Generalizing Giuga's conjecture

TL;DR

This paper extends Giuga's conjecture by exploring a generalized congruence involving powers and characterizes composite numbers satisfying it, linking them to Giuga Numbers and their properties.

Contribution

It introduces a generalized congruence framework for Giuga's conjecture and characterizes composite solutions in terms of Giuga Numbers and divisibility conditions.

Findings

A pair (n,k) satisfies the generalized congruence iff n is a Giuga Number and a divisibility condition holds.

New characterizations of Giuga Numbers are established.

The work links the generalized congruence to properties of the Carmichael function.

Abstract

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)/\gcd(\lambda(n),n-1)$ divides $k$. In passing, we establish some new characterizations of Giuga Numbers.