A prime number of the form \( M_p = 2^p-1 \)
where \(p\) is prime.
- The Mersenne number \( M_p \) is prime for exponent \(p\) =
| 2 |
3 |
5 |
7 |
13 |
| 17 |
19 |
31 |
61 |
89 |
| 107 |
127 |
521 |
607 |
1279 |
| 2203 |
2281 |
3217 |
4253 |
4423 |
| 9689 |
9941 |
11 213 |
19 937 |
21 701 |
| 23 209 |
44 497 |
86 243 |
110 503 |
132 049 |
| 216 091 |
756 839 |
859 433 |
1 257 787 |
1 398 269 |
| 2 976 221 |
3 021 377 |
6 972 593 |
13 466 917 |
20 996 011 |
| 24 036 583 |
25 964 951 |
30 402 457 |
32 582 657 |
37 156 667 |
| 42 643 801 |
43 112 609 |
57 885 161 |
... |
74 207 281 |
| ... |
77 232 917 |
... |
82 589 933 |
... |
For all other prime exponents less than
66,121,409
\( M_p \) is known to be composite (tested and double-checked), and for those less than
114,290,467 it is probably composite (tested at least once). (As of 12 Dec 2023)
... it is the
greatest that will ever be discovered for, as they are merely curious
without being useful, it is not likely that any person will attempt to
find one beyond it.
--
Peter
Barlow, 1811, on M31 = 2,147,483,647
- Mersenne numbers have a particulary simple test for primality, the
Lucas-Lehmer test.
- The number-theoretic interest in Mersenne primes comes from the
following theorem: if \(m \) and \(n \) are natural numbers,
with \(n \) greater than \(1 \), and if \(m^n-1 \) is prime, then \(m \) is \(2 \) and \(n \) is prime.
$$
n>1 \land m^n \in \mbox{prime} \implies m=2 \land n \in \mbox{prime}
$$
Marin Mersenne
References and reviews
Factors of Mersenne Numbers
Not all the non-prime Mersennes have been completely factored. Some
factoring data is
- M11 = 2,047 = 23
× 89
- M23 = 8,388,607 =
47 × 178,481
- M29 = 233 × 1,103 × 2,089
- M37 = 223 × 616,318,177
- M41 = 13,367 × 164,511,353
- M43 = 431 × 9,719 ×
2,099,863
- M47 = 2,351 × 4,513 ×
13,264,529
- M53 = 6,361 × 69,431 ×
20,394,401
- M59 = 179,951 × 3,203,431,780,337
- M67 = 193,707,721 ×
761,838,257,287
- M71 = 228,479 × 48,544,121 ×
212,885,833
- M73 = 439 × 2,298,041 ×
9,361,973,132,609
- M79 = 2,687 × 202,029,703 ×
1,113,491,139,767
- M83 = 167 ×
57,912,614,113,275,649,087,721
- M97 = 11,447 ×
13,842,607,235,828,485,645,766,393
- M101 = 7,432,339,208,719 × ...
- M103 = 2,550,183,799 × ...
- M109 = 745,988,807 × ...
- M113 = 3,391 × 23,279 ×
65,993 × 1,868,569 × ...
- 2125-1 = 31 × 601 × 1801
× 269,089,806,001 × 4,710,883,168,879,506,001
- M131 = 263 × ...
- M137 = 32,032,215,596,496,435,569 ×
5,439,042,183,600,204,290,159
- M139 = 5,625,767,248,687 × ...
- M149 = 86,656,268,566,282,183,151 ×
...
- M151 = 18,121 × 55,871 ×165,799
× 2,332,951 × ...
- M157 = 852,133,201 ×
60,726,444,167 × 1,654,058,017,289 × ...
- M163 = 150,287 × 704,161 ×
110,211,473 × 27,669,118,297 × ...
- M167 = 2,349,023 × ...
- M173 = 730,753 × 1,505,447 ×
70,084,436,712,553,223 × ...
- M179 = 359 × 1,433 ×...
- M181 = 43,441 × 1,164,193 ×
7,648,337 × ...
- M191 = 383 × 7,068,569,257 ×
39,940,132,241 × 332,584,516,519,201 × ...
- M193 = 13,821,503 ×
61,654,440,233,248,340,616,559 × ...
- M197 = 7,487 × ...
- M199 = 164,504,919,713 × ...
- M211 = 15,193 ×
60,272,956,433,838,849,161 × ...
- M223 = 18,287 × 196,687 ×
1,466,449 × 2,916,841 × 1,469,495,262,398,780,123,809 ×
...
- M227 = 26,986,333,437,777,017 ×
...
- M229 = 1,504,073 × 20,492,753 ×
59,833,457,464,970,183 × ...
- M233 = 1,399 × 135,607 ×
622,577 × ...
- M239 = 479 × 1,913 × 5,737 ×
176,383 × 134,000,609 ×
7,110,008,717,824,458,123,105,014,279,253,754,096,863,768,062,879
- M241 = 22,000,409 × ...
- M251 = 503 × 54217 ×
178,230,287,214,063,289,511 × 61,676,882,198,695,257,501,367 ×
...
- M257 = 535,006,138,814,359 ×
1,155,685,395,246,619,182,673,033 × ...
- M263 = 23,671 ×
13,572,264,529,177 × 120,226,360,536,848,498,024,035,943 ×
...
More
of this factorisation data, including the known information
for all exponents less than 200,000. Some other data taken from
Robert
Munafo's Large Number Notes page.