Idempotence in Carmichael and Giuga numbers and its application to the Agoh-Giuga conjecture

Source abstract

This article analyzes the idempotence of Giuga numbers and Carmichael numbers. The first sections will lead us to define an arithmetic function through which we will find a new characterization of Giuga numbers. The following sections will lead us to generalize Carmichael numbers, finding numbers that are Fermat pseudoprimes in certain specific cases, and to formulate various statements and open questions regarding their properties. Finally, we will find a new alternative condition, based on idempotence, that a number must satisfy to be a counterexample to the Agoh-Giuga conjecture and which, therefore, presumably, no number will ever satisfy. We will almost always work under the premise that n is square-free, since both numbers always have that property, but in certain situations we will extend some of the results to the general case.