What is Fermat's little theorem?

Fermat's little theorem is a fundamental result in elementary number theory stating that if $p$ is a prime number, the $p$-th power of any integer $a$ is congruent to $a$ modulo $p$. Named after Pierre de Fermat, who stated it in 1640 without proof, it forms the basis for primality tests. Leonhard Euler later published the first known proof of the theorem in 1736.

What we know

• For any prime number $p$ and integer $a$, the number $a^p - a$ is an integer multiple of $p$. [2]
• If $a$ is coprime to $p$, the theorem is equivalent to $a^{p-1} \equiv 1 \pmod p$. [2]
• Pierre de Fermat stated the theorem in 1640. [2]
• Leonhard Euler published the first proof of Fermat's little theorem in 1736. [4]

Where to go next

• whoWho was Pierre de Fermat?
• the bigger pictureWhat is Euler's theorem?
• what followedHow does the Fermat primality test work?
• how we knowWhat are some proofs of Fermat's little theorem?
• compared withWhat is Fermat's Last Theorem?
• an unexpected connectionHow does Euler's totient function connect to modular arithmetic?Euler generalized Fermat's little theorem into a broader theorem using his totient function for numbers that are not necessarily prime.

Sources

• [1]Fermat's little theorem (Wikidata Q188295) · CC0 1.0
• [2]Fermat's little theorem · CC BY-SA 4.0
• [3]Fermat's theorem · CC BY-SA 4.0
• [4]Euler's theorem · CC BY-SA 4.0
• [5]Proofs of Fermat's little theorem · CC BY-SA 4.0