Generalizing Giuga's conjecture

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In 1950 G. Giuga studied the congruence ∑ⱼ₌₁ⁿ⁻¹ jⁿ⁻¹ ≡ -1 (mod n) and conjectured that it was only satisfied by prime numbers. In this work we generalize Giuga's ideas considering, for each k ∈ N, the congruence ∑ⱼ₌₁ⁿ⁻¹ jᵏ⁽ⁿ⁻¹⁾ ≡ -1 (mod n). It particular, it is proved that a pair (n,k)∈ N² (with composite n) satisfies the congruence if and only if n is a Giuga Number and λ(n)/(λ(n),n-1) divides k. In passing, we establish some new characterizations of Giuga Numbers.