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,, 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.