Lenstra, H. W.Manasse, M. S.Pollard, J. M.Lenstra, Arjen K.2011-03-282011-03-282011-03-28199310.1090/S0025-5718-1993-1182953-4https://infoscience.epfl.ch/handle/20.500.14299/65681In this paper we exhibit the full prime factorization of the ninth Fermat number F9 = 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 Q(fifth-root 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.Fermat NumberFactoring AlgorithmFactoring AlgorithmPrimalitytext::journal::journal article::research article