mersenne Primes: History, Theorems and Lists
6452:2, 5, 7, 107, 383, 17359, 21929, 26393, ...
5933:3, 11, 17, 173, 839, 971, 40867, 45821, ...
5795:2, 19, 1021, 5077, 34031, 46099, 65707, ...
4772:) (notice this OEIS sequence does not allow
1400:) is a perfect number. In the 18th century,
30:
6478:2, 3, 13, 31, 59, 131, 223, 227, 1523, ...
3429:Factorization of composite Mersenne numbers
3401:List of Mersenne primes and perfect numbers
3243:and therefore 2 is a quadratic residue mod
3007:does not hold. By Fermat's little theorem,
2040:) were found early in the 20th century, by
12093:
11925:
11911:
11903:
11840:
11826:
11818:
11767:
11731:
11695:
11659:
11619:
11293:
11258:
11244:
11133:
10876:
10859:
10774:
10729:
10606:
10564:
10452:
10318:
10309:
10296:
10243:
10200:Possessing a specific set of other numbers
10195:
10129:
10081:
10019:
9923:
9906:
9892:
9884:
9140:
8625:
8611:
8603:
7885:"A research of Mersenne and Fermat primes"
6680:, a difference of two consecutive perfect
5811:2, 3, 7, 19, 31, 67, 89, 9227, 43891, ...
4915:{\displaystyle {\frac {a^{n}-b^{n}}{a-b}}}
4589:It is a conjecture that for every integer
4359:. In these cases, such numbers are called
3626:193,707,721 × 761,838,257,287 (12 digits)
2904:, there is at least one prime of the form
1516:
349:Are there infinitely many Mersenne primes?
29:
8519:The Cunningham project, factorization of
8265:
8256:
8243:
8234:
8185:
8176:
8163:
8154:
7957:
7904:Encyclopedia of Cryptography and Security
7370:The Mathematics Department and the Mark 1
7308:Mathematics, queen and servant of science
7288:
6536:2, 3, 7, 13, 47, 89, 139, 523, 1051, ...
6276:2, 3, 11, 163, 191, 269, 1381, 1493, ...
5869:2, 7, 19, 167, 173, 223, 281, 21647, ...
4892:
4879:
4872:
4870:
4468:
4461:
4459:
4431:The other way to deal with the fact that
4373:is an Eisenstein prime for the following
3838:and has its even leg a power of 2 (
3223:, so −2 would be a quadratic residue mod
2999:is a Mersenne prime, then the congruence
2802:. As a result, for all positive integers
2278:was the first with more than 10,000, and
2145:
1843:
1828:
1819:
1748:2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257.
1491:
1465:
1435:
1420:
1196:
1174:
1158:
1145:
1112:
1069:
1047:
1031:
1018:
985:
956:
948:
946:
822:
803:
797:
770:
748:
742:
690:
684:
622:
620:
436:
430:
393:
380:
374:
8427:with hyperlinks to original publications
8339:GIMPS, known factors of Mersenne numbers
7774:"M1277 Mersenne number exponent details"
7260:(4th ed.). Oxford University Press.
7258:An Introduction to the Theory of Numbers
6520:2, 3, 7, 89, 101, 293, 4463, 70067, ...
6007:2, 3, 7, 127, 283, 883, 1523, 4001, ...
5099:
3756:Mersenne numbers in nature and elsewhere
3615:179,951 × 3,203,431,780,337 (13 digits)
3487:
1247:= 2 − 1 = 15 = 3 × 5 = (2 − 1) × (1 + 2)
243:) and the resulting Mersenne primes are
7676:Advances in Cryptology – ASIACRYPT 2014
7149:
6893:
4862:Another generalized Mersenne number is
353:(more unsolved problems in mathematics)
7800:Famous Puzzles of Great Mathematicians
6244:2, 7, 11, 17, 37, 521, 877, 2423, ...
5051:, there are infinitely many values of
4599:, there are infinitely many values of
4497:{\displaystyle {\frac {b^{n}-1}{b-1}}}
4280:is a Gaussian prime for the following
3911:natural number, and can be written as
2790: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
7936:The Prime Glossary: Gaussian Mersenne
6196:4, 7, 67, 73, 1091, 1483, 10937, ...
3112:Mersenne numbers is pairwise coprime.
2214:University of California, Los Angeles
1983:in 1588. After nearly two centuries,
363:Lenstra–Pomerance–Wagstaff conjecture
7:
11880:Great Internet Mersenne Prime Search
8495:– Factorization of Mersenne numbers
7707:Henri Lifchitz and Renaud Lifchitz.
7107:with this property: in these cases,
6835:Great Internet Mersenne Prime Search
6404:3, 103, 271, 523, 23087, 69833, ...
6119:2, 3, 5, 37, 599, 38393, 51431, ...
2675:is composite. By contraposition, if
2368:, discovered a 48th Mersenne prime,
2339:Great Internet Mersenne Prime Search
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:
7273:"On the factoring of large numbers"
6103:2, 31, 103, 617, 10253, 10691, ...
3637:228,479 × 48,544,121 × 212,885,833
3255:, −1 is a quadratic nonresidue mod
2630:is composite, hence can be written
2413:made the find on December 7, 2018.
2398:, 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
12303:Unsolved problems in number theory
7488:Chang, Kenneth (21 January 2016).
7460:Brook, Robert (January 19, 2016).
7382:Maugh II, Thomas H. (2008-09-27).
6420:2, 7, 53, 67, 71, 443, 26497, ...
2932:is an odd prime, then every prime
2692:is an odd prime, then every prime
1772:(which are composite) and omitted
745:
687:
482:(which is also prime) will divide
25:
12199:Indefinite and fictitious numbers
8461:Prime curios: 2305843009213693951
8451:Known factors of Mersenne numbers
7906:. Springer US. pp. 509–510.
7850:"JPL Small-Body Database Browser"
7434:Cooper, Curtis (7 January 2016).
7409:"Largest Prime Number Discovered"
6228:5, 31, 271, 929, 2789, 4153, ...
4858:Other generalized Mersenne primes
3339:(3, 2). That is, Mersenne number
2210:Western Automatic Computer (SWAC)
1355:, generalized shift register and
161:. That is, it is a prime number
12298:Eponymous numbers in mathematics
11801:
11409:Perfect digit-to-digit invariant
9001:Supersingular (moonshine theory)
4308:As for all Gaussian primes, the
3932:, it is a Mersenne number. When
3165:: 11 and 23 are both prime, and
2044:in 1911 and 1914, respectively.
27:Prime number of the form (2^n)-1
8276:{\displaystyle (a^{n}+b^{n})/c}
8196:{\displaystyle (a^{n}-b^{n})/c}
7290:10.1090/S0002-9904-1903-01079-9
7271:Cole, F. N. (1 December 1903).
6900:This number is the same as the
6504:53, 421, 647, 1601, 35527, ...
2600:which is not prime. Therefore,
2424:Theorems about Mersenne numbers
2216:(UCLA), under the direction of
1911:The first four Mersenne primes
344:Unsolved problem in mathematics
8996:Supersingular (elliptic curve)
8445:Will Edgington's Mersenne Page
8262:
8236:
8182:
8156:
7825:"Wheat and Chessboard Problem"
7802:. AMS Bookstore. p. 197.
7635:Will Edgington's Mersenne Page
7137:can be factored algebraically.
7019:. Thus, in this case the pair
6491:2, 3, 17, 41, 43, 59, 83, ...
3215:. Supposing latter true, then
3191:. By Fermat's little theorem,
2878:This fact leads to a proof of
2452: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:
1:
12114:Conway chained arrow notation
10248:Expressible via specific sums
8777:2 ± 2 ± 1
7611:"GIMPS - The Math - PrimeNet"
6705:, because it is divisible by
3443:integer factorization records
3395:List of known Mersenne primes
3127:are both prime (meaning that
2982:A Mersenne prime cannot be a
2700: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
8431:report about Mersenne primes
7912:10.1007/978-1-4419-5906-5_32
7684:10.1007/978-3-662-45611-8_19
4698:is prime are (starting with
4266:which will then be called a
4140:, so unless the latter is a
3781:wheat and chessboard problem
3760:In the mathematical problem
3733:272225893536...454145691647
3722:103845937170...992658440191
3711:649037107316...312041152511
3700:101412048018...973625643007
3689:253530120045...993406410751
3678:158456325028...187087900671
3667:967140655691...033397649407
3604:6,361 × 69,431 × 20,394,401
3018:. Therefore, one can write
2760:, 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 (
11337:Multiplicative digital root
8528:= 2, 3, 5, 6, 7, 10, 11, 12
8425:Mersenne prime bibliography
8316:Encyclopedia of Mathematics
7754:"Exponent Status for M1277"
5160:(some large terms are only
5101:For more information, see
3764:, solving a puzzle with an
3593:2,351 × 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^
12334:
12209:Largest known prime number
7798:Petković, Miodrag (2009).
6841:Largest known prime number
6651:is even, then the numbers
6465:7, 1163, 4007, 10159, ...
5917:3, 5, 7, 4703, 30113, ...
4740:, they are (starting with
4424:
4361:Eisenstein Mersenne primes
4331:Eisenstein Mersenne primes
4004:Generalized Mersenne prime
4001:
3435:special number field sieve
3398:
2722:. A composite example is
1502:{\displaystyle P=M(M+1)/2}
1366:
579:). Since for these primes
320:largest known prime number
12262:
12194:Extended real number line
12109:Knuth's up-arrow notation
11855:
11797:
11780:
11766:
11744:
11730:
11708:
11694:
11672:
11658:
11631:
11618:
11414:Perfect digital invariant
11257:
11243:
11151:
11132:
10989:Superior highly composite
10875:
10858:
10786:
10773:
10741:
10728:
10616:
10605:
10308:
10295:
10253:
10242:
10205:
10194:
10142:
10128:
10091:
10080:
10033:
10018:
9936:
9922:
9872:
7184:"GIMPS Milestones Report"
5763:3, 53, 83, 487, 743, ...
4220:If we regard the ring of
3656:604462909807314587353087
3088:are natural numbers then
2364:, a mathematician at the
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
12293:Classes of prime numbers
12119:Steinhaus–Moser notation
11027:Euler's totient function
10811:Euler–Jacobi pseudoprime
10086:Other polynomial numbers
9383:Mega (1,000,000+ digits)
9252:Arithmetic progression (
8399:"31 and Mersenne Primes"
8227:PRP records, search for
8147:PRP records, search for
7346:Horace S. Uhler (1952).
7325:"h2g2: Mersenne Numbers"
7190:. Mersenne Research, Inc
7054:must be prime. That is,
6629:47, 401, 509, 8609, ...
4991:can be 4 if and only if
4216:Gaussian Mersenne primes
3582:431 × 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
10841:Somer–Lucas pseudoprime
10831:Lucas–Carmichael number
10666:Lazy caterer's sequence
8333:GIMPS Milestones Report
7312:. McGraw-Hill New York.
4969:is always divisible by
4507:be prime. (The integer
4438:is always divisible by
4268:Gaussian Mersenne prime
4133:is always divisible by
3862:and its inradius to be
3645:9444732965739290427391
3634:2361183241434822606847
3297:has no solutions where
2742:Fermat's little theorem
2707:. This holds even when
2666:(2) + (2) + ... + 2 + 1
675:is a prime, it must be
12162:Fast-growing hierarchy
11875:Double Mersenne number
10716:Wedderburn–Etherington
10116:Lucky numbers of Euler
9538:Industrial-grade prime
8915:Newman–Shanks–Williams
8433:– detection in detail
8277:
8197:
7563:christianchronicle.org
7440:Mersenne Research, Inc
7162:Mersenne Research, Inc
6824:Double Mersenne number
6809:Erdős–Borwein constant
4916:
4498:
4181:is either 2 or 0. But
4101:to primes of the form
3876:Mersenne–Fermat number
3870:Mersenne–Fermat primes
3623:147573952589676412927
2912:less than or equal to
2886:, all primes dividing
2551:. 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
12219:Long and short scales
12157:Grzegorczyk hierarchy
11004:Prime omega functions
10821:Frobenius pseudoprime
10611:Combinatorial numbers
10480:Centered dodecahedral
10273:Primary pseudoperfect
9875:List of prime numbers
9333:Sophie Germain/Safe (
8278:
8198:
7237:Mersenne's conjecture
5032:are not both perfect
4917:
4499:
4122:). However (see also
4064:; another example is
3992:cyclotomic polynomial
3803:number 8191 is named
3770:-disc tower requires
3571:13,367 × 164,511,353
3055:= 1 + 2 + 2 + ... + 2
2971:quadratic reciprocity
2852:. Furthermore, since
2732:89 = 1 + 4 × (2 × 11)
2396: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
11860:Mersenne conjectures
11463:-composition related
11263:Arithmetic functions
10865:Arithmetic functions
10801:Elliptic pseudoprime
10485:Centered icosahedral
10465:Centered tetrahedral
9057:(10 − 1)/9
8233:
8153:
7335:on December 5, 2014.
6814:Mersenne conjectures
4869:
4458:
4339:, being of the form
3549:233 × 1,103 × 2,089
3454:factors allowed) is
3288:Mihăilescu's theorem
3133:Sophie Germain prime
3075:which is impossible.
