##  [Euler's Theorem](/eulers-theorem) 

  ##  [Euler's Theorem](https://puremath.quantumdictionary.io/eulers-theorem-0) 

  

 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A number-theoretic result: for integers a and n with gcd(a,n)=1, a^{φ(n)} ≡ 1 (mod n), where φ(n) is Euler's totient function giving the order of the unit group (Z/nZ)× when counted multiplicatively.

 

 

 

 

 





 

 



 ##  [Euler's Theorem](https://algebra.quantumdictionary.io/eulers-theorem-1) 

  

 [![Algebra](/sites/default/files/styles/large/public/2026-01/Algebra.png.webp?itok=3pHxBnUF)](/topic-specific-dictionaries/pure-mathematics/algebra)

- Pure Mathematics -

**Algebra Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A generalization of Fermat's result: for integer n≥1 and integer a with gcd(a,n)=1, a^{φ(n)} ≡ 1 (mod n), where φ(n) is Euler's totient function counting units modulo n.