It is not known whether the double-Mersenne number
MM_{61} = 2^{261 − 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 MM_{61} for primality. At
694127911065419642 digits it is far too large for the usual Lucas-Lehmer test.
However, if MM_{61} 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
2 ^{2p − 1} − 1 for p = 61, 89, 107 or 127**, please go to
http://www.doublemersennes.org.

Let *d* be a prime divisor of MM_{61}. We know that

*d* = *N* (2^{61} − 1) + 1

for some *N* congruent to 0 or 2 (mod 8). We use a sieving process. We
start with a long list of *N*s and we reject all those *N*s 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 MM_{61}. For this test we do not have
to operate on MM_{61} itself. All calculations are done modulo *d*.
We start with *x* = 2 and compute *x*^{2} (mod *d*) sixty-one times to
obtain 2^{261} (mod *d*). If the final result is 2, then *d*
divides MM_{61}.

Of course, MM_{61} is not the only double-Mersenne number. The first
four, MM_{2} = 7, MM_{3} = 127, MM_{5} = 2147483647, and
MM_{7} = 170141183460469231731687303715884105727, are prime.
The next four, MM_{13}, MM_{17}, MM_{19} and
MM_{31} have known factors. And that's the
extent of our knowledge. All other double-Mersenne numbers are 'status unknown',
and MM_{61} happens to be the smallest. A number of people (including the
author of this page) have tried finding divisors of MM_{61}, MM_{89},
MM_{107}, MM_{127}, MM_{521}, MM_{607} and others, but so
far without success.

**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

***

Tony Forbes 6 September 2012.