MM61

A search for a factor of 2261 − 1 − 1

It is not known whether the double-Mersenne number MM61 = 2261 − 1 − 1 is prime or composite, and, just as with other Mersenne numbers, it is interesting to resolve this question one way or another.

There is no hope of testing MM61 for primality. At 694127911065419642 digits it is far too large for the usual Lucas-Lehmer test. However, if MM61 is composite (and it very probably is), it might have a factor which is small enough to be discoverable. The MM61 project is a coordinated search for such a factor.

After many years my involvement in this project has ceased. I will continue to collect data from current assignments but I am no longer handing out ranges to test.

The MM61 project is now managed by Luigi Morelli, who will coordinate the running of new and far superior GPU-based software written by George Woltman.

If you have a PC equipped with an nVidia graphics card having computing capabilities ≥ 2.0 and are interested in helping to find a factor of 22p − 1 − 1 for p = 61, 89, 107 or 127, please go to http://www.doublemersennes.org.

Background

Let d be a prime divisor of MM61. We know that

d = N (261 − 1) + 1

for some N congruent to 0 or 2 (mod 8). We use a sieving process. We start with a long list of Ns and we reject all those Ns for which the corresponding d is divisible by a prime q = 3, 5, 7, 11, ... up to some suitable limit. For each N that survives we compute d and see if it divides into MM61. For this test we do not have to operate on MM61 itself. All calculations are done modulo d. We start with x = 2 and compute x2 (mod d) sixty-one times to obtain 2261 (mod d). If the final result is 2, then d divides MM61.

Of course, MM61 is not the only double-Mersenne number. The first four, MM2 = 7, MM3 = 127, MM5 = 2147483647, and MM7 = 170141183460469231731687303715884105727, are prime. The next four, MM13, MM17, MM19 and MM31 have known factors. And that's the extent of our knowledge. All other double-Mersenne numbers are 'status unknown', and MM61 happens to be the smallest. A number of people (including the author of this page) have tried finding divisors of MM61, MM89, MM107, MM127, MM521, MM607 and others, but so far without success.

Further Information

PROGRESS to September 2012

The double-Mersenne project: http://www.doublemersennes.org

A forum for discussions about Mersenne numbers: mersenneforum.org

General information about prime numbers: The Largest Known Primes

Mersenne numbers: The Great Internet Mersenne Prime Search

Double-Mersenne numbers: Will Edgington's pages

Fermat numbers: Wilfrid Keller's page

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Tony Forbes 6 September 2012.