Breaking Mersenne Numbers
# Lattice Siever, Mersenne Number (2^751)-1

### Dear Lattice Siever,

This is a project that breaks a 227-digit number, using
the Special Number Field Sieve (SNFS). This specific
Mersenne number, M751, is the smallest Mersenne number (with
prime exponent) for which there is no known prime factor. We
know that M751 is composite by the Lucas-Lehmer test, used
to find Mersenne primes (in the Great Internet Mersenne Prime
Search, www.mersenne.org, for example). But an extensive
effort to find small factors was unsuccessful.
The program collects data used to build a large matrix, precisely
the same method used to break RSA-keys. Building and solving the
matrix problem for M751 provides a test of the methods used to
break RSA-keys, and is roughly as hard as breaking an RSA-key with
150-digits. Depending upon the results
of preliminary work presently in progress, we may continue with
the next RSA Challenge number, RSA-576
(cf.
The New RSA Factoring Challenge ).

This calculation is in a preliminary stage, not yet ready to
distribute.