Fermat’s Theorem And Its Consequences

Abstract Fermat’s theorem. In this chapter we apply the general ideas of Ch. V to the proof of a series of classical theorems, due mainly to Fermat, Euler, Legendre, and Gauss. Theorem 71 (Fermat’s theorem). If p is prime, and pf a, then The congruences (6.1.1) and (6.1.2) are equivalent when pf a; and (6.1.1) is trivial when p a, since then ap0 a. Hence Theorems 70 and 71 are equivalent. Theorem 71 is a particular case of the more general Theorem 72 (The Fermat–Euler theorem). If (a, m)= 1, then If xruns through a complete system of residues prime to m, then, by Theorem 58, axalso runs through such a system. Hence, taking the product of each set, we have Since every number xis prime to m, their product is prime to m; and hence, by Theorem 55,