A generalization of Fermat's little theorem

Abstract

Fermat's Little Theorem states that if p is a prime number and gcd (x,p) = 1, then xp−1 ≡ 1 (modp) If the requirement that gcd (x,p) = 1 is dropped, we can say xp ≡ x(modp)for any integer x. Euler generalized Fermat's Theorem in the following way: if gcd (x,n) = 1 then xφ(n) ≡ 1(modn), where φ is the Euler phi-function. It is clear that Euler's result cannot be extended to all integers x in the same way Fermat's Theorem can; that is, the congruence xφ(n)+1 ≡ x(modn)is not always valid. In this note we show exactly when the congruence xφ(n)+1 ≡ x(modn) is valid.