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 57,179,977 \( M_p \) is known to be composite (tested and double-checked), and for those less than 104,286,673 it is probably composite (tested at least once). (As of 5 Sept 2021)

... 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 *M*_{31} = 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.
$$ n>1 \land m^n \in \mbox{prime} \implies m=2 \land n \in \mbox{prime} $$

- Each Mersenne prime corresponds to an even perfect number.

- The GREAT Internet Mersenne Prime Search -- help find another Mersenne prime!
- Chris Caldwell's Mersenne Primes page -- history, lists, theorems, conjectures, ...
- Luke Welsh's Marin Mersenne page -- biographies, prime number lists, algorithms, bibliography, ...

- A short biography of Marin Mersenne
- Mersenne on the Moon

- Conway and Guy.
*The Book of Numbers*. chapter 5 - Giblin.
*Primes and Programming*. section 9.3 - Hardy & Wright.
*Theory of Numbers*. section 2.5 - Stepney. Prime Candidates

Not all the non-prime Mersennes have been completely factored. Some factoring data is

- M
_{11}= 2,047 = 23 × 89 - M
_{23}= 8,388,607 = 47 × 178,481 - M
_{29}= 233 × 1,103 × 2,089 - M
_{37}= 223 × 616,318,177 - M
_{41}= 13,367 × 164,511,353 - M
_{43}= 431 × 9,719 × 2,099,863 - M
_{47}= 2,351 × 4,513 × 13,264,529 - M
_{53}= 6,361 × 69,431 × 20,394,401 - M
_{59}= 179,951 × 3,203,431,780,337 - M
_{67}= 193,707,721 × 761,838,257,287 - M
_{71}= 228,479 × 48,544,121 × 212,885,833 - M
_{73}= 439 × 2,298,041 × 9,361,973,132,609 - M
_{79}= 2,687 × 202,029,703 × 1,113,491,139,767 - M
_{83}= 167 × 57,912,614,113,275,649,087,721 - M
_{97}= 11,447 × 13,842,607,235,828,485,645,766,393 - M
_{101}= 7,432,339,208,719 × ... - M
_{103}= 2,550,183,799 × ... - M
_{109}= 745,988,807 × ... - M
_{113}= 3,391 × 23,279 × 65,993 × 1,868,569 × ... *2*^{125}-1 = 31 × 601 × 1801 × 269,089,806,001 × 4,710,883,168,879,506,001- M
_{131}= 263 × ... - M
_{137}= 32,032,215,596,496,435,569 × 5,439,042,183,600,204,290,159 - M
_{139}= 5,625,767,248,687 × ... - M
_{149}= 86,656,268,566,282,183,151 × ... - M
_{151}= 18,121 × 55,871 ×165,799 × 2,332,951 × ... - M
_{157}= 852,133,201 × 60,726,444,167 × 1,654,058,017,289 × ... - M
_{163}= 150,287 × 704,161 × 110,211,473 × 27,669,118,297 × ... - M
_{167}= 2,349,023 × ... - M
_{173}= 730,753 × 1,505,447 × 70,084,436,712,553,223 × ... - M
_{179}= 359 × 1,433 ×... - M
_{181}= 43,441 × 1,164,193 × 7,648,337 × ... - M
_{191}= 383 × 7,068,569,257 × 39,940,132,241 × 332,584,516,519,201 × ... - M
_{193}= 13,821,503 × 61,654,440,233,248,340,616,559 × ... - M
_{197}= 7,487 × ... - M
_{199}= 164,504,919,713 × ... - M
_{211}= 15,193 × 60,272,956,433,838,849,161 × ... - M
_{223}= 18,287 × 196,687 × 1,466,449 × 2,916,841 × 1,469,495,262,398,780,123,809 × ... - M
_{227}= 26,986,333,437,777,017 × ... - M
_{229}= 1,504,073 × 20,492,753 × 59,833,457,464,970,183 × ... - M
_{233}= 1,399 × 135,607 × 622,577 × ... - M
_{239}= 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 - M
_{241}= 22,000,409 × ... - M
_{251}= 503 × 54217 × 178,230,287,214,063,289,511 × 61,676,882,198,695,257,501,367 × ... - M
_{257}= 535,006,138,814,359 × 1,155,685,395,246,619,182,673,033 × ... - M
_{263}= 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.