2962:is a square root of
2770:is also a factor of
2720:31 = 1 + 3 × (2 × 5)
2575: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
12234:Orders of magnitude
12104:Scientific notation
11389:Kaprekar's constant
10909:Colossally abundant
10796:Catalan pseudoprime
10696:Schröder–Hipparchus
10475:Centered octahedral
10351:Centered heptagonal
10341:Centered pentagonal
10331:Centered triangular
9931:and related numbers
9366: ± 7, ...
8893:By integer sequence
8678:(2 + 1)/3
7823:Weisstein, Eric W.
7653:Caldwell, Chris K.
7565:. January 12, 2018.
7414:Scientific American
7352:Scripta Mathematica
7223:Chris K. Caldwell,
7089:th powers for some
6876:Gillies' conjecture
5102:
4832:For negative bases
4734:For negative bases
4210:Eisenstein integers
4030:with small integer
3612:576460752303423487
3281:strong pseudoprimes
2919:, for some integer
2864:is odd. Therefore,
2822:. Therefore, since
2782:is not a factor of
2022:in 1883. Two more (
1410:odd perfect numbers
909:prime factorization
32:
12152:Ackermann function
11807:Mathematics portal
11749:Aronson's sequence
11495:Smarandache–Wellin
11252:-dependent numbers
10959:Primitive abundant
10846:Strong pseudoprime
10836:Perrin pseudoprime
10816:Fermat pseudoprime
10756:Wolstenholme prime
10580:Squared triangular
10366:Centered decagonal
10361:Centered nonagonal
10356:Centered octagonal
10346:Centered hexagonal
9548:Formula for primes
9181: + 2 or
9113:Smarandache–Wellin
8586:Weisstein, Eric W.
8567:Weisstein, Eric W.
8551:2016-02-02 at the
8539:2016-03-04 at the
8483:2013-05-02 at the
8471:2014-11-05 at the
8273:
8193:
7852:. Ssd.jpl.nasa.gov
7829:Mathworld. Wolfram
7640:2014-10-14 at the
7579:. January 5, 2018.
7495:The New York Times
7164:. 21 December 2018
6939:quadratic equation
6684:th powers, and if
5170:are checked up to
5100:
5036:th powers for any
4912:
4494:
4224:, we get the case
3560:223 × 616,318,177
3309:are integers with
3217:2 = (2) ≡ −2 (mod
2598:0 − 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
12308:Integer sequences
12280:
12279:
12170:
12169:
11900:
11899:
11815:
11814:
11793:
11792:
11762:
11761:
11726:
11725:
11690:
11689:
11654:
11653:
11614:
11613:
11610:
11609:
11429:
11428:
11239:
11238:
11128:
11127:
11124:
11123:
11070:Aliquot sequences
10881:Divisor functions
10854:
10853:
10826:Lucas pseudoprime
10806:Euler pseudoprime
10791:Carmichael number
10769:
10768:
10724:
10723:
10601:
10600:
10597:
10596:
10593:
10592:
10554:
10553:
10442:
10441:
10399:Square triangular
10291:
10290:
10238:
10237:
10190:
10189:
10124:
10123:
10076:
10075:
10014:
10013:
9881:
9880:
9492:Carmichael number
9427:Composite numbers
9362: ± 3, 8
9358: ± 1, 4
9321: ± 1, …
9317: ± 1, 4
9313: ± 1, 2
9303:
9302:
8848:3·2 − 1
8753:2·3 + 1
8667:Double Mersenne (
8570:"Mersenne number"
8311:"Mersenne number"
7921:978-1-4419-5905-8
7848:Alan Chamberlin.
7809:978-0-8218-4814-2
7709:"PRP Top Records"
7693:978-3-662-45607-1
7388:Los Angeles Times
7235:The Prime Pages,
7209:Caldwell, Chris.
7085:are both perfect
6638:
6637:
5366:2, 3 (no others)
5047:is not a perfect
4910:
4492:
4416:Divide an integer
4222:Gaussian integers
4206:Gaussian integers
3740:
3739:
3601:9007199254740991
3508:Factorization of
3227:. However, since
2728:23 = 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
12325:
12094:
12024:Eddington number
11969:Hundred thousand
11927:
11920:
11913:
11904:
11892:Mersenne Twister
11842:
11835:
11828:
11819:
11805:
11768:
11737:Natural language
11732:
11696:
11664:Generated via a
11660:
11620:
11525:Digit-reassembly
11490:Self-descriptive
11294:
11259:
11245:
11196:Lucas–Carmichael
11186:Harmonic divisor
11134:
11060:Sparsely totient
11035:Highly cototient
10944:Multiply perfect
10934:Highly composite
10877:
10860:
10775:
10730:
10711:Telephone number
10607:
10565:
10546:Square pyramidal
10528:Stella octangula
10453:
10319:
10310:
10302:Figurate numbers
10297:
10244:
10196:
10130:
10082:
10020:
9924:
9908:
9901:
9894:
9885:
9412:Eisenstein prime
9367:
9343:
9322:
9294:
9266:
9246:
9230:
9214:
9209: + 6,
9205: + 2,
9190:
9185: + 4,
9166:
9141:
9058:
9021:Highly cototient
8883:
8882:
8876:
8866:
8849:
8840:
8825:
8802:
8801:·2 − 1
8790:
8789:·2 + 1
8778:
8769:
8754:
8745:
8732:
8717:
8702:
8690:
8689:·2 + 1
8679:
8670:
8661:
8652:
8627:
8620:
8613:
8604:
8599:
8598:
8589:"Mersenne prime"
8580:
8579:
8529:
8509:
8503:
8491:
8436:
8421:
8419:
8418:
8409:. Archived from
8389:
8361:
8324:
8298:
8296:
8284:
8282:
8280:
8279:
8274:
8269:
8261:
8260:
8248:
8247:
8224:
8218:
8216:
8204:
8202:
8200:
8199:
8194:
8189:
8181:
8180:
8168:
8167:
8144:
8138:
8135:
8131:
8122:
8116:
8113:
8109:
8096:
8090:
8087:
8083:
8070:
8064:
8061:
8057:
8044:
8038:
8035:
8031:
8022:
8016:
8013:
8009:
8000:
7994:
7991:
7987:
7979:
7970:
7964:
7963:
7961:
7949:
7943:
7934:Chris Caldwell:
7932:
7926:
7925:
7899:
7893:
7892:
7887:. Archived from
7881:
7875:
7874:
7867:
7861:
7860:
7858:
7857:
7845:
7839:
7838:
7836:
7835:
7820:
7814:
7813:
7795:
7789:
7788:
7786:
7784:
7770:
7764:
7763:
7761:
7760:
7750:
7744:
7743:
7741:
7739:
7725:
7719:
7718:
7716:
7715:
7704:
7698:
7697:
7671:
7665:
7664:
7650:
7644:
7632:
7626:
7625:
7623:
7621:
7615:www.mersenne.org
7607:
7601:
7600:
7598:
7597:
7587:
7581:
7580:
7573:
7567:
7566:
7555:
7549:
7548:
7546:
7545:
7539:www.mersenne.org
7531:
7525:
7524:
7519:. Archived from
7513:
7507:
7506:
7504:
7502:
7485:
7479:
7478:
7476:
7474:
7457:
7451:
7450:
7448:
7446:
7431:
7425:
7424:
7422:
7421:
7404:
7398:
7397:
7395:
7394:
7379:
7373:
7366:
7360:
7359:
7343:
7337:
7336:
7331:. Archived from
7321:
7315:
7313:
7311:
7301:
7295:
7294:
7292:
7268:
7262:
7261:
7246:
7240:
7233:
7227:
7221:
7215:
7214:
7206:
7200:
7199:
7197:
7195:
7180:
7174:
7173:
7171:
7169:
7154:
7138:
7136:
7135:
7133:
7132:
7123:
7120:
7106:
7102:
7095:
7088:
7084:
7080:
7075:
7069:
7067:
7057:
7053:
7042:
7030:
7018:
7000:
6998:
6997:
6988:
6985:
6970:
6964:
6962:
6932:
6928:
6924:
6898:
6819:Mersenne twister
6777:
6766:
6755:
6743:
6732:
6721:
6714:
6704:
6697:
6693:
6683:
6679:
6668:
6654:
6650:
6646:
5221:
5220:
5214:
5205:
5201:
5200:
5194:
5185:
5184:
5182:
5173:
5169:
5168:
5157:
5156:
5155:
5153:
5152:
5143:
5140:
5125:
5124:
5117:
5116:
5110:
5109:
5103:
5096:
5084:
5083:
5081:
5080:
5071:
5068:
5054:
5046:
5039:
5035:
5031:
5027:
5023:
5011:
5001:
4990:
4986:
4982:
4978:
4968:
4958:
4943:
4932:
4928:
4921:
4919:
4918:
4913:
4911:
4909:
4898:
4897:
4896:
4884:
4883:
4873:
4848:
4837:
4823:
4812:
4811:
4809:
4808:
4802:
4799:
4788:
4778:
4767:
4756:
4750:
4746:
4739:
4725:
4714:
4708:
4704:
4697:
4696:
4694:
4693:
4687:
4684:
4673:
4664:
4663:
4661:
4660:
4654:
4651:
4641:value such that
4640:
4634:
4629:is prime. (When
4628:
4627:
4625:
4624:
4618:
4615:
4604:
4594:
4578:
4568:
4562:
4547:
4535:
4525:
4519:
4512:
4503:
4501:
4500:
4495:
4493:
4491:
4480:
4473:
4472:
4462:
4450:
4444:
4437:
4406:
4388:
4378:
4372:
4358:
4348:
4337:Eisenstein prime
4321:
4296:
4285:
4279:
4261:
4253:
4243:
4233:
4195:
4188:
4184:
4180:
4170:
4160:of integers (on
4139:
4132:
4121:
4114:
4107:
4100:
4093:
4074:
4068:, in this case,
4067:
4063:
4044:
4038:, in this case,
4037:
4034:. An example is
4026:is a low-degree
4025:
4014:
3989:
3983:
3960:
3956:
3949:
3938:
3931:
3923:
3910:
3904:
3898:
3897:
3895:
3894:
3891:
3888:
3865:
3861:
3853:
3849:
3841:
3791:
3778:
3769:
3747:
3590:140737488355327
3516:
3505:
3495:
3488:
3480:
3464:
3460:
3459:
3440:
3419:
3389:
3379:
3369:
3338:
3322:
3315:
3308:
3304:
3300:
3296:
3273:
3266:
3258:
3254:
3251:is congruent to
3250:
3246:
3242:
3239:is congruent to
3238:
3234:
3231:is congruent to
3230:
3226:
3222:
3214:
3206:
3198:
3190:
3182:
3172:
3169:, so 23 divides
3168:
3158:
3154:
3146:
3138:
3130:
3126:
3118:
3107:
3103:
3095:
3091:
3087:
3083:
3074:
3071:, and therefore
3070:
3063:
3056:
3053:
3052:
3050:
3049:
3046:
3043:
3034:
3027:
3017:
3006:
2998:
2976:
2968:
2961:
2957:
2943:
2940:is congruent to
2939:
2935:
2931:
2922:
2918:
2911:
2903:
2893:
2890:are larger than
2889:
2885:
2880:Euclid's theorem
2874:
2863:
2860:, which is odd,
2859:
2855:
2851:
2840:
2833:
2829:
2825:
2821:
2817:
2813:
2809:
2805:
2801:
2797:
2793:
2789:
2785:
2781:
2777:
2773:
2769:
2765:
2759:
2755:
2751:
2747:
2733:
2729:
2725:
2721:
2717:
2710:
2706:
2699:
2695:
2691:
2678:
2674:
2670:
2660:
2657:
2654:
2650:
2643:
2639:
2629:
2619:
2615:
2606:
2599:
2595:
2588:
2581:
2574:
2567:
2560:
2550:
2543:
2532:
2525:
2514:
2503:
2492:
2472:
2465:
2458:
2451:
2447:
2433:
2408:
2401:
2390:
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:
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:
21:
12333:
12332:
12328:
12327:
12326:
12324:
12323:
12322:
12318:Perfect numbers
12313:Mersenne primes
12283:
12282:
12281:
12276:
12258:
12214:List of numbers
12182:
12180:
12178:
12176:
12166:
12123:
12089:
12083:
12054:Graham's number
12044:Skewes's number
11946:
11944:
11942:
11936:
11931:
11901:
11896:
11865:Mersenne's laws
11851:
11846:
11816:
11811:
11789:
11785:Strobogrammatic
11776:
11758:
11740:
11722:
11704:
11686:
11668:
11650:
11627:
11606:
11590:
11549:Divisor-related
11544:
11504:
11455:
11425:
11362:
11346:
11325:
11292:
11265:
11253:
11235:
11147:
11146:related numbers
11120:
11097:
11064:
11055:Perfect totient
11021:
10998:
10929:Highly abundant
10871:
10850:
10782:
10765:
10737:
10720:
10706:Stirling second
10612:
10589:
10550:
10532:
10489:
10438:
10375:
10336:Centered square
10304:
10287:
10249:
10234:
10201:
10186:
10138:
10137:defined numbers
10120:
10087:
10072:
10043:Double Mersenne
10029:
10010:
9932:
9918:
9916:natural numbers
9912:
9882:
9877:
9868:
9562:First 60 primes
9557:
9521:
9421:
9404:Complex numbers
9398:
9372:
9350:
9334:
9309:
9308:Bi-twin chain (
9299:
9273:
9253:
9237:
9221:
9197:
9173:
9157:
9132:
9118:Strobogrammatic
9056:
9030:
8924:
8888:
8880:
8874:
8873:
8856:
8847:
8832:
8809:
8797:
8785:
8776:
8761:
8752:
8739:
8731:# + 1
8729:
8724:
8716:# ± 1
8714:
8709:
8701:! ± 1
8697:
8685:
8677:
8669:2 − 1
8668:
8660:2 − 1
8659:
8651:2 + 1
8650:
8637:
8631:
8584:
8583:
8565:
8564:
8561:
8559:MathWorld links
8553:Wayback Machine
8541:Wayback Machine
8520:
8505:
8501:
8496:
8485:Wayback Machine
8473:Wayback Machine
8434:
8416:
8414:
8396:
8376:
8368:
8351:
8343:
8328:GIMPS home page
8309:
8306:
8301:
8286:
8252:
8239:
8231:
8230:
8228:
8225:
8221:
8206:
8172:
8159:
8151:
8150:
8148:
8145:
8141:
8133:
8125:
8123:
8119:
8111:
8099:
8097:
8093:
8085:
8073:
8071:
8067:
8059:
8047:
8045:
8041:
8033:
8025:
8023:
8019:
8011:
8003:
8001:
7997:
7989:
7981:
7973:
7971:
7967:
7951:
7950:
7946:
7933:
7929:
7922:
7901:
7900:
7896:
7883:
7882:
7878:
7869:
7868:
7864:
7855:
7853:
7847:
7846:
7842:
7833:
7831:
7822:
7821:
7817:
7810:
7797:
7796:
7792:
7782:
