The Factorization of the Ninth Fermat Number

Abstract

In this paper we exhibit the full prime factorization of the ninth Fermat number <!-- MATH ${F_9} = {2^{512}} + 1$ --> . It is the product of three prime factors that have 7, 49, and 99 decimal digits. We found the two largest prime factors by means of the number field sieve, which is a factoring algorithm that depends on arithmetic in an algebraic number field. In the present case, the number field used was <!-- MATH ${\mathbf{Q}}(\sqrt[5]{2})$ --> . The calculations were done on approximately 700 workstations scattered around the world, and in one of the final stages a supercomputer was used. The entire factorization took four months.