A Note on Numbers

Abstract

A Fermat number is a number of the form F_n=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 and a Fermat prime is a prime Fermat number. Fermat composites and Fermat primes are characterized via divisibility in [4] and in [5]. 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 (E.). For every integer n > 0 such that n ≡ 1 mod [2], we have Fn-1 ≡ 4 mod[7]; and for every integer n ≥ 2, we have Fn−1 ≡ 1 mod[j], where j ∈ {3, 5}. Result (E.) immediately implies that there are infinitely many composite numbers of the form 2 + Fn. Result (E.) also implies that the only prime of the form 4 + Fn is 7 and the only primes of the form 8 + Fn are twin primes 11 and 13. That being said, using result (E.) and a special case of a Theorem of Dirichlet, we explain why it is natural to conjecture that there are infinitely many primes of the form 2 + Fn.