7780:
7778:www.mersenne.ca
7772:
7771:
7767:
7758:
7756:
7752:
7751:
7747:
7737:
7735:
7733:www.mersenne.ca
7727:
7726:
7722:
7713:
7711:
7706:
7705:
7701:
7694:
7673:
7672:
7668:
7652:
7651:
7647:
7642:Wayback Machine
7633:
7629:
7619:
7617:
7609:
7608:
7604:
7595:
7593:
7589:
7588:
7584:
7575:
7574:
7570:
7557:
7556:
7552:
7543:
7541:
7533:
7532:
7528:
7515:
7514:
7510:
7500:
7498:
7487:
7486:
7482:
7472:
7470:
7459:
7458:
7454:
7444:
7442:
7433:
7432:
7428:
7419:
7417:
7406:
7405:
7401:
7392:
7390:
7381:
7380:
7376:
7367:
7363:
7345:
7344:
7340:
7323:
7322:
7318:
7303:
7302:
7298:
7270:
7269:
7265:
7248:
7247:
7243:
7234:
7230:
7222:
7218:
7208:
7207:
7203:
7193:
7191:
7182:
7181:
7177:
7167:
7165:
7156:
7155:
7151:
7147:
7142:
7141:
7124:
7121:
7112:
7111:
7109:
7108:
7104:
7097:
7090:
7086:
7082:
7078:
7076:
7072:
7059:
7055:
7044:
7032:
7020:
6989:
6986:
6977:
6976:
6974:
6973:
6971:
6967:
6941:
6930:
6926:
6909:
6904:
6899:
6895:
6890:
6885:
6881:Williams number
6846:Wieferich prime
6789:
6773:
6757:
6753:
6739:
6723:
6719:
6706:
6699:
6695:
6694:is prime, then
6685:
6681:
6670:
6659:
6652:
6648:
6641:
5210:
5208:
5207:
5203:
5190:
5188:
5187:
5178:
5176:
5175:
5171:
5166:
5165:
5162:probable primes
5159:
5144:
5141:
5132:
5131:
5129:
5128:
5127:
5122:
5121:
5114:
5113:
5107:
5106:
5086:
5072:
5069:
5060:
5059:
5057:
5056:
5052:
5041:
5037:
5033:
5029:
5025:
5013:
5003:
4992:
4988:
4984:
4980:
4970:
4960:
4945:
4938:
4930:
4926:
4899:
4888:
4875:
4874:
4867:
4866:
4860:
4844:
4833:
4819:
4803:
4800:
4794:
4793:
4791:
4790:
4784:
4773:
4763:
4752:
4748:
4741:
4735:
4721:
4710:
4706:
4699:
4688:
4685:
4679:
4678:
4676:
4675:
4669:
4655:
4652:
4646:
4645:
4643:
4642:
4636:
4630:
4619:
4616:
4610:
4609:
4607:
4606:
4600:
4595:which is not a
4590:
4584:
4574:
4564:
4557:
4543:
4541:
4531:
4521:
4514:
4508:
4481:
4464:
4463:
4456:
4455:
4446:
4439:
4432:
4429:
4423:
4418:
4402:
4384:
4374:
4366:
4350:
4340:
4333:
4317:
4292:
4281:
4273:
4255:
4249:
4244:, and can ask (
4235:
4225:
4218:
4198:complex numbers
4190:
4186:
4182:
4176:
4165:
4154:
4152:Complex numbers
4134:
4127:
4116:
4109:
4102:
4098:
4076:
4069:
4065:
4046:
4039:
4035:
4016:
4009:
4006:
4000:
3998:Generalizations
3985:
3980:
3966:
3958:
3954:
3944:
3933:
3926:
3913:
3906:
3900:
3892:
3889:
3883:
3882:
3880:
3879:
3872:
3864:2 − 1
3863:
3860:4 + 1
3859:
3852:4 − 1
3851:
3847:
3839:
3789:
3784:
3776:
3771:
3765:
3758:
3743:
3514:
3509:
3503:
3498:
3491:
3476:
3471:
3462:
3457:
3455:
3438:
3431:
3415:
3403:
3397:
3387:
3381:
3377:
3371:
3368:
3358:
3348:
3340:
3337:
3331:
3328:Lucas sequences
3317:
3310:
3306:
3302:
3298:
3291:
3290:, the equation
3272:
3268:
3260:
3256:
3252:
3248:
3244:
3240:
3236:
3232:
3228:
3224:
3216:
3208:
3200:
3192:
3184:
3180:
3170:
3166:
3156:
3148:
3144:
3136:
3128:
3120:
3116:
3105:
3101:
3100:if and only if
3093:
3089:
3085:
3081:
3072:
3065:
3057:
3054:
3047:
3044:
3041:
3040:
3038:
3036:
3029:
3019:
3008:
3000:
2993:
2984:Wieferich prime
2974:
2963:
2959:
2951:
2941:
2937:
2933:
2929:
2920:
2917:
2913:
2905:
2898:
2891:
2887:
2883:
2865:
2861:
2857:
2856:is a factor of
2853:
2842:
2835:
2834:is a factor of
2831:
2827:
2826:is a factor of
2823:
2819:
2818:is a factor of
2815:
2814:if and only if
2811:
2810:is a factor of
2807:
2803:
2799:
2798:is a factor of
2795:
2791:
2787:
2783:
2779:
2775:
2771:
2767:
2761:
2757:
2756:is a factor of
2753:
2749:
2748:is a factor of
2745:
2731:
2727:
2724:2 − 1 = 23 × 89
2723:
2719:
2715:
2708:
2701:
2697:
2693:
2689:
2676:
2672:
2669:
2665:
2661:
2658:
2655:
2652:
2645:
2641:
2631:
2627:
2626:: Suppose that
2617:
2616:is prime, then
2613:
2601:
2597:
2590:
2583:
2576:
2569:
2562:
2552:
2545:
2534:
2527:
2516:
2505:
2494:
2480:
2467:
2460:
2459:is prime, then
2453:
2449:
2443:
2429:
2426:
2406:
2399:
2389:
2383:
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:
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:
1824:
1816:
1815:
1812:
1806:
1797:
1791:
1788:
1782:
1779:
1773:
1770:
1764:
1761:
1755:
1515:
1462:
1461:
1431:
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:
1192:
1170:
1154:
1141:
1134:
1130:
1108:
1096:
1095:
1065:
1043:
1027:
1014:
1007:
1003:
981:
971:
952:
943:
942:
936:
933:
929:
919:
912:
904:
898:
894:
888:
885:
879:
875:
869:
865:
859:
855:
849:
845:
839:
818:
799:
794:
793:
766:
744:
739:
738:
730:
726:
686:
681:
680:
676:
672:
624:
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:
432:
427:
426:
389:
376:
371:
370:
367:order of growth
356:
355:
350:
347:
343:
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:
18:524287 (number)
15:
12:
11:
5:
12331:
12329:
12321:
12320:
12315:
12310:
12305:
12300:
12295:
12285:
12284:
12278:
12277:
12275:
12274:
12269:
12263:
12260:
12259:
12257:
12256:
12251:
12246:
12244:Power of three
12241:
12236:
12231:
12226:
12224:Number systems
12221:
12216:
12211:
12206:
12201:
12196:
12191:
12185:
12183:
12179:(alphabetical
12172:
12171:
12168:
12167:
12165:
12164:
12159:
12154:
12149:
12148:
12147:
12142:
12135:Hyperoperation
12131:
12129:
12125:
12124:
12122:
12121:
12116:
12111:
12106:
12100:
12098:
12091:
12085:
12084:
12082:
12081:
12076:
12071:
12066:
12061:
12056:
12051:
12049:Moser's number
12046:
12041:
12036:
12034:Shannon number
12031:
12026:
12021:
12016:
12011:
12006:
12001:
11996:
11991:
11986:
11981:
11976:
11971:
11966:
11961:
11956:
11950:
11948:
11938:
11937:
11932:
11930:
11929:
11922:
11915:
11907:
11898:
11897:
11895:
11894:
11889:
11888:
11887:
11882:
11877:
11870:Mersenne prime
11867:
11862:
11856:
11853:
11852:
11849:Marin Mersenne
11847:
11845:
11844:
11837:
11830:
11822:
11813:
11812:
11810:
11809:
11798:
11795:
11794:
11791:
11790:
11788:
11787:
11781:
11778:
11777:
11771:
11764:
11763:
11760:
11759:
11757:
11756:
11751:
11745:
11742:
11741:
11735:
11728:
11727:
11724:
11723:
11721:
11720:
11718:Sorting number
11715:
11713:Pancake number
11709:
11706:
11705:
11699:
11692:
11691:
11688:
11687:
11685:
11684:
11679:
11673:
11670:
11669:
11663:
11656:
11655:
11652:
11651:
11649:
11648:
11643:
11638:
11632:
11629:
11628:
11625:Binary numbers
11623:
11616:
11615:
11612:
11611:
11608:
11607:
11605:
11604:
11598:
11596:
11592:
11591:
11589:
11588:
11583:
11578:
11573:
11568:
11563:
11558:
11552:
11550:
11546:
11545:
11543:
11542:
11537:
11532:
11527:
11522:
11516:
11514:
11506:
11505:
11503:
11502:
11497:
11492:
11487:
11482:
11477:
11472:
11466:
11464:
11457:
11456:
11454:
11453:
11452:
11451:
11440:
11438:
11435:P-adic numbers
11431:
11430:
11427:
11426:
11424:
11423:
11422:
11421:
11411:
11406:
11401:
11396:
11391:
11386:
11381:
11376:
11370:
11368:
11364:
11363:
11361:
11360:
11354:
11352:
11351:Coding-related
11348:
11347:
11345:
11344:
11339:
11333:
11331:
11327:
11326:
11324:
11323:
11318:
11313:
11308:
11302:
11300:
11291:
11290:
11289:
11288:
11286:Multiplicative
11283:
11272:
11270:
11255:
11254:
11250:Numeral system
11248:
11241:
11240:
11237:
11236:
11234:
11233:
11228:
11223:
11218:
11213:
11208:
11203:
11198:
11193:
11188:
11183:
11178:
11173:
11168:
11163:
11158:
11152:
11149:
11148:
11137:
11130:
11129:
11126:
11125:
11122:
11121:
11119:
11118:
11113:
11107:
11105:
11099:
11098:
11096:
11095:
11090:
11085:
11080:
11074:
11072:
11066:
11065:
11063:
11062:
11057:
11052:
11047:
11042:
11040:Highly totient
11037:
11031:
11029:
11023:
11022:
11020:
11019:
11014:
11008:
11006:
11000:
10999:
10997:
10996:
10991:
10986:
10981:
10976:
10971:
10966:
10961:
10956:
10951:
10946:
10941:
10936:
10931:
10926:
10921:
10916:
10911:
10906:
10901:
10896:
10894:Almost perfect
10891:
10885:
10883:
10873:
10872:
10863:
10856:
10855:
10852:
10851:
10849:
10848:
10843:
10838:
10833:
10828:
10823:
10818:
10813:
10808:
10803:
10798:
10793:
10787:
10784:
10783:
10778:
10771:
10770:
10767:
10766:
10764:
10763:
10758:
10753:
10748:
10742:
10739:
10738:
10733:
10726:
10725:
10722:
10721:
10719:
10718:
10713:
10708:
10703:
10701:Stirling first
10698:
10693:
10688:
10683:
10678:
10673:
10668:
10663:
10658:
10653:
10648:
10643:
10638:
10633:
10628:
10623:
10617:
10614:
10613:
10610:
10603:
10602:
10599:
10598:
10595:
10594:
10591:
10590:
10588:
10587:
10582:
10577:
10571:
10569:
10562:
10556:
10555:
10552:
10551:
10549:
10548:
10542:
10540:
10534:
10533:
10531:
10530:
10525:
10520:
10515:
10510:
10505:
10499:
10497:
10491:
10490:
10488:
10487:
10482:
10477:
10472:
10467:
10461:
10459:
10450:
10444:
10443:
10440:
10439:
10437:
10436:
10431:
10426:
10421:
10416:
10411:
10406:
10401:
10396:
10391:
10385:
10383:
10377:
10376:
10374:
10373:
10368:
10363:
10358:
10353:
10348:
10343:
10338:
10333:
10327:
10325:
10316:
10306:
10305:
10300:
10293:
10292:
10289:
10288:
10286:
10285:
10280:
10275:
10270:
10265:
10260:
10254:
10251:
10250:
10247:
10240:
10239:
10236:
10235:
10233:
10232:
10227:
10222:
10217:
10212:
10206:
10203:
10202:
10199:
10192:
10191:
10188:
10187:
10185:
10184:
10179:
10174:
10169:
10164:
10159:
10154:
10149:
10143:
10140:
10139:
10133:
10126:
10125:
10122:
10121:
10119:
10118:
10113:
10108:
10103:
10098:
10092:
10089:
10088:
10085:
10078:
10077:
10074:
10073:
10071:
10070:
10065:
10060:
10055:
10050:
10045:
10040:
10034:
10031:
10030:
10023:
10016:
10015:
10012:
10011:
10009:
10008:
10003:
9998:
9993:
9988:
9983:
9978:
9973:
9968:
9963:
9958:
9953:
9948:
9943:
9937:
9934:
9933:
9927:
9920:
9919:
9913:
9911:
9910:
9903:
9896:
9888:
9879:
9878:
9873:
9870:
9869:
9867:
9866:
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:
9565:
9563:
9559:
9558:
9556:
9555:
9550:
9545:
9540:
9535:
9533:Probable prime
9529:
9527:
9526:Related topics
9523:
9522:
9520:
9519:
9514:
9509:
9507:Sphenic number
9504:
9499:
9494:
9489:
9488:
9487:
9482:
9477:
9472:
9467:
9462:
9457:
9452:
9447:
9442:
9431:
9429:
9423:
9422:
9420:
9419:
9417:Gaussian prime
9414:
9408:
9406:
9400:
9399:
9397:
9396:
9395:
9385:
9380:
9378:
9374:
9373:
9371:
9370:
9346:
9342: + 1
9330:
9325:
9304:
9301:
9300:
9298:
9297:
9269:
9249:
9245: + 6
9233:
9229: + 4
9217:
9213: + 8
9193:
9189: + 6
9169:
9165: + 2
9152:
9150:
9138:
9134:
9133:
9131:
9130:
9125:
9120:
9115:
9110:
9105:
9100:
9095:
9090:
9085:
9080:
9075:
9070:
9065:
9060:
9052:
9047:
9041:
9039:
9032:
9031:
9029:
9028:
9023:
9018:
9013:
9008:
9003:
8998:
8993:
8988:
8983:
8978:
8973:
8968:
8963:
8958:
8953:
8948:
8943:
8932:
8930:
8926:
8925:
8923:
8922:
8917:
8912:
8907:
8902:
8896:
8894:
8890:
8889:
8887:
8886:
8869:
8865: − 1
8852:
8843:
8828:
8805:
8793:
8781:
8772:
8757:
8748:
8744: + 1
8735:
8727:
8720:
8712:
8705:
8693:
8681:
8673:
8664:
8655:
8645:
8643:
8639:
8638:
8632:
8630:
8629:
8622:
8615:
8607:
8601:
8600:
8581:
8560:
8557:
8556:
8555:
8543:
8531:
8516:
8511:
8499:
8487:
8475:
8463:
8458:
8453:
8448:
8442:
8437:
8428:
8422:
8397:Grime, James.
