An Original note on Fermat numbers, on numbers of the form Wn and on numbers of the form 10k + 8 + Fn [ where Wn ∈ {22 + Fn, 2 n + Fn}, n is an integer ≥ 0, Fn is a Fermat number and k is an integer ≥ 0]

Abstract

A Fermat number is a number of the form Fn = 2^2^ n+ 1, where n is an integer ≥ 0. A Fermat composite (see [1] or [2] or [4] ) is a non prime Fermat number. Fermat composites and Fermat primes are characterized via divisibility in [4] and [5] (A Fermat prime (see [1] or [2] or [4] ) is a prime Fermat number). It is known (see [4]) that for every j ∈ {0, 1, 2, 3, 4}, Fj is a Fermat prime and it is also known (see [2] or [3]) that F5 and F6 are Fermat composites. In this paper, we show [via elementary arithmetic congruences] the following result (T.). For every integer n ≥ 2, Fn − 1 ≡ 1 mod[j] (where j ∈ {3, 5}). Result (T) immediately implies that for every fixed integer k ≥ 0, there exists at most two primes of the form 10k + 8 + Fn [in particular , for every fixed integer k ≥ 0, the numbers of the form 10k + 8 + Fn (where n is an integer ≥ 2) are all composites]. Result (T.) also implies that there are infinitely many composite numbers of the form 2n + Fn and there exists no prime number of form 22+Fn. Result (T.) coupled with a special case of a Theorem of Dirichlet help us to explain why it is natural to conjecture that there are infinitely many Fermat primes.