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The structure of the group ${(\mathbb{Z}/n\mathbb{Z})}^{☆}$ and Fermat’s little theorem are the basis for some of the best-known primality testing algorithms. Many related concepts arise: Euler’s totient function and Carmichael’s lambda function, Fermat pseudoprimes, Carmichael and cyclic numbers, Lehmer’s totient problem, Giuga’s conjecture, etc. In this paper, we present and study analogues to some of the previous concepts arising when we consider the underlying group ${𝒢}_n:=\{a+b\mathrm{i}\in \mathbb{Z}\left[\mathrm{i}\right]/n\mathbb{Z}\left[\mathrm{i}\right]:a^2+b^2\equiv 1(modn)\}$. In particular, we characterize Gaussian Carmichael numbers...

This paper is devoted to the study of the volcanoes of ℓ-isogenies of elliptic curves over a finite field, focusing on their height as well as on the location of curves across its different levels. The core of the paper lies on the relationship between the ℓ-Sylow subgroup of an elliptic curve and the level of the volcano where it is placed. The particular case ℓ = 3 is studied in detail, giving an algorithm to determine the volcano of 3-isogenies of a given elliptic curve. Experimental results...