8394:
8372:
8366:
8347:
8341:
8336:
8330:
8325:
8305:
8304:External links
8302:
8300:
8299:
8272:
8268:
8264:
8259:
8255:
8251:
8246:
8242:
8238:
8219:
8192:
8188:
8184:
8179:
8175:
8171:
8166:
8162:
8158:
8139:
8117:
8091:
8065:
8039:
8017:
7995:
7965:
7944:
7927:
7920:
7894:
7891:on 2012-05-29.
7876:
7871:"OEIS A016131"
7862:
7840:
7815:
7808:
7790:
7765:
7745:
7720:
7699:
7692:
7666:
7645:
7627:
7602:
7582:
7568:
7550:
7526:
7523:on 2016-09-03.
7508:
7480:
7452:
7426:
7399:
7374:
7368:Brian Napper,
7361:
7338:
7316:
7296:
7283:(3): 134–138.
7263:
7241:
7228:
7216:
7201:
7175:
7148:
7146:
7143:
7140:
7139:
7070:
6965:
6907:
6892:
6891:
6889:
6886:
6884:
6883:
6878:
6873:
6868:
6863:
6858:
6853:
6851:Wagstaff prime
6848:
6843:
6838:
6832:
6826:
6821:
6816:
6811:
6806:
6801:
6796:
6790:
6788:
6785:
6784:
6783:
6750:
6749:
6636:
6635:
6630:
6627:
6624:
6620:
6619:
6617:
6614:
6611:
6607:
6606:
6601:
6598:
6595:
6591:
6590:
6585:
6582:
6579:
6575:
6574:
6569:
6566:
6563:
6559:
6558:
6553:
6550:
6547:
6543:
6542:
6537:
6534:
6531:
6527:
6526:
6521:
6518:
6515:
6511:
6510:
6505:
6502:
6499:
6495:
6494:
6492:
6489:
6486:
6482:
6481:
6479:
6476:
6473:
6469:
6468:
6466:
6463:
6460:
6456:
6455:
6453:
6450:
6447:
6443:
6442:
6437:
6434:
6431:
6427:
6426:
6421:
6418:
6415:
6411:
6410:
6405:
6402:
6399:
6395:
6394:
6389:
6386:
6383:
6379:
6378:
6373:
6370:
6367:
6363:
6362:
6357:
6354:
6351:
6347:
6346:
6341:
6338:
6335:
6331:
6330:
6325:
6322:
6319:
6315:
6314:
6309:
6306:
6303:
6299:
6298:
6293:
6290:
6287:
6283:
6282:
6277:
6274:
6271:
6267:
6266:
6261:
6258:
6255:
6251:
6250:
6245:
6242:
6239:
6235:
6234:
6229:
6226:
6223:
6219:
6218:
6213:
6210:
6207:
6203:
6202:
6197:
6194:
6191:
6187:
6186:
6184:
6181:
6178:
6174:
6173:
6168:
6165:
6162:
6158:
6157:
6152:
6149:
6146:
6142:
6141:
6136:
6133:
6130:
6126:
6125:
6120:
6117:
6114:
6110:
6109:
6104:
6101:
6098:
6094:
6093:
6088:
6085:
6082:
6078:
6077:
6072:
6069:
6066:
6062:
6061:
6056:
6053:
6050:
6046:
6045:
6040:
6037:
6034:
6030:
6029:
6024:
6021:
6018:
6014:
6013:
6008:
6005:
6002:
5998:
5997:
5992:
5989:
5986:
5982:
5981:
5979:
5976:
5973:
5969:
5968:
5963:
5960:
5957:
5953:
5952:
5950:
5949:2 (no others)
5947:
5944:
5940:
5939:
5934:
5931:
5928:
5924:
5923:
5918:
5915:
5912:
5908:
5907:
5902:
5899:
5896:
5892:
5891:
5886:
5883:
5880:
5876:
5875:
5870:
5867:
5864:
5860:
5859:
5854:
5851:
5848:
5844:
5843:
5841:
5840:2 (no others)
5838:
5835:
5831:
5830:
5828:
5827:3 (no others)
5825:
5822:
5818:
5817:
5812:
5809:
5806:
5802:
5801:
5796:
5793:
5790:
5786:
5785:
5780:
5777:
5774:
5770:
5769:
5764:
5761:
5758:
5754:
5753:
5748:
5745:
5742:
5738:
5737:
5732:
5729:
5726:
5722:
5721:
5716:
5713:
5710:
5706:
5705:
5700:
5697:
5694:
5690:
5689:
5684:
5681:
5678:
5674:
5673:
5668:
5665:
5662:
5658:
5657:
5652:
5649:
5646:
5642:
5641:
5636:
5633:
5630:
5626:
5625:
5620:
5617:
5614:
5610:
5609:
5604:
5601:
5598:
5594:
5593:
5588:
5585:
5582:
5578:
5577:
5572:
5569:
5566:
5562:
5561:
5556:
5553:
5550:
5546:
5545:
5540:
5537:
5534:
5530:
5529:
5524:
5521:
5518:
5514:
5513:
5508:
5505:
5502:
5498:
5497:
5492:
5489:
5486:
5482:
5481:
5476:
5473:
5470:
5466:
5465:
5460:
5457:
5454:
5450:
5449:
5444:
5441:
5438:
5434:
5433:
5428:
5425:
5422:
5418:
5417:
5412:
5409:
5406:
5402:
5401:
5396:
5393:
5390:
5386:
5385:
5380:
5377:
5374:
5370:
5369:
5367:
5364:
5361:
5357:
5356:
5354:
5353:2 (no others)
5351:
5348:
5344:
5343:
5338:
5335:
5332:
5328:
5327:
5322:
5319:
5316:
5312:
5311:
5306:
5303:
5300:
5296:
5295:
5290:
5287:
5284:
5280:
5279:
5274:
5271:
5268:
5264:
5263:
5258:
5254:
5251:
5247:
5246:
5241:
5237:
5234:
5230:
5229:
5223:
5118:
5111:
4923:
4922:
4908:
4905:
4902:
4895:
4891:
4887:
4882:
4878:
4859:
4856:
4855:
4854:
4830:
4829:
4781:
4780:
4732:
4731:
4587:
4586:
4554:
4553:
4505:
4504:
4490:
4487:
4484:
4479:
4476:
4471:
4467:
4425:Main article:
4422:
4421:Repunit primes
4419:
4417:
4414:
4413:
4412:
4395:
4394:
4332:
4329:
4328:
4327:
4303:
4302:
4264:Gaussian prime
4217:
4214:
4153:
4150:
4124:theorems above
4002:Main article:
3999:
3996:
3978:
3963:
3962:
3878:is defined as
3871:
3868:
3832:right triangle
3787:
3774:
3762:Tower of Hanoi
3757:
3754:
3738:
3737:
3734:
3731:
3727:
3726:
3723:
3720:
3716:
3715:
3712:
3709:
3705:
3704:
3701:
3698:
3694:
3693:
3690:
3687:
3683:
3682:
3679:
3676:
3672:
3671:
3668:
3665:
3661:
3660:
3657:
3654:
3650:
3649:
3646:
3643:
3639:
3638:
3635:
3632:
3628:
3627:
3624:
3621:
3617:
3616:
3613:
3610:
3606:
3605:
3602:
3599:
3595:
3594:
3591:
3588:
3584:
3583:
3580:
3579:8796093022207
3577:
3573:
3572:
3569:
3568:2199023255551
3566:
3562:
3561:
3558:
3555:
3551:
3550:
3547:
3544:
3540:
3539:
3536:
3533:
3529:
3528:
3525:
3522:
3518:
3517:
3512:
3506:
3501:
3496:
3469:
3452:probable prime
3448:primality test
3430:
3427:
3426:
3425:
3399:Main article:
3396:
3393:
3392:
3391:
3385:
3375:
3363:
3353:
3344:
3333:
3324:
3284:
3283:to the base 2.
