How many Mersenne Primes are currently known, and what are they?

Answer 1

There are currently 43 known Mersenne Primes (of the form 2^p-1, where p is prime). All numbers up to the 39th prime have been exhaustively searched (see the mersenne.org status page) but there are gaps beyond that so it is possible that there is/are more Mersenne Primes between some of the later numbers. I've denoted those with ? in the sequence numbering.

The numbers get pretty large, so after awhile all I can give you is the number in the form 2^p-1 and the number of digits in the number. You can go look up the long form if you really need it. :)

1) 2^2-1 = 3
2) 2^3-1 = 7
3) 2^5-1 = 31
4) 2^7-1 = 127
5) 2^13-1 = 8191
6) 2^17-1 = 131071
7) 2^19-1 = 524287
8) 2^31-1 = 2147483647
9) 2^61-1 = 2 305 843 009 213 693 951
10) 2^89-1 = 618 970 019 642 690 137 449 562 111
11) 2^107-1 = 162 259 276 829 213 363 391 578 010 288 127
12) 2^127-1 = 170 141 183 460 469 231 731 687 303 715 884 105 727
13) 2^521-1 = 157 digits
14) 2^607-1 = 183 digits
15) 2^1,279-1 = 386 digits
16) 2^2,203-1 = 664 digits
17) 2^2,281-1 = 687 digits
18) 2^3,217-1 = 969 digits
19) 2^4,253-1 = 1,281 digits
20) 2^4,423-1 = 1,332 digits
21) 2^9,689-1 = 2,917 digits
22) 2^9,941-1 = 2,993 digits
23) 2^11,213-1 = 3,376 digits
24) 2^19,937-1 = 6,002 digits
25) 2^21,701-1 = 6,533 digits
26) 2^23,209-1 = 6,987 digits
27) 2^44,497-1 = 13,395 digits
28) 2^86,243-1 = 25,962 digits
29) 2^110,503-1 = 33,265 digits
30) 2^132,049-1 = 39,751 digits
31) 2^216,091-1 = 65,050 digits
32) 2^756,839-1 = 227,832 digits
33) 2^859,433-1 = 258,716 digits
34) 2^1,257,787-1 = 378,632 digits
35) 2^1,398,269-1 = 420,921 digits
36) 2^2,976,221-1 = 895,932 digits
37) 2^3,021,377-1 = 909,526 digits
38) 2^6,972,593-1 = 2,098,960 digits
39) 2^13,466,917-1 = 4,053,946 digits
40?) 2^20,996,011-1 = 6,320,430 digits
41?) 2^24,036,583-1 = 7,235,733 digits
42?) 2^25,964,951-1 = 7,816,230 digits
43?) 2^30,402,457-1 = 9,152,052 digits

And if you want to know the English name of that 43rd known Mersenne prime, it starts, "three hundred fifteen tremillia millia quin quagin millia sescendo octo gintillion, four hundred sixteen tremillia millia quin quagin millia sescenun octo gintillion..." (See the last link for the full name.)

Answer 2

As of this writing, there are 43 known Mersenne Primes. They are:

1: 2 ^ 2 - 1
2: 2 ^ 3 - 1
3: 2 ^ 5 - 1
4: 2 ^ 7 - 1
5: 2 ^ 13 - 1
6: 2 ^ 17 - 1
7: 2 ^ 19 - 1
8: 2 ^ 31 - 1
9: 2 ^ 61 - 1
10: 2 ^ 89 - 1
11: 2 ^ 107 - 1
12: 2 ^ 127 - 1
13: 2 ^ 521 - 1
14: 2 ^ 607 - 1
15: 2 ^ 1,279 - 1
16: 2 ^ 2,203 - 1
17: 2 ^ 2,281 - 1
18: 2 ^ 3,217 - 1
19: 2 ^ 4,253 - 1
20: 2 ^ 4,423 - 1
21: 2 ^ 9,689 - 1
22: 2 ^ 9,941 - 1
23: 2 ^ 11,213 - 1
24: 2 ^ 19,937 - 1
25: 2 ^ 21,701 - 1
26: 2 ^ 23,209 - 1
27: 2 ^ 44,497 - 1
28: 2 ^ 86,243 - 1
29: 2 ^ 110,503 - 1
30: 2 ^ 132,049 - 1
31: 2 ^ 216,091 - 1
32: 2 ^ 756,839 - 1
33: 2 ^ 859,433 - 1
34: 2 ^ 1,257,787 - 1
35: 2 ^ 1,398,269 - 1
36: 2 ^ 2,976,221 - 1
37: 2 ^ 3,021,377 - 1
38: 2 ^ 6,972,593 - 1
39: 2 ^ 13,466,917 - 1
40: 2 ^ 20,996,011 - 1
41: 2 ^ 24,036,583 - 1
42: 2 ^ 25,964,951 - 1
43: 2 ^ 30,402,457 - 1

Note that the primes up to the 38th have been confirmed to be the lowest 38 Mersenne Primes, but those above it are not nessarily so. That is to say, additional Mersenne Primes may yet be discovered "in between" the numbers listed. This is an artifact of the methodology used to discover these primes, since Mersenne Primes #35 and up have all been discovered as part of the GIMPS (Great Internet Mersenne Prime Search) project. See http://www.mersenne.org for details.