A Curious Proof of Fermat's Little Theorem

Abstract

Fermat's little theorem states that for p prime and a e Z, p divides ap a. This result is of huge importance in elementary and algebraic number theory. For instance, with its help we obtain the so-called Frobenius automorphism of a finite field ¥pn over ¥p. This theorem has many interesting and sometimes unexpected proofs. One classical proof is based upon properties of binomial coefficients. In fact, (d + ')p dp 1 = EfJi1 {')#. Since (p) = jr^ is divisible by p for 1 < i < p 1, then {d + 1) dp 1 is divisible by p. Summing this over d = 1 , 2, . . . , a 1 , we obtain the desired result. Another classical proof is based upon Lagrange's theorem, which states that the order of an element of a finite group divides the group order. Applying this theorem to the multiplicative group of a finite field ¥p we obtain the result immediately. Several other proofs can be found at [2]. Nevertheless, in all of these proofs one or another analogue of the Euclidean algorithm (hence arithmetic) is being used. In this short note we present a proof which was found as a side result of another, unrelated problem (which is the case, maybe, with many such curious proofs). Surprisingly, arithmetic, group theory, and the properties of binomial coefficients do not manifest at all.