Variations on Giuga Numbers and Giuga’s Congruence

Abstract

A $k$ -strong Giuga number is a composite integer such that $∑_{j = 1}^{n − 1} j^{n − 1} ≡  − 1 (mod n)$. We consider the congruence $∑_{j = 1}^{n − 1} j^{k(n − 1)} ≡  − 1 (mod n)$ for each $k ϵ ℕ$ (thus extending Giuga’s ideas for $k = 1$). In particular, it is proved that a pair $(n, k)$ with composite n satisfies this congruence if and only if $n$ is a Giuga number and $⋋(n) | k(n − 1)$. In passing, we establish some new characterizations of Giuga numbers and study some properties of the numbers n satisfying $⋋(n) | k(n − 1)$.