Six Primes in Generalized Fermat Numbers Gerold Br¨andli and J¨org Waldvogel November 28, 2023 1 Introduction and Definitions This section refreshes some basic knowledge about the Fermat numbers and defines the generalization used in this paper. The mathematician Pierre de Fermat introduced the integers F n =2 2 n +1(n ≥ 0) and conjectured them all to be prime. Euler found a divisor of F 5 , namely 641. Since then the Fermat numbers beyond the first five primes {3, 5, 17, 257, 65537} have been widely studied. The existence of a sixth prime is very improbable ([2]), but its non- existence is not yet proven. There exist several ways to generalize the Fermat numbers. A simple one is described by Dubner and Gallot [4] as Definition 1.1. Generalized Fermat Numbers (GFN) are defined as F b,n = b 2 n +1, b positive integer and n ≥ 0. In the search for primes one can additionally require b to be even and not to be the square of a lesser b, to avoid a double visit of the same GFN. This definition of GFN includes with b = 2 the Fermat primes. In the search for very large primes, Mersenne numbers of the form 2 n - 1 and Fermat numbers are in the focus of the specialists. During many years Mersenne numbers were easier to factorize. In 1994, R. Crandall and B. Fagin discovered the Discrete Weighted Transforms to speed up the multiplication and applied it to Mersenne numbers. In 1998, Y. Gallot remarked that the new method is also applicabale to Fermat numbers and even to GFN [5]. Since then many new large primes have been found [9]. At present the largest known primes are Mersenne numbers, followed by the not so known Proth numbers of form k 2 e + 1 [8], and then by GFN. The focus of this paper is to count the number of primes in GFN for fixed b and it shows what can be done using more traditional software. The F b,n have – like the Fermat numbers – a double exponentiation, they grow very fast. The writing b 2 n + 1 is used instead of more general b e + 1 with integer e> 0 to skip composites on the search for prime F b,n : 1