A number-theoretic statement that for a prime p and integer a, a^p ≡ a (mod p); equivalently, if a is not divisible by p then a^(p-1) ≡ 1 (mod p).

Principle

Arises from the fact that the multiplicative group of integers modulo a prime p is cyclic of order p−1, so exponentiation respects group order leading to these congruences.

Demonstration

For p=7 and a=3, compute 3^6 = 729 ≡ 1 (mod 7), verifying the theorem; for a multiple of p the congruence a^p ≡ a is trivially true.

Misapplication

Using the theorem to certify primality without checking composites that satisfy the congruence for many bases (Carmichael numbers) or applying the reduced-form congruence modulo a composite modulus without conditions.

Consequence

Provides a fast modular exponentiation identity used in primality tests, modular inverses computation when modulus is prime, and in many algorithms in computational number theory and cryptography.

Reversal

Euler's theorem generalizes this statement to composite moduli by replacing p−1 with Euler's totient φ(n); reversing the statement weakens to group-order statements in arbitrary finite rings.

Boundary

Valid only when the modulus is prime for the simple exponent p or for integers coprime to p in the p−1 exponent form; does not imply primality nor hold for arbitrary composite moduli.

Semantic Tension

Competes conceptually with Euler's theorem and Carmichael counterexamples: Fermat's congruence is a strong, easy-to-check property for primes but insufficient alone for primality testing in the presence of pseudoprimes.

Synthesis

A foundational congruence linking prime moduli and exponentiation: powers of integers reduce predictably modulo primes because of the finite multiplicative group's order.