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}:=\left \{ a+bi\in \mathbb{Z}\left [ i\right ]:a^{2}+b^{2}\equiv 1\left ( mod n \right ) \right \}. In particular, we characterize Gaussian Carmichael numbers via a Korselt’s criterion and present their relation with Gaussian cyclic numbers. Finally, we present the relation between Gaussian Carmichael number and 1-Williams numbers for numbers n ≡ 3 (mod 4). There are also no known composite numbers less than 1018 in this family that are both pseudoprime to base 1 + 2i and 2-pseudoprime., José María Grau, Antonio M. Oller-Marcén, Manuel Rodríguez, Daniel Sadornil., and Obsahuje seznam literatury
Let $d$ be a fixed positive integer. A Lucas $d$-pseudoprime is a Lucas pseudoprime $N$ for which there exists a Lucas sequence $U(P,Q)$ such that the rank of $N$ in $U(P,Q)$ is exactly $(N - \varepsilon (N))/d$, where $\varepsilon $ is the signature of $U(P,Q)$. We prove here that all but a finite number of Lucas $d$-pseudoprimes are square free. We also prove that all but a finite number of Lucas $d$-pseudoprimes are Carmichael-Lucas numbers.