Euler's totient theorem

In number theory, Euler's totient theorem (also known as the Fermat–Euler theorem) states that if n and a are coprime, (meaning that the only number that divides n and a is 1), then the following equivalence relation holds:^[1]

${\displaystyle a^{\varphi (n)}\equiv 1{\pmod {n}}}$

where ${\displaystyle \varphi (n)}$ is Euler's totient function.

Euler's theorem is a more refined theorem of Fermat's little theorem, which Pierre de Fermat had published in 1640, a hundred years prior. Fermat's theorem remained unproven until the work of 18th-century Swiss mathematician Leonhard Euler.^[2]