Giuga number

Definition

A composite number $n$ is termed a Giuga number if and only if, for every prime number $p$ dividing $n$, we have that $p$ divides $(n/p)-1$, or equivalently, that $p^2$ divides $n-p$.

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.