Fermat's Little Theorem Calculator

Fermat's Little Theorem connects prime numbers, powers, and modular arithmetic. Its standard form says that if p is prime and a is not divisible by p, then a^(p - 1) leaves remainder 1 when divided by p. In congruence notation, a^(p - 1) is congruent to 1 modulo p. For a = 2 and p = 7, the calculation is 2^6 = 64, and 64 leaves remainder 1 after division by 7. An equivalent form states that a^p is congruent to a modulo p for every integer a when p is prime. This version also covers the case where p divides a, because both sides then have remainder zero. The standard form needs the greatest common divisor of a and p to equal 1. The calculator checks primality and this coprime condition before evaluating the selected congruence. Directly calculating a large power can produce an enormous intermediate number. Instead, the tool uses binary modular exponentiation, also called repeated squaring. It repeatedly squares the current base while reducing modulo p after every multiplication. This preserves the exact remainder, avoids unwieldy full powers, and needs only a logarithmic number of exponent steps. The method is fundamental in practical cryptographic software. The theorem can prove that a number is composite: if a coprime base fails the congruence, the proposed modulus cannot be prime. Passing one Fermat test, however, does not prove primality. Some composites pass for particular bases, and Carmichael numbers pass the standard test for every base coprime to them. Reliable primality testing therefore uses stronger deterministic checks or tests such as Miller-Rabin with carefully selected bases. Fermat's result supports many areas of number theory and computing. It helps reduce exponents in modular calculations, motivates primality tests, and contributes to the mathematical foundations of public-key cryptography. RSA relies more directly on Euler's related theorem, but Fermat's theorem explains the prime-modulus behavior at its core. Modular inverses can also be found with a^(p - 2) modulo p when p is prime and a is nonzero modulo p. Use this calculator to verify classroom examples and explore modular patterns, not to certify large cryptographic primes. Inputs are limited to JavaScript safe integers for reliable primality checking, while the exponentiation itself uses exact integer arithmetic. A successful result confirms that the chosen prime and base satisfy the selected theorem form; it does not by itself establish that an unchecked composite candidate is prime.