A Note on Numbers

A Fermat number is a number of the form Fₙ=2^ (2ⁿ) +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 mod7; and for every integer n ≥ 2, we have Fn−1 ≡ 1 modj, 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.