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...

The Fermat equation is solved in integral two by two matrices of determinant one as well as in finite order integral three by three matrices.

Let Fₙ be the Fibonacci sequence defined by F₀=0, F₁=1, $F_{n+1}=Fₙ+F_{n-1}\left(n\ge 1\right)$. It is well known that $F_{p-(5/p)}\equiv 0\left(modp\right)$ for any odd prime p, where (-) denotes the Legendre symbol. In 1960 D. D. Wall [13] asked whether $p²|F_{p-(5/p)}$ is always impossible; up to now this is still open. In this paper the sum ${\sum }_{k\equiv r\left(mod10\right)}\left(\genfrac{}{}{0pt}{}{n}{k}\right)$ is expressed in terms of Fibonacci numbers. As applications we obtain a new formula for the Fibonacci quotient $F_{p-(5/p)}/p$ and a criterion for the relation $p|F_{(p-1)/4}$ (if p ≡ 1 (mod 4), where p ≠ 5 is an odd prime. We also prove that the affirmative answer to...

We give an automata-theoretic description of the algebraic closure of the rational function field ${𝔽}_q\left(t\right)$ over a finite field ${𝔽}_q$, generalizing a result of Christol. The description occurs within the Hahn-Mal’cev-Neumann field of “generalized power series” over ${𝔽}_q$. In passing, we obtain a characterization of well-ordered sets of rational numbers whose base $p$ expansions are generated by a finite automaton, and exhibit some techniques for computing in the algebraic closure; these include an adaptation to positive...

Let g ≥ 2 be an integer and ${}_g\subset \mathbb{N}$ be the set of repdigits in base g. Let ${}_g$ be the set of Diophantine triples with values in ${}_g$; that is, ${}_g$ is the set of all triples (a,b,c) ∈ ℕ³ with c < b < a such that ab + 1, ac + 1 and bc + 1 lie in the set ${}_g$. We prove effective finiteness results for the set ${}_g$.

Ce papier présente les récents progrès concernant les fonctions zêta des hauteurs associées à la conjecture de Manin. En particulier, des exemples où on peut prouver un prolongement méromorphe de ces fonctions sont détaillés.

We present a density result for the norm of the fundamental unit in a real quadratic order that follows from an equidistribution assumption for the infinite Frobenius elements in the class groups of these orders.

A set of m positive integers with the property that the product of any two of them is the predecessor of a perfect square is called a Diophantine m-tuple. Much work has been done attempting to prove that there exist no Diophantine quintuples. In this paper we give stringent conditions that should be met by a putative Diophantine quintuple. Among others, we show that any Diophantine quintuple a,b,c,d,e with a < b < c < d < e$satisfies$d < 1.55·1072$and$b < 6.21·1035$when4a<b,whileforb<4aonehaseither$c = a + b + 2√(ab+1) and $d<1.96·{10}^{53}$...