Proofs, Generalizations and Applications of Fermat’s Little Theorem

Abstract

This paper introduces Fermat’s little theorem (FLT), which says that any integer raised to power is congruent to modulo . This paper will give several proofs of FLT, using methods including number theory and group theory, together with generalizations of FLT in different directions. FLT is an important result in number theory and group theory. It has multiple generalizations and corollaries, and one of its corollaries is the foundation of RSA cryptography. The effort made trying to prove FLT stimulated researches in many fields in mathematics, and FLT is crucial and fundamental in research of modern cryptography.