Breaking Mersenne Numbers
# Lattice Siever, Mersenne Number (2^727)-1

### Dear Lattice Siever,

This is a project that breaks a 219-digit number, using
the Special Number Field Sieve (SNFS). This specific
Mersenne number, M727, is the smallest Mersenne number (with
prime exponent) for which there is no known prime factor. We
know that M727 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 a very extensive
effort to find small factors, estimated to be sufficient to
find an average prime factor with 50-digits, 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 M727 provides a test of the methods used to
break RSA-keys, and is roughly as hard as breaking an RSA-key with
460-bits (140-digits, such as RSA-140).

This calculation was not distributed.