Fermat’s little theorem is a cornerstone of number theory and cryptography. It powers primality tests (if a^(n−1) mod n ≠ 1, n is definitely composite) and explains why RSA decryption works.
How to use the Fermat’s Little Theorem Calculator
- a
- p (prime)
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
The calculator first checks that p is prime, then computes the powers with fast modular exponentiation, so even large exponents are handled exactly.
Formula
p prime: aᵖ ≡ a (mod p); if p ∤ a: a^(p−1) ≡ 1 (mod p)
Worked example
Inputs — a: 3; p (prime): 7.
Results — Theorem applied = a and p are coprime, so a^(p−1) ≡ 1 (mod p); aᵖ mod p = 3⁷ ≡ 3 ≡ 3 (mod 7); a^(p−1) mod p = 3⁶ ≡ 1 (mod 7).
Key terms
- Prime
- A whole number greater than 1 with no divisors other than 1 and itself.
Frequently asked questions
What does the theorem say?
For a prime p, aᵖ − a is divisible by p; if p does not divide a, then a^(p−1) leaves remainder 1.
Does the converse hold?
No – Carmichael numbers such as 561 pass the test without being prime.
How is it used in practice?
In the Fermat primality test and in RSA.
What if p divides a?
Then a^(p−1) ≡ 0 (mod p), but aᵖ ≡ a still holds.