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,, 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.