3277:
3276:
3275:
3270:
3174:
3167:11 = 2 × 4 + 3
3113:
3078:
3077:
3076:
3073:−1 = 0 (mod p)
2980:
2979:
2978:
2926:
2925:
2924:
2915:
2895:
2876:
2735:
2718:is prime, and
2686:
2685:
2684:
2679:is prime then
2667:
2663:
2610:
2609:
2608:
2425:
2422:
2418:Probable prime
2411:Ocala, Florida
2387:
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:
24:
14:
13:
10:
9:
6:
4:
3:
2:
12330:
12319:
12316:
12314:
12311:
12309:
12306:
12304:
12301:
12299:
12296:
12294:
12291:
12290:
12288:
12273:
12270:
12268:
12265:
12264:
12261:
12255:
12252:
12250:
12247:
12245:
12242:
12240:
12237:
12235:
12232:
12230:
12227:
12225:
12222:
12220:
12217:
12215:
12212:
12210:
12207:
12205:
12204:Infinitesimal
12202:
12200:
12197:
12195:
12192:
12190:
12187:
12186:
12184:
12173:
12163:
12160:
12158:
12155:
12153:
12150:
12146:
12143:
12141:
12138:
12137:
12136:
12133:
12132:
12130:
12126:
12120:
12117:
12115:
12112:
12110:
12107:
12105:
12102:
12101:
12099:
12095:
12092:
12086:
12080:
12077:
12075:
12074:Rayo's number
12072:
12070:
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:
11951:
11949:
11939:
11935:
11934:Large numbers
11928:
11923:
11921:
11916:
11914:
11909:
11908:
11905:
11893:
11890:
11886:
11883:
11881:
11878:
11876:
11873:
11872:
11871:
11868:
11866:
11863:
11861:
11858:
11857:
11854:
11850:
11843:
11838:
11836:
11831:
11829:
11824:
11823:
11820:
11808:
11804:
11800:
11799:
11796:
11786:
11783:
11782:
11779:
11774:
11769:
11765:
11755:
11752:
11750:
11747:
11746:
11743:
11738:
11733:
11729:
11719:
11716:
11714:
11711:
11710:
11707:
11702:
11697:
11693:
11683:
11680:
11678:
11675:
11674:
11671:
11667:
11661:
11657:
11647:
11644:
11642:
11639:
11637:
11634:
11633:
11630:
11626:
11621:
11617:
11603:
11600:
11599:
11597:
11593:
11587:
11584:
11582:
11579:
11577:
11576:Polydivisible
11574:
11572:
11569:
11567:
11564:
11562:
11559:
11557:
11554:
11553:
11551:
11547:
11541:
11538:
11536:
11533:
11531:
11528:
11526:
11523:
11521:
11518:
11517:
11515:
11512:
11507:
11501:
11498:
11496:
11493:
11491:
11488:
11486:
11483:
11481:
11478:
11476:
11473:
11471:
11468:
11467:
11465:
11462:
11458:
11450:
11447:
11446:
11445:
11442:
11441:
11439:
11436:
11432:
11420:
11417:
11416:
11415:
11412:
11410:
11407:
11405:
11402:
11400:
11397:
11395:
11392:
11390:
11387:
11385:
11382:
11380:
11377:
11375:
11372:
11371:
11369:
11365:
11359:
11356:
11355:
11353:
11349:
11343:
11340:
11338:
11335:
11334:
11332:
11330:Digit product
11328:
11322:
11319:
11317:
11314:
11312:
11309:
11307:
11304:
11303:
11301:
11299:
11295:
11287:
11284:
11282:
11279:
11278:
11277:
11274:
11273:
11271:
11269:
11264:
11260:
11256:
11251:
11246:
11242:
11232:
11229:
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:
11166:Erdős–Nicolas
11164:
11162:
11159:
11157:
11154:
11153:
11150:
11145:
11141:
11135:
11131:
11117:
11114:
11112:
11109:
11108:
11106:
11104:
11100:
11094:
11091:
11089:
11086:
11084:
11081:
11079:
11076:
11075:
11073:
11071:
11067:
11061:
11058:
11056:
11053:
11051:
11048:
11046:
11043:
11041:
11038:
11036:
11033:
11032:
11030:
11028:
11024:
11018:
11015:
11013:
11010:
11009:
11007:
11005:
11001:
10995:
10992:
10990:
10987:
10985:
10984:Superabundant
10982:
10980:
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:
10886:
10884:
10882:
10878:
10874:
10870:
10866:
10861:
10857:
10847:
10844:
10842:
10839:
10837:
10834:
10832:
10829:
10827:
10824:
10822:
10819:
10817:
10814:
10812:
10809:
10807:
10804:
10802:
10799:
10797:
10794:
10792:
10789:
10788:
10785:
10781:
10776:
10772:
10762:
10759:
10757:
10754:
10752:
10749:
10747:
10744:
10743:
10740:
10736:
10731:
10727:
10717:
10714:
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:
10618:
10615:
10608:
10604:
10586:
10583:
10581:
10578:
10576:
10573:
10572:
10570:
10566:
10563:
10561:
10560:4-dimensional
10557:
10547:
10544:
10543:
10541:
10539:
10535:
10529:
10526:
10524:
10521:
10519:
10516:
10514:
10511:
10509:
10506:
10504:
10501:
10500:
10498:
10496:
10492:
10486:
10483:
10481:
10478:
10476:
10473:
10471:
10470:Centered cube
10468:
10466:
10463:
10462:
10460:
10458:
10454:
10451:
10449:
10448:3-dimensional
10445:
10435:
10432:
10430:
10427:
10425:
10422:
10420:
10417:
10415:
10412:
10410:
10407:
10405:
10402:
10400:
10397:
10395:
10392:
10390:
10387:
10386:
10384:
10382:
10378:
10372:
10369:
10367:
10364:
10362:
10359:
10357:
10354:
10352:
10349:
10347:
10344:
10342:
10339:
10337:
10334:
10332:
10329:
10328:
10326:
10324:
10320:
10317:
10315:
10314:2-dimensional
10311:
10307:
10303:
10298:
10294:
10284:
10281:
10279:
10276:
10274:
10271:
10269:
10266:
10264:
10261:
10259:
10258:Nonhypotenuse
10256:
10255:
10252:
10245:
10241:
10231:
10228:
10226:
10223:
10221:
10218:
10216:
10213:
10211:
10208:
10207:
10204:
10197:
10193:
10183:
10180:
10178:
10175:
10173:
10170:
10168:
10165:
10163:
10160:
10158:
10155:
10153:
10150:
10148:
10145:
10144:
10141:
10136:
10131:
10127:
10117:
10114:
10112:
10109:
10107:
10104:
10102:
10099:
10097:
10094:
10093:
10090:
10083:
10079:
10069:
10066:
10064:
10061:
10059:
10056:
10054:
10051:
10049:
10046:
10044:
10041:
10039:
10036:
10035:
10032:
10027:
10021:
10017:
10007:
10004:
10002:
9999:
9997:
9996:Perfect power
9994:
9992:
9989:
9987:
9986:Seventh power
9984:
9982:
9979:
9977:
9974:
9972:
9969:
9967:
9964:
9962:
9959:
9957:
9954:
9952:
9949:
9947:
9944:
9942:
9939:
9938:
9935:
9930:
9925:
9921:
9917:
9909:
9904:
9902:
9897:
9895:
9890:
9889:
9886:
9876:
9871:
9865:
9862:
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:
9566:
9564:
9560:
9554:
9551:
9549:
9546:
9544:
9543:Illegal prime
9541:
9539:
9536:
9534:
9531:
9530:
9528:
9524:
9518:
9515:
9513:
9510:
9508:
9505:
9503:
9500:
9498:
9495:
9493:
9490:
9486:
9483:
9481:
9478:
9476:
9473:
9471:
9468:
9466:
9463:
9461:
9458:
9456:
9453:
9451:
9448:
9446:
9443:
9441:
9438:
9437:
9436:
9433:
9432:
9430:
9428:
9424:
9418:
9415:
9413:
9410:
9409:
9407:
9405:
9401:
9394:
9391:
9390:
9389:
9388:Largest known
9386:
9384:
9381:
9379:
9375:
9369:
9365:
9361:
9357:
9353:
9347:
9345:
9341:
9337:
9331:
9329:
9326:
9324:
9320:
9316:
9312:
9306:
9305:
9296:
9293:
9290: +
9289:
9285:
9281:
9278: −
9277:
9270:
9268:
9264:
9260:
9257: +
9256:
9250:
9248:
9244:
9240:
9234:
9232:
9228:
9224:
9218:
9216:
9212:
9208:
9204:
9200:
9194:
9192:
9188:
9184:
9180:
9176:
9170:
9168:
9164:
9160:
9154:
9153:
9151:
9149:
9147:
9142:
9139:
9135:
9129:
9126:
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:
9053:
9051:
9048:
9046:
9043:
9042:
9040:
9037:
9033:
9027:
9024:
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:
8941:
8937:
8934:
8933:
8931:
8927:
8921:
8918:
8916:
8913:
8911:
8908:
8906:
8903:
8901:
8898:
8897:
8895:
8891:
8885:
8879:
8870:
8868:
8864:
8860:
8853:
8851:
8844:
8842:
8839:
8836: +
8835:
8829:
8827:
8824:
8821: −
8820:
8816:
8813: −
8812:
8806:
8804:
8800:
8794:
8792:
8788:
8782:
8780:
8773:
8771:
8768:
8765: +
8764:
8758:
8756:
8749:
8747:
8743:
8738:Pythagorean (
8736:
8734:
8730:
8721:
8719:
8715:
8706:
8704:
8700:
8694:
8692:
8688:
8682:
8680:
8674:
8672:
8665:
8663:
8656:
8654:
8647:
8646:
8644:
8640:
8635:
8628:
8623:
8621:
8616:
8614:
8609:
8608:
8605:
8596:
8595:
8590:
8587:
8582:
8577:
8576:
8571:
8568:
8563:
8562:
8558:
8554:
8550:
8547:
8544:
8542:
8538:
8535:
8532:
8530:
8527:
8523:
8517:
8515:
8512:
8508:
8502:
8494:
8488:
8486:
8482:
8479:
8476:
8474:
8470:
8467:
8464:
8462:
8459:
8457:
8454:
8452:
8449:
8446:
8443:
8441:
8438:
8432:
8429:
8426:
8423:
8413:on 2013-05-31
8412:
8408:
8404:
8400:
8395:
8392:
8388:
8384:
8380:
8375:
8371:
8367:
8364:
8359:
8355:
8350:
8346:
8342:
8340:
8337:
8334:
8331:
8329:
8326:
8322:
8318:
8317:
8312:
8308:
8307:
8303:
8297:
8294:
8290:
8270:
8266:
8257:
8253:
8249:
8244:
8240:
8223:
8220:
8217:
8214:
8210:
8190:
8186:
8177:
8173:
8169:
8164:
8160:
8143:
8140:
8137:
8129:
8121:
8118:
8115:
8107:
8103:
8095:
8092:
8089:
8081:
8077:
8069:
8066:
8063:
8055:
8051:
8043:
8040:
8037:
8029:
8021:
8018:
8015:
8007:
7999:
7996:
7993:
7985:
7977:
7969:
7966:
7960:
7955:
7948:
7945:
7941:
7938:(part of the
7937:
7931:
7928:
7923:
7917:
7913:
7909:
7905:
7898:
7895:
7890:
7886:
7880:
7877:
7872:
7866:
7863:
7851:
7844:
7841:
7830:
7826:
7819:
7816:
7811:
7805:
7801:
7794:
7791:
7779:
7775:
7769:
7766:
7755:
7749:
7746:
7734:
7730:
7724:
7721:
7710:
7703:
7700:
7695:
7689:
7685:
7681:
7677:
7670:
7667:
7662:
7661:
7656:
7649:
7646:
7643:
7639:
7636:
7631:
7628:
7616:
7612:
7606:
7603:
7592:
7586:
7583:
7578:
7572:
7569:
7564:
7560:
7554:
7551:
7540:
7536:
7530:
7527:
7522:
7518:
7512:
7509:
7497:
7496:
7491:
7484:
7481:
7469:
7468:
7467:New Scientist
7463:
7456:
7453:
7441:
7437:
7430:
7427:
7416:
7415:
7410:
7403:
7400:
7389:
7385:
7378:
7375:
7371:
7365:
7362:
7357:
7353:
7349:
7342:
7339:
7334:
7330:
7326:
7320:
7317:
7310:
7309:
7300:
7297:
7291:
7286:
7282:
7278:
7274:
7267:
7264:
7259:
7255:
7254:Wright, E. M.
