The Twenty-Second Fermat Number is Composite

Abstract

Abstract. We have shown by machine proof that F22 = 22 +1 is composite. In addition, we reenacted Young and Buell's 1988 resolution of F20 as composite, finding agreement with their final Selfridge-Hurwitz residues. We also resolved the character of all extant cofactors of Fn , n < 22, finding no new primes, and ruling out prime powers.

Theorems and Open Problems That Concern Decidable Sets

“…Theorem 4. Conjecture 1 proves the following implication: if there exists an integer x 1 2 such that x 2 1 + 1 is prime and greater than f (7), then the set P n 2 +1 is infinite. Proof.…”