On the congruence ${\sum }_{j=1}^{n-1}{j}^{k(n-1)}\equiv -1(\mathrm{mod}n)$. k-strong Giuga and k-Carmichael numbers

Abstract

In this work we consider the congruence ${\sum }_{j=1}^{n-1}{j}^{k(n-1)}\equiv -1(\mathrm{mod}n)$ for each $k\in \mathbb{N}$, thus extending Giuga's ideas for $k=1$. In particular, it is proved that a pair $(n,k)\in {\mathbb{N}}^2$ satisfies this congruence if and only if $n$ is prime or a Giuga Number and $\lambda (n)\mid k(n-1)$. In passing, we establish new characterizations of Giuga numbers and we study some properties of the numbers $n$ satisfying $\lambda (n)\mid k(n-1)$.

arXiv commentarXiv admin note: text overlap with arXiv:1103.3483

References in corpus (2)

• The Equivalence of Giuga's and Agoh's Conjectures
• Giuga Numbers and the arithmetic derivative