Fermat test with gaussian base and Gaussian pseudoprimes

TL;DR

This paper explores Gaussian analogues of classical primality concepts, characterizing Gaussian Carmichael numbers and their relations with cyclic and Williams numbers, extending primality testing ideas into Gaussian integers.

Contribution

It introduces Gaussian Carmichael numbers, provides a Korselt's criterion for them, and studies their properties and relations with other pseudoprimes in the Gaussian integer setting.

Findings

Gaussian Carmichael numbers characterized via Korselt's criterion

No known composite Gaussian pseudoprimes less than 10^18 for specific bases

Relations established between Gaussian Carmichael numbers and Gaussian cyclic and Williams numbers

Abstract

The structure of the group $(\mathbb{Z}/n\mathbb{Z})^\star$ and Fermat's little theorem are the basis for some of 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 $\mathcal{G}_n:=\{a+bi\in\mathbb{Z}[i]/n\mathbb{Z}[i] : a^2+b^2\equiv 1\ \textrm{$\pmod n$}\}$. In particular we characterize Gaussian Carmichael numbers via a Korselt's criterion and we present their relation with Gaussian cyclic numbers. Finally, we present the relation between Gaussian Carmichael number and 1-Williams numbers for numbers $n \equiv 3 \pmod{4}$. There are also no known composite numbers less than $10^{18}$ in this family that are both pseudoprime to base $1+2i$ and 2-pseudoprime.