# Giuga number

From Number

## Definition

A composite number is termed a **Giuga number** if and only if it satisfies the following equivalent conditions:

- For every prime number dividing , we have that divides , or equivalently, that divides .
- We have the following congruence:

where denotes the Euler totient function of .

Note that all prime numbers satisfy the stated condition but we deliberately exclude them by imposing the restriction of being composite.

## Facts

- Giuga number is square-free
- One of the equivalent formulations of the Agoh-Giuga conjecture is that there is no natural number that is both a
**Giuga number**and a Carmichael number.