AI 中文总结
本文通过定义算术函数刻画Giuga数，推广Carmichael数至特定Fermat伪素数，并基于幂等性给出Agoh-Giuga猜想反例的新必要条件，推测无整数满足。
AI 中文摘要
本文分析了Giuga数和Carmichael数的幂等性。前几节将引导我们定义一个算术函数，通过该函数我们将找到Giuga数的一个新刻画。接下来的几节将引导我们推广Carmichael数，找到在某些特定情况下是Fermat伪素数的数，并针对其性质提出各种陈述和开放问题。最后，我们将基于幂等性找到一个数要成为Agoh-Giuga猜想的反例所必须满足的新替代条件，因此，该条件推测没有任何数会满足。我们几乎总是在n是无平方因子的前提下工作，因为这两类数总是具有该性质，但在某些情况下，我们会将部分结果推广到一般情形。
英文摘要
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.
