A converse of Fermat's Little Theorem

Abstract

As the name of the paper implies, a converse of Fermat's Little Theorem (FLT) is stated and proved. FLT states the following: if p is any prime, and x any integer, then xp ≡ x (mod p). There is already a well-known converse of FLT, known as Lehmer's Theorem, which is as follows: if x is an integer coprime with m, such that xm −1 ≡ 1 (mod m), and if there exists no integer e < m − 1 such that xe ≡ 1 (mod m), then m is prime. The new converse in question states the following: if p is any prime and xp ≡ x (mod p), where x is known only to be algebraic, then x must be an integer (mod p).