A Short Note on Numbers of the Form K + Fn, Where K ? {2, 4, 8} and Fn is a Fermat Number

Abstract

A Fermat number is a number of the form , where n is an integer 0. A Fermat composite (see Dickson [1] or Hardy and Wright [2] or Ikorong [3]) 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 Ikorong [3] and in Ikorong [4]. It is known (see Ikorong [3]) that for every j ∈ {0, 1, 2, 3, 4}, Fj is a Fermat prime and it is also known (see Hardy and Wright [2] or Paul [5]) that F5 and F6 are Fermat composites 641×6700417, and since 2013, it is known that +1 is Fermat composite number). 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 on arithmetic progression, we conjecture that there are infinitely many primes of the form 2+Fn.