7251:
7245:
7242:
7238:
7232:
7229:
7226:
7220:
7217:
7212:
7205:
7202:
7189:
7185:
7179:
7176:
7163:
7159:
7153:
7150:
7144:
7131:
7127:
7119:
7115:
7101:
7093:
7074:
7071:
7066:
7062:
7051:
7047:
7040:
7036:
7028:
7024:
7016:
7012:
7008:
7004:
6996:
6992:
6984:
6980:
6969:
6966:
6960:
6956:
6952:
6948:
6944:
6940:
6936:
6922:
6918:
6914:
6910:
6903:
6897:
6894:
6887:
6882:
6879:
6877:
6874:
6872:
6871:Solinas prime
6869:
6867:
6864:
6862:
6861:Woodall prime
6859:
6857:
6854:
6852:
6849:
6847:
6844:
6842:
6839:
6836:
6833:
6830:
6827:
6825:
6822:
6820:
6817:
6815:
6812:
6810:
6807:
6805:
6802:
6800:
6799:Fermat number
6797:
6795:
6792:
6791:
6786:
6781:
6776:
6770:
6769:
6768:
6767:is prime are
6765:
6761:
6747:
6742:
6736:
6735:
6734:
6733:is prime are
6731:
6727:
6716:
6713:
6709:
6702:
6692:
6688:
6678:
6674:
6666:
6662:
6656:
6644:
6634:
6631:
6628:
6625:
6622:
6621:
6618:
6615:
6612:
6609:
6608:
6605:
6602:
6599:
6596:
6593:
6592:
6589:
6586:
6583:
6580:
6577:
6576:
6573:
6570:
6567:
6564:
6561:
6560:
6557:
6554:
6551:
6548:
6545:
6544:
6541:
6538:
6535:
6532:
6529:
6528:
6525:
6522:
6519:
6516:
6513:
6512:
6509:
6506:
6503:
6500:
6497:
6496:
6493:
6490:
6487:
6484:
6483:
6480:
6477:
6474:
6471:
6470:
6467:
6464:
6461:
6458:
6457:
6454:
6451:
6448:
6445:
6444:
6441:
6438:
6435:
6432:
6429:
6428:
6425:
6422:
6419:
6416:
6413:
6412:
6409:
6406:
6403:
6400:
6397:
6396:
6393:
6390:
6387:
6384:
6381:
6380:
6377:
6374:
6371:
6368:
6365:
6364:
6361:
6358:
6355:
6352:
6349:
6348:
6345:
6342:
6339:
6336:
6333:
6332:
6329:
6326:
6323:
6320:
6317:
6316:
6313:
6310:
6307:
6304:
6301:
6300:
6297:
6294:
6291:
6288:
6285:
6284:
6281:
6278:
6275:
6272:
6269:
6268:
6265:
6262:
6259:
6256:
6253:
6252:
6249:
6246:
6243:
6240:
6237:
6236:
6233:
6230:
6227:
6224:
6221:
6220:
6217:
6214:
6211:
6208:
6205:
6204:
6201:
6198:
6195:
6192:
6189:
6188:
6185:
6182:
6179:
6176:
6175:
6172:
6169:
6166:
6163:
6160:
6159:
6156:
6153:
6150:
6147:
6144:
6143:
6140:
6137:
6134:
6131:
6128:
6127:
6124:
6121:
6118:
6115:
6112:
6111:
6108:
6105:
6102:
6099:
6096:
6095:
6092:
6089:
6086:
6083:
6080:
6079:
6076:
6073:
6070:
6067:
6064:
6063:
6060:
6057:
6054:
6051:
6048:
6047:
6044:
6041:
6038:
6035:
6032:
6031:
6028:
6025:
6022:
6019:
6016:
6015:
6012:
6009:
6006:
6003:
6000:
5999:
5996:
5993:
5990:
5987:
5984:
5983:
5980:
5977:
5974:
5971:
5970:
5967:
5964:
5961:
5958:
5955:
5954:
5951:
5948:
5945:
5942:
5941:
5938:
5935:
5932:
5929:
5926:
5925:
5922:
5919:
5916:
5913:
5910:
5909:
5906:
5903:
5900:
5897:
5894:
5893:
5890:
5887:
5884:
5881:
5878:
5877:
5874:
5871:
5868:
5865:
5862:
5861:
5858:
5855:
5852:
5849:
5846:
5845:
5842:
5839:
5836:
5833:
5832:
5829:
5826:
5823:
5820:
5819:
5816:
5813:
5810:
5807:
5804:
5803:
5800:
5797:
5794:
5791:
5788:
5787:
5784:
5781:
5778:
5775:
5772:
5771:
5768:
5765:
5762:
5759:
5756:
5755:
5752:
5749:
5746:
5743:
5740:
5739:
5736:
5733:
5730:
5727:
5724:
5723:
5720:
5717:
5714:
5711:
5708:
5707:
5704:
5701:
5698:
5695:
5692:
5691:
5688:
5685:
5682:
5679:
5676:
5675:
5672:
5669:
5666:
5663:
5660:
5659:
5656:
5653:
5650:
5647:
5644:
5643:
5640:
5637:
5634:
5631:
5628:
5627:
5624:
5621:
5618:
5615:
5612:
5611:
5608:
5605:
5602:
5599:
5596:
5595:
5592:
5589:
5586:
5583:
5580:
5579:
5576:
5573:
5570:
5567:
5564:
5563:
5560:
5557:
5554:
5551:
5548:
5547:
5544:
5541:
5538:
5535:
5532:
5531:
5528:
5525:
5522:
5519:
5516:
5515:
5512:
5509:
5506:
5503:
5500:
5499:
5496:
5493:
5490:
5487:
5484:
5483:
5480:
5477:
5474:
5471:
5468:
5467:
5464:
5461:
5458:
5455:
5452:
5451:
5448:
5445:
5442:
5439:
5436:
5435:
5432:
5429:
5426:
5423:
5420:
5419:
5416:
5413:
5410:
5407:
5404:
5403:
5400:
5397:
5394:
5391:
5388:
5387:
5384:
5381:
5378:
5375:
5372:
5371:
5368:
5365:
5362:
5359:
5358:
5355:
5352:
5349:
5346:
5345:
5342:
5339:
5336:
5333:
5330:
5329:
5326:
5323:
5320:
5317:
5314:
5313:
5310:
5307:
5304:
5301:
5298:
5297:
5294:
5291:
5288:
5285:
5282:
5281:
5278:
5275:
5272:
5269:
5266:
5265:
5262:
5259:
5255:
5252:
5249:
5248:
5245:
5242:
5238:
5235:
5232:
5231:
5227:
5224:
5218:
5213:
5209:5 < |
5198:
5193:
5181:
5163:
5151:
5147:
5139:
5135:
5119:
5112:
5105:
5104:
5098:
5094:
5090:
5079:
5075:
5067:
5063:
5050:
5045:
5021:
5017:
5010:
5006:
4999:
4995:
4977:
4973:
4967:
4963:
4957:
4953:
4949:
4941:
4936:
4906:
4903:
4900:
4893:
4889:
4885:
4880:
4876:
4865:
4864:
4863:
4857:
4852:
4847:
4841:
4840:
4839:
4836:
4827:
4822:
4816:
4815:
4814:
4813:is prime are
4806:
4797:
4787:
4776:
4771:
4766:
4760:
4759:
4758:
4755:
4744:
4738:
4729:
4724:
4718:
4717:
4716:
4713:
4702:
4691:
4682:
4672:
4666:
4658:
4649:
4639:
4633:
4622:
4613:
4603:
4598:
4597:perfect power
4593:
4582:
4577:
4572:
4571:
4570:
4567:
4560:
4551:
4546:
4539:
4534:
4529:
4528:
4527:
4524:
4517:
4511:
4488:
4485:
4482:
4477:
4474:
4469:
4465:
4454:
4453:
4452:
4449:
4442:
4435:
4428:
4420:
4415:
4410:
4405:
4400:
4399:
4398:
4392:
4387:
4382:
4381:
4380:
4377:
4370:
4364:
4362:
4357:
4353:
4347:
4343:
4338:
4330:
4325:
4320:
4315:
4314:
4313:
4311:
4306:
4300:
4295:
4289:
4288:
4287:
4284:
4277:
4271:
4269:
4265:
4259:
4252:
4247:
4242:
4238:
4232:
4228:
4223:
4215:
4213:
4211:
4207:
4203:
4199:
4193:
4179:
4174:
4168:
4163:
4159:
4151:
4149:
4147:
4143:
4137:
4130:
4125:
4119:
4112:
4105:
4095:
4091:
4087:
4083:
4079:
4072:
4061:
4057:
4053:
4049:
4042:
4033:
4029:
4023:
4019:
4012:
4005:
3997:
3995:
3993:
3988:
3981:
3974:
3970:
3953:
3952:
3951:
3947:
3942:
3941:Fermat number
3936:
3929:
3924:
3921:
3917:
3909:
3903:
3886:
3877:
3869:
3867:
3857:
3845:
3837:
3833:
3830:, an integer
3829:
3824:
3822:
3818:
3817:31 Euphrosyne
3814:
3810:
3806:
3805:8191 Mersenne
3802:
3798:
3793:
3790:
3782:
3777:
3768:
3763:
3755:
3753:
3751:
3746:
3735:
3732:
3729:
3728:
3724:
3721:
3718:
3717:
3713:
3710:
3707:
3706:
3702:
3699:
3696:
3695:
3691:
3688:
3685:
3684:
3680:
3677:
3674:
3673:
3669:
3666:
3663:
3662:
3658:
3655:
3652:
3651:
3647:
3644:
3641:
3640:
3636:
3633:
3630:
3629:
3625:
3622:
3619:
3618:
3614:
3611:
3608:
3607:
3603:
3600:
3597:
3596:
3592:
3589:
3586:
3585:
3581:
3578:
3575:
3574:
3570:
3567:
3564:
3563:
3559:
3557:137438953471
3556:
3553:
3552:
3548:
3545:
3542:
3541:
3538:47 × 178,481
3537:
3534:
3531:
3530:
3526:
3523:
3520:
3519:
3515:
3507:
3504:
3497:
3494:
3490:
3489:
3486:
3484:
3479:
3473:
3468:
3453:
3449:
3444:
3436:
3428:
3423:
3418:
3412:
3411:
3410:
3408:
3402:
3394:
3384:
3374:
3366:
3362:
3356:
3352:
3347:
3343:
3336:
3329:
3325:
3320:
3313:
3295:
3289:
3285:
3282:
3278:
3264:
3247:. Also since
3220:
3212:
3204:
3196:
3188:
3178:
3175:
3164:
3161:
3160:
3152:
3142:
3134:
3124:
3114:
3111:
3099:
3079:
3069:
3061:
3032:
3026:
3022:
3015:
3011:
3004:
2996:
2992:: We show if
2991:
2988:
2987:
2985:
2981:
2972:
2967:
2955:
2949:
2946:
2945:
2936:that divides
2927:
2909:
2901:
2896:
2881:
2877:
2872:
2868:
2849:
2845:
2838:
2778:is prime and
2764:
2743:
2739:
2736:
2714:For example,
2713:
2712:
2705:
2696:that divides
2687:
2682:
2648:
2638:
2634:
2625:
2622:
2621:
2611:
2604:
2593:
2586:
2579:
2572:
2565:
2559:
2555:
2548:
2541:
2537:
2533:is prime, so
2530:
2523:
2519:
2512:
2509:− 1 ≡ 0 (mod
2508:
2501:
2497:
2490:
2487:
2483:
2478:
2475:
2474:
2470:
2463:
2456:
2446:
2441:
2440:
2439:
2437:
2432:
2423:
2421:
2419:
2414:
2412:
2403:
2397:
2392:
2386:
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:
1899:
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:
1512:
1510:
1496:
1492:
1485:
1482:
1479:
1473:
1470:
1467:
1444:
1441:
1436:
1432:
1428:
1425:
1413:
1411:
1407:
1403:
1391:
1387:
1381:
1377:
1370:
1362:
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:
337:
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:
12239:Power of two
12229:Number names
11964:Ten thousand
11869:
11540:Transposable
11404:Narcissistic
11311:Digital root
11231:Super-Poulet
11191:Jordan–Pólya
11140:prime factor
11045:Noncototient
11012:Almost prime
10994:Superperfect
10969:Refactorable
10964:Quasiperfect
10939:Hyperperfect
10780:Pseudoprimes
10751:Wall–Sun–Sun
10686:Ordered Bell
10656:Fuss–Catalan
10568:non-centered
10518:Dodecahedral
10495:non-centered
10381:non-centered
10283:Wolstenholme
10052:
10028:× 2 ± 1
10025:
10024:Of the form
9991:Eighth power
9971:Fourth power
9497:Almost prime
9455:Euler–Jacobi
9363:
9359:
9355:
9351:
9349:Cunningham (
9339:
9335:
9318:
9314:
9310:
9291:
9287:
9283:
9279:
9275:
9274:consecutive
9262:
9258:
9254:
9242:
9238:
9226:
9222:
9210:
9206:
9202:
9198:
9196:Quadruplet (
9186:
9182:
9178:
9174:
9162:
9158:
9145:
9093:Full reptend
8951:Wolstenholme
8946:Wall–Sun–Sun
8877:
8862:
8858:
8837:
8833:
8822:
8818:
8814:
8810:
8798:
8786:
8766:
8762:
8741:
8725:
8710:
8698:
8686:
8657:
8634:Prime number
8592:
8573:
8525:
8521:
8506:
8497:
8415:. Retrieved
8411:the original
8402:
8386:
8382:
8378:
8373:
8369:
8357:
8353:
8348:
8344:
8314:
8292:
8288:
8222:
8212:
8208:
8142:
8127:
8120:
8105:
8101:
8094:
8079:
8075:
8068:
8053:
8049:
8042:
8027:
8020:
8005:
7998:
7983:
7975:
7968:
7947:
7930:
7903:
7897:
7889:the original
7879:
7865:
7854:. Retrieved
7843:
7832:. Retrieved
7828:
7818:
7799:
7793:
7781:. Retrieved
7777:
7768:
7757:. Retrieved
7748:
7736:. Retrieved
7732:
7723:
7712:. Retrieved
7702:
7675:
7669:
7658:
7648:
7630:
7618:. Retrieved
7614:
7605:
7594:. Retrieved
7585:
7571:
7562:
7553:
7542:. Retrieved
7538:
7529:
7521:the original
7517:"Milestones"
7511:
7499:. Retrieved
7493:
7483:
7471:. Retrieved
7465:
7455:
7443:. Retrieved
7439:
7429:
7418:. Retrieved
7412:
7402:
7391:. Retrieved
7387:
7377:
7364:
7355:
7351:
7341:
7333:the original
7328:
7319:
7307:
7299:
7280:
7276:
7266:
7257:
7250:Hardy, G. H.
