Variations on Giuga Numbers and Giuga’s Congruence - Grau, José María (Ukrains’kyi Matematychnyi Zhurnal, 2015)
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).