Generalizing Giuga's conjecture

In 1950 G. Giuga studied the congruence ${\sum }_{j=1}^{n-1}{j}^{n-1}\equiv -1$ (mod $n$) and conjectured that it was only satisfied by prime numbers. In this work we generalize Giuga’s ideas considering, for each $k\in \mathbb{N}$, the congruence ${\sum }_{j=1}^{n-1}{j}^{k(n-1)}\equiv -1$ (mod $n$). It particular, it is proved that a pair $(n,k)\in {\mathbb{N}}^2$ (with composite $n$) satisfies the congruence if and only if $n$ is a Giuga Number and $\lambda (n)\mathrm{/}\gcd (\lambda (n),n-1)$ divides $k$. In passing, we establish some new characterizations of Giuga Numbers.