# 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]

## References

1. "Euler's Totient Function and Euler's Theorem". www.doc.ic.ac.uk. Retrieved 2021-04-12.
2. "Art of Problem Solving". artofproblemsolving.com. Retrieved 2021-04-12.