Factors of generalized Fermat numbers

Abstract

Generalized Fermat numbers have the form F b , m = b 2 m + 1 {F_{b,m}} = {b^{{2^m}}} + 1 . Their odd prime factors are of the form k ⋅ 2 n + 1 k \cdot {2^n} + 1 , k odd, n > m n > m . It is shown that each prime is a factor of some F b , m {F_{b,m}} for approximately 1 / k 1/k bases b, independent of n. Divisors of generalized Fermat numbers of base 6, base 10, and base 12 are tabulated. Three new factors of standard Fermat numbers are included.