THE FACTORIZATION OF THE NINTH FERMAT NUMBER

Dedicated to the nietnori vi D. II Lehiner ABsTRAcT. In this paper we exhibit the full prime factorization of the ninth Fermat number F9 = 2512 + I. 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 Q(/). The calculations were done on approximately 700 worksta tions scattered around the world, and in one of the final stages a supercomputer was used. The entire factorization took four months