7244:
7231:
7219:
7204:
7192:. Retrieved
7188:Mersenne.org
7187:
7178:
7166:. Retrieved
7161:
7152:
7129:
7125:
7117:
7113:
7099:
7091:
7073:
7049:
7045:
7038:
7034:
7026:
7022:
7014:
7010:
7006:
7002:
6994:
6990:
6982:
6978:
6968:
6958:
6954:
6950:
6946:
6942:
6920:
6916:
6912:
6905:
6902:Lucas number
6896:
6856:Cullen prime
6804:Power of two
6763:
6759:
6751:
6729:
6725:
6717:
6711:
6707:
6700:
6690:
6686:
6676:
6672:
6664:
6660:
6657:
6642:
6639:
5216:
5215:| <
5211:
5196:
5191:
5179:
5149:
5145:
5137:
5133:
5092:
5088:
5077:
5073:
5065:
5061:
5049:fourth power
5043:
5019:
5015:
5008:
5004:
4997:
4993:
4975:
4971:
4965:
4961:
4955:
4951:
4947:
4939:
4924:
4861:
4834:
4831:
4804:
4795:
4785:
4782:
4774:
4753:
4742:
4736:
4733:
4711:
4700:
4689:
4680:
4670:
4667:
4656:
4647:
4637:
4631:
4620:
4611:
4601:
4591:
4588:
4565:
4558:
4555:
4522:
4515:
4509:
4506:
4447:
4440:
4433:
4430:
4396:
4375:
4368:
4365:
4360:
4355:
4351:
4345:
4341:
4336:
4334:
4307:
4304:
4282:
4275:
4272:
4267:
4257:
4250:
4248:) for which
4240:
4236:
4230:
4226:
4219:
4202:real numbers
4191:
4177:
4166:
4162:real numbers
4155:
4145:
4135:
4128:
4117:
4110:
4103:
4096:
4089:
4085:
4081:
4077:
4070:
4059:
4055:
4051:
4047:
4040:
4032:coefficients
4021:
4017:
4010:
4007:
3986:
3976:
3972:
3968:
3964:
3945:
3934:
3927:
3919:
3915:
3912:
3907:
3901:
3885:
3875:
3873:
3825:
3801:minor planet
3794:
3785:
3772:
3766:
3759:
3741:
3510:
3499:
3492:
3474:
3466:
3432:
3406:
3404:
3382:
3372:
3364:
3360:
3354:
3350:
3345:
3341:
3334:
3318:
3311:
3293:
3262:
3218:
3210:
3209:2 ≡ −1 (mod
3202:
3199:, so either
3194:
3186:
3176:
3162:
3150:
3122:
3067:
3059:
3035:, therefore
3030:
3024:
3020:
3013:
3009:
3002:
2994:
2989:
2965:
2953:
2947:
2907:
2899:
2870:
2866:
2847:
2843:
2836:
2762:
2737:
2703:
2680:
2646:
2636:
2632:
2623:
2602:
2591:
2584:
2577:
2570:
2563:
2557:
2553:
2546:
2539:
2535:
2528:
2521:
2517:
2510:
2506:
2499:
2495:
2488:
2481:
2476:
2468:
2461:
2454:
2444:
2427:
2415:
2404:
2393:
2384:
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
12249:Power of 10
12189:Busy beaver
11994:Quintillion
11989:Quadrillion
11561:Extravagant
11556:Equidigital
11511:permutation
11470:Palindromic
11444:Automorphic
11342:Sum-product
11321:Sum-product
11276:Persistence
11171:Erdős–Woods
11093:Untouchable
10974:Semiperfect
10924:Hemiperfect
10585:Tesseractic
10523:Icosahedral
10503:Tetrahedral
10434:Dodecagonal
10135:Recursively
10006:Prime power
9981:Sixth power
9976:Fifth power
9956:Power of 10
9914:Classes of
9480:Somer–Lucas
9435:Pseudoprime
9073:Truncatable
9045:Palindromic
8929:By property
8708:Primorial (
8696:Factorial (
8510:up to 1280)
8435:(in German)
8407:Brady Haran
8403:Numberphile
8391:math thesis
8285:, that is,
8205:, that is,
7940:Prime Pages
7738:5 September
7660:Prime Pages
7407:Tia Ghose.
7168:21 December
7058:must be in
6866:Proth prime
4838:, they are
4783:Least base
4751:if no such
4709:if no such
4569:values of:
4526:values of:
4254:the number
4200:instead of
3821:127 Johanna
3201:2 ≡ 1 (mod
3193:2 ≡ 1 (mod
3062:mod (2 − 1)
3001:2 ≡ 1 (mod
2952:2 ≡ 2 (mod
2596:, however,
2526:. 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
12287:Categories
12254:Sagan Unit
12088:Expression
12039:Googolplex
12004:Septillion
11999:Sextillion
11945:numerical
11773:Graphemics
11646:Pernicious
11500:Undulating
11475:Pandigital
11449:Trimorphic
11050:Nontotient
10899:Arithmetic
10513:Octahedral
10414:Heptagonal
10404:Pentagonal
10389:Triangular
10230:Sierpiński
10152:Jacobsthal
9951:Power of 3
9946:Power of 2
9517:Pernicious
9512:Interprime
9272:Balanced (
9063:Permutable
9038:-dependent
8855:Williams (
8751:Pierpont (
8676:Wagstaff
8658:Mersenne (
8642:By formula
8440:GIMPS wiki
8417:2013-04-06
8136:= 2 to 200
8114:= 1 to 107
8062:= 1 to 160
8036:= 2 to 160
8014:= 2 to 160
7959:1503.07688
7856:2011-05-21
7834:2023-02-11
7759:2021-07-21
7714:2022-09-05
7596:2019-01-01
7544:2018-01-03
7501:22 January
7473:19 January
7445:22 January
7420:2013-02-07
7393:2011-05-21
7358:: 122–131.
7194:5 December
7145:References
6756:such that
6722:such that
5183:| ≤ 5
5126:such that
5055:such that
5024:such that
4937:integers,
4789:such that
4674:such that
4665:is prime)
4605:such that
4028:polynomial
3939:, it is a
3856:hypotenuse
3546:536870911
3110:pernicious
2975:±1 (mod 8)
2942:±1 (mod 8)
2869:≡ 1 (mod 2
2794:such that
2716:2 − 1 = 31
2711:is prime.
2620:is prime.
2388:37,156,667
603:, and the
569:(sequence
261:2147483647
132:− 1 where
12145:Pentation
12140:Tetration
12128:Operators
12097:Notations
12019:Decillion
12014:Nonillion
12009:Octillion
11941:Examples
11530:Parasitic
11379:Factorion
11306:Digit sum
11298:Digit sum
11116:Fortunate
11103:Primorial
11017:Semiprime
10954:Practical
10919:Descartes
10914:Deficient
10904:Betrothed
10746:Wieferich
10575:Pentatope
10538:pyramidal
10429:Decagonal
10424:Nonagonal
10419:Octagonal
10409:Hexagonal
10268:Practical
10215:Congruent
10147:Fibonacci
10111:Loeschian
9553:Prime gap
9502:Semiprime
9465:Frobenius
9172:Triplet (
8971:Ramanujan
8966:Fortunate
8936:Wieferich
8900:Fibonacci
8831:Leyland (
8796:Woodall (
8775:Solinas (
8760:Quartan (
8594:MathWorld
8575:MathWorld
8321:EMS Press
8170:−
8088:= 1 to 40
7992:= 2 to 50
6640:Note: if
5228:sequence
5195:| =
4959:. (Since
4904:−
4886:−
4563:, we get
4520:, we get
4486:−
4475:−
4066:2 − 2 − 1
4036:2 − 2 + 1
3965:In fact,
3959:MF(59, 2)
3840:≥ 4
3836:primitive
3253:3 (mod 4)
3241:7 (mod 8)
3233:3 (mod 4)
3145:3 (mod 4)
3141:congruent
2846:≡ 1 (mod
2683:is prime.
2662:= (2 − 1)
2659:= (2) − 1
2498:≡ 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
12177:articles
12175:Related
12079:Infinity
11984:Trillion
11959:Thousand
11602:Friedman
11535:Primeval
11480:Repdigit
11437:-related
11384:Kaprekar
11358:Meertens
11281:Additive
11268:dynamics
11176:Friendly
11088:Sociable
11078:Amicable
10889:Abundant
10869:dynamics
10691:Schröder
10681:Narayana
10651:Eulerian
10641:Delannoy
10636:Dedekind
10457:centered
10323:centered
10210:Amenable
10167:Narayana
10157:Leonardo
10053:Mersenne
10001:Powerful
9941:Achilles
9445:Elliptic
9220:Cousin (
9137:Patterns
9128:Tetradic
9123:Dihedral
9088:Primeval
9083:Delicate
9068:Circular
9055:Repunit
8846:Thabit (
8784:Cullen (
8723:Euclid (
8649:Fermat (
8549:Archived
8537:Archived
8481:Archived
8469:Archived
8110:for odd
7638:Archived
7329:BBC News
7256:(1959).
7096:or when
7031:must be
6933:are the
6925:, since
6831:/ MPrime
6787:See also
6698:must be
6669:, it is
5164:, these
5158:is prime
5120:numbers
4757:exists)
4715:exists)
4015:, where
3984:, where
3844:inradius
3834:that is
3828:geometry
3797:asteroid
3535:8388607
3527:23 × 89
3461:, where
3330:. It is
3292:2 − 1 =
3267:divides
3155:divides
3064:. Hence
2774:. Since
2752:. Since
2726:, where
2561:, hence
2549:− 1 = ±1
2308:, where
2092:, where
2080:divides
1738:scholar
924:A basic
913:≥
737:divides
671:. Since
64:Infinite
61:of terms
12272:History
12090:methods
12064:SSCG(3)
12059:TREE(3)
11979:Billion
11974:Million
11954:Hundred
11775:related
11739:related
11703:related
11701:Sorting
11586:Vampire
11571:Harshad
11513:related
11485:Repunit
11399:Lychrel
11374:Dudeney
11226:Størmer
11221:Sphenic
11206:Regular
11144:divisor
11083:Perfect
10979:Sublime
10949:Perfect
10676:Motzkin
10631:Catalan
10172:Padovan
10106:Leyland
10101:Idoneal
10096:Hilbert
10068:Woodall
9440:Catalan
9377:By size
9148:-tuples
9078:Minimal
8981:Regular
8872:Mills (
8808:Cuban (
8684:Proth (
8636:classes
8323:, 2001
8283:
8229:
8203:
8149:
7783:24 June
7620:29 June
7314:p. 228.
7134:
7110:
7065:A027861
7063::
6999:
6975:
6937:of the
6837:(GIMPS)
6829:Prime95
6794:Repunit
6778:in the
6775:A222119
6762:+ 1) −
6744:in the
6741:A058013
6728:+ 1) −
6675:+ 1) −
6633:A213216
6604:A128341
6588:A057178
6572:A004064
6556:A128348
6540:A273814
6524:A062578
6508:A185239
6440:A128340
6424:A224501
6408:A128070
6392:A125957
6376:A057177
6360:A005808
6344:A210506
6328:A128027
6312:A216181
6296:A128347
6280:A273598
6264:A273599
6248:A273600
6232:A273601
6216:A062577
6200:A217095
6171:A128069
6155:A001562
6139:A004023
6123:A128026
6107:A273403
6091:A062576
6075:A187819
6059:A301369
6043:A128339
6027:A211409
6011:A125956
5995:A057175
5978:(none)
5966:A173718
5937:A128346
5921:A273010
5905:A059803
5889:A181141
5873:A128338
5857:A128068
5815:A128025
5799:A128345
5783:A062574
5767:A187805
5751:A128337
5735:A218373
5719:A128067
5703:A125955
5687:A057173
5671:A004063
5655:A215487
5639:A128024
5623:A213073
5607:A128344
5591:A062573
5575:A128336
5559:A057172
5543:A004062
5527:A062572
5511:A128335
5495:A122853
5479:A082387
5463:A057171
5447:A004061
5431:A082182
5415:A121877
5399:A059802
5383:A128066
5341:A059801
5325:A057469
5309:A007658
5293:A028491
5277:A057468
5261:A000978
5244:A000043
5154:
5130:
5082:
5058:
4935:coprime
4849:in the
4846:A103795
4824:in the
4821:A066180
4810:
4792:
4768:in the
4765:A084742
4726:in the
4723:A084740
4695:
4677:
4662:
4644:
4626:
4608:
4579:in the
4576:A057178
4548:in the
4545:A004022
4536:in the
4533:A004023
4427:Repunit
4407:in the
4404:A066413
4389:in the
4386:A066408
4322:in the
4319:A182300
4297:in the
4294:A057429
4204:, like
4175:, then
4156:In the
3990:is the
3925:. When
3905:prime,
3896:
3881:
3748:in the
3745:A046800
3481:in the
3478:A244453
3420:in the
3417:A000043
3163:Example
3147:, then
3135:), and
3098:coprime
3051:
3039:
3033:| 2 − 1
2997:= 2 − 1
2656:= 2 − 1
2651:. Then
2515:. Thus
2493:. Then
2434:in the
2431: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
12181:order)
12029:Googol
11641:Odious
11566:Frugal
11520:Cyclic
11509:Digit-
11216:Smooth
11201:Pronic
11161:Cyclic
11138:Other
11111:Euclid
10761:Wilson
10735:Primes
10394:Square
10263:Polite
10225:Riesel
10220:Knödel
10182:Perrin
10063:Thabit
10048:Fermat
10038:Cullen
9961:Square
9929:Powers
9485:Strong
9475:Perrin
9460:Fermat
9236:Sexy (
9156:Twin (
9098:Unique
9026:Unique
8986:Strong
8976:Pillai
8956:Wilson
8920:Perrin
8356:) − (3
8078:+ 1, −
7918:
7806:
7690:
7094:> 1
7037:+ 1, −
6972:Since
6752:Least
6718:Least
6645:< 0
5189:|
5177:|
5172:100000
4942:> 1
4668:Least
4354:= 1 −
4344:= 1 +
4239:= 1 −
4229:= 1 +
4194:> 0
4164:), if
4120:> 1
4075:, and
4045:, and
3948:> 1
3858:to be
3854:, the
3813:7 Iris
3809:3 Juno
3321:> 1
3314:> 1
3305:, and
3179:: Let
3023:− 1 =
2969:. By
2964:2 mod
2902:> 2
2649:> 1
2566:= 0, 1
2538:− 1 =
2520:− 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
12267:Names
12069:BH(3)
11947:order
11682:Prime
11677:Lucky
11666:sieve
11595:Other
11581:Smith
11461:Digit
11419:Happy
11394:Keith
11367:Other
11211:Rough
11181:Giuga
10646:Euler
10508:Cubic
10162:Lucas
10058:Proth
9470:Lucas
9450:Euler
9103:Happy
9050:Emirp
9016:Higgs
9011:Super
8991:Stern
8961:Lucky
8905:Lucas
8524:± 1,
8365:(PDF)
8130:, −1)
8104:+ 2,
8052:+ 1,
8030:, −1)
7986:, −1)
7954:arXiv
7077:When
6935:roots
6888:Notes
6658:When
5204:20000
4954:<
4950:<
4925:with
4561:= −12
4451:make
4371:) − 1
4367:(1 +
4310:norms
4278:) − 1
4274:(1 +
4262:is a
4260:) − 1
4256:(1 +
4187:0 − 1
4183:2 − 1
4171:is a
4108:(for
4099:2 − 1
3899:with
3893:2 − 1
3799:with
3524:2047
3439:2 − 1
3370:with
3177:Proof
3171:2 − 1
3157:2 − 1
3131:is a
3106:2 − 1
3102:2 − 1
3048:2 − 1
3042:2 − 1
2990:Proof
2958:, so
2948:Proof
2938:2 − 1
2888:2 − 1
2858:2 − 1
2828:2 − 1
2812:2 − 1
2800:2 − 1
2784:2 − 1
2772:2 − 1
2758:2 − 1
2750:2 − 1
2740:: By
2738:Proof
2709:2 − 1
2698:2 − 1
2677:2 − 1
2673:2 − 1
2653:2 − 1
2640:with
2624:Proof
2614:2 − 1
2589:. If
2504:, so
2484:≡ 1 (
2477:Proof
2407:2 − 1
2400:2 − 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
11885:List
11636:Evil
11316:Self
11266:and
11156:Blum
10867:and
10671:Lobb
10626:Cake
10621:Bell
10371:Star
10278:Ulam
10177:Pell
9966:Cube
9393:list
9328:Chen
9108:Self
9036:Base
9006:Good
8940:pair
8910:Pell
8861:−1)·
8490:OEIS
8393:(PS)
8352:= (8
8132:for
8084:for
8058:for
8032:for
8010:for
8008:, 1)
7988:for
7980:and
7978:, 1)
7916:ISBN
7804:ISBN
7785:2022
7740:2022
7688:ISBN
7622:2021
7503:2016
7475:2016
7447:2016
7196:2020
7170:2018
7081:and
7061:OEIS
7052:+ 1)
7043:and
6929:and
6780:OEIS
6746:OEIS
6647:and
6626:−11
6501:−10
5226:OEIS
5206:for
5174:for
5040:and
5028:and
5002:and
4944:and
4933:any
4851:OEIS
4826:OEIS
4770:OEIS
4745:= −2
4728:OEIS
4581:OEIS
4550:OEIS
4538:OEIS
4518:= 10
4409:OEIS
4391:OEIS
4349:and
4324:OEIS
4299:OEIS
4246:WLOG
4234:and
4208:and
4173:unit
4158:ring
4142:unit
4115:and
4084:) =
4073:= 64
4054:) =
4043:= 32
3975:) =
3957:and
3950:are
3819:and
3795:The
3750:OEIS
3730:131
3719:113
3708:109
3697:103
3686:101
3483:OEIS
3470:1277
3422:OEIS
3380:and
3316:and
3119:and
3104:and
3096:are
3092:and
3084:and
3066:p |
3037:0 ≡
2730:and
2644:and
2573:= 1.
2513:− 1)
2502:− 1)
2491:− 1)
2448:and
2436:OEIS
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
11754:Ban
11142:or
10661:Lah
9864:281
9859:277
9854:271
9849:269
9844:263
9839:257
9834:251
9829:241
9824:239
9819:233
9814:229
9809:227
9804:223
9799:211
9794:199
9789:197
9784:193
9779:191
9774:181
9769:179
9764:173
9759:167
9754:163
9749:157
9744:151
9739:149
9734:139
9729:137
9724:131
9719:127
9714:113
9709:109
9704:107
9699:103
9694:101
9354:, 2
9338:, 2
9259:a·n
8817:)/(
8291:, −
7908:doi
7680:doi
7285:doi
7048:+ (
7001:= (
6961:= 0
6945:− (
6703:+ 1
6667:+ 1
6623:12
6613:−7
6610:12
6597:−5
6594:12
6581:−1
6578:12
6562:12
6546:12
6530:12
6517:11
6514:12
6498:11
6488:−9
6485:11
6475:−8
6472:11
6462:−7
6459:11
6449:−6
6446:11
6433:−5
6430:11
6417:−4
6414:11
6401:−3
6398:11
6385:−2
6382:11
6369:−1
6366:11
6350:11
6334:11
6318:11
6302:11
6286:11
6270:11
6254:11
6238:11
6222:11
6209:10
6206:11
6193:−9
6190:10
6180:−7
6177:10
6164:−3
6161:10
6148:−1
6145:10
6129:10
6113:10
6097:10
6081:10
6068:−8
6052:−7
6036:−5
6020:−4
6004:−2
5988:−1
5882:−7
5866:−5
5850:−3
5837:−1
5760:−6
5744:−5
5728:−4
5712:−3
5696:−2
5680:−1
5568:−5
5552:−1
5504:−4
5488:−3
5472:−2
5456:−1
5376:−3
5363:−1
5318:−2
5302:−1
5253:−1
5219:− 1
5199:− 1
5186:or
5000:= 1
4807:− 1
4798:− 1
4777:= 2
4703:= 2
4692:− 1
4683:− 1
4659:− 1
4650:− 1
4623:− 1
4614:− 1
4443:− 1
4436:− 1
4169:− 1
4138:− 1
4131:− 1
4126:),
4113:≠ 2
4106:− 1
4092:− 1
4062:+ 1
4013:(2)
3982:(2)
3967:MF(
3937:= 2
3930:= 1
3914:MF(
3887:− 1
3826:In
3783:is
3752:).
3675:97
3664:83
3653:79
3642:73
3631:71
3620:67
3609:59
3598:53
3587:47
3576:43
3565:41
3554:37
3543:29
3532:23
3521:11
3485:).
3388:= 1
3378:= 0
3359:- 2
3349:= 3
3265:+ 1
3207:or
3189:+ 1
3183:be
3153:+ 1
3143:to
3139:is
3125:+ 1
3115:If
3080:If
3016:− 1
2928:If
2841:so
2839:− 1
2688:If
2671:so
2612:If
2605:= 2
2594:= 0
2587:= 0
2582:or
2580:= 2
2544:or
2542:− 1
2531:− 1
2524:− 1
2486:mod
2471:= 1
2466:or
2464:= 2
2457:− 1
2442:If
2438:).
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.
12289::
11943:in
9689:97
9684:89
9679:83
9674:79
9669:73
9664:71
9659:67
9654:61
9649:59
9644:53
9639:47
9634:43
9629:41
9624:37
9619:31
9614:29
9609:23
9604:19
9599:17
9594:13
9589:11
9286:,
9282:,
9261:,
9241:,
9225:,
9201:,
9177:,
9161:,
8591:.
8572:.
8405:.
8401:.
8381:+
8377:=
8358:qy
8319:,
8313:,
8211:,
7914:.
7827:.
7776:.
7731:.
7686:.
7657:.
7613:.
7561:.
7537:.
7492:.
7464:.
7438:.
7411:.
7386:.
7356:18
7354:.
7350:.
7327:.
7281:10
7279:.
7275:.
7252:;
7186:.
7160:.
7128:−
7116:−
7100:ab
7098:−4
7025:,
7013:+
7009:)(
7005:+
6993:−
6981:−
6959:ab
6957:+
6949:+
6921:ab
6919:,
6915:+
6715:.
6710:−
6689:−
6663:=
6565:1
6549:5
6533:7
6353:1
6337:2
6321:3
6305:4
6289:5
6273:6
6257:7
6241:8
6225:9
6132:1
6116:3
6100:7
6084:9
6065:9
6049:9
6033:9
6017:9
6001:9
5985:9
5975:1
5972:9
5959:2
5956:9
5946:4
5943:9
5930:5
5927:9
5914:7
5911:9
5898:8
5895:9
5879:8
5863:8
5847:8
5834:8
5824:1
5821:8
5808:3
5805:8
5792:5
5789:8
5776:7
5773:8
5757:7
5741:7
5725:7
5709:7
5693:7
5677:7
5664:1
5661:7
5648:2
5645:7
5632:3
5629:7
5616:4
5613:7
5600:5
5597:7
5584:6
5581:7
5565:6
5549:6
5536:1
5533:6
5520:5
5517:6
5501:5
5485:5
5469:5
5453:5
5440:1
5437:5
5424:2
5421:5
5408:3
5405:5
5392:4
5389:5
5373:4
5360:4
5350:1
5347:4
5334:3
5331:4
5315:3
5299:3
5286:1
5283:3
5270:2
5267:3
5250:2
5236:1
5233:2
5222:)
5202:,
5148:−
5136:−
5097:.
5091:,
5076:−
5064:−
5044:ab
5042:−4
5018:,
5007:+
4996:+
4974:−
4964:−
4929:,
4747:,
4705:,
4583:),
4552:).
4540:),
4379::
4363:.
4326:).
4286::
4270:.
4212:.
4094:.
4088:−
4058:−
3994:.
3971:,
3918:,
3874:A
3866:.
3815:,
3811:,
3792:.
3788:64
3409::
3367:-2
3357:-1
3301:,
3235:,
3159:.
3058:≡
3025:mλ
3012:|
2986:.
2950::
2944:.
2910:+1
2908:kp
2830:,
2806:,
2786:,
2766:,
2744:,
2637:ab
2635:=
2556:=
2479::
2473:.
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
11926:e
11919:t
11912:v
11841:e
11834:t
11827:v
10026:a
9907:e
9900:t
9893:v
9584:7
9579:5
9574:3
9569:2
9368:)
9364:p
9360:p
9356:p
9352:p
9344:)
9340:p
9336:p
9323:)
9319:n
9315:n
9311:n
9295:)
9292:n
9288:p
9284:p
9280:n
9276:p
9267:)
9263:n
9255:p
9247:)
9243:p
9239:p
9231:)
9227:p
9223:p
9215:)
9211:p
9207:p
9203:p
9199:p
9191:)
9187:p
9183:p
9179:p
9175:p
9167:)
9163:p
9159:p
9146:k
8942:)
8938:(
8884:)
8881:⌋
8878:A
8875:⌊
8867:)
8863:b
8859:b
8857:(
8850:)
8841:)
8838:y
8834:x
8826:)
8823:y
8819:x
8815:y
8811:x
8803:)
8799:n
8791:)
8787:n
8779:)
8770:)
8767:y
8763:x
8755:)
8746:)
8742:n
8740:4
8733:)
8728:n
8726:p
8718:)
8713:n
8711:p
8703:)
8699:n
8691:)
8687:k
8671:)
8662:)
8653:)
8626:e
8619:t
8612:v
8597:.
8578:.
8526:b
8522:b
8507:n
8504:(
8500:n
8498:M
8420:.
8387:y
8385:·
8383:d
8379:x
8374:q
8370:M
8360:)
8354:x
8349:q
8345:M
8295:)
8293:b
8289:a
8287:(
8271:c
8267:/
8263:)
8258:n
8254:b
8250:+
8245:n
8241:a
8237:(
8215:)
8213:b
8209:a
8207:(
8191:c
8187:/
8183:)
8178:n
8174:b
8165:n
8161:a
8157:(
8134:x
8128:x
8126:(
8112:x
8108:)
8106:x
8102:x
8100:(
8086:x
8082:)
8080:x
8076:x
8074:(
8060:x
8056:)
8054:x
8050:x
8048:(
8034:x
8028:x
8026:(
8012:x
8006:x
8004:(
7990:x
7984:x
7982:(
7976:x
7974:(
7962:.
7956::
7942:)
7924:.
7910::
7859:.
7837:.
7812:.
7787:.
7762:.
7742:.
7717:.
7696:.
7682::
7663:.
7624:.
7599:.
7547:.
7505:.
7477:.
7449:.
7423:.
7396:.
7372:.
7293:.
7287::
7239:.
7213:.
7198:.
7172:.
7130:b
7126:a
7122:/
7118:b
7114:a
7105:n
7092:r
7087:r
7083:b
7079:a
7068:.
7056:x
7050:x
7046:x
7041:)
7039:x
7035:x
7033:(
7029:)
7027:b
7023:a
7021:(
7017:)
7015:b
7011:a
7007:b
7003:a
6995:b
6991:a
6987:/
6983:b
6979:a
6963:.
6955:x
6953:)
6951:b
6947:a
6943:x
6931:b
6927:a
6923:)
6917:b
6913:a
6911:(
6908:n
6906:U
6782:)
6764:b
6760:b
6758:(
6754:b
6748:)
6730:b
6726:b
6724:(
6720:n
6712:b
6708:a
6701:b
6696:a
6691:b
6687:a
6682:n
6677:b
6673:b
6671:(
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