On a system of equations with primes

Leonetti, Paolo; Tringali, Salvatore

Journal de théorie des nombres de Bordeaux, Tome 26 (2014) no. 2, pp. 399-413.

Étant donné un entier $n\ge 3$, soient $u_1,...,u_n$ des entiers $\ge 2$ et premiers entre eux deux à deux, soit $𝒟$ une famille de sous-ensembles propres et non vides de $\{1,...,n\}$ qui contient un nombre “suffisant” des éléments, et soit $\epsilon$ une fonction $𝒟\to \{\pm 1\}$. Est-ce qu’il existe au moins un nombre premier $q$ tel que $q$ divise le nombre ${\prod }_{i\in I}{u}_i-\epsilon \left(I\right)$ pour un certain $I\in 𝒟$, mais $q$ ne divise pas $u_1\cdots u_n$ ? Nous donnons une réponse positive à cette question dans le cas où les $u_i$ sont des puissances de nombres premiers et on impose certaines restrictions sur $\epsilon$ et $𝒟$.

Nous utilisons ce résultat pour prouver que, si ${\epsilon }_0\in \{\pm 1\}$ et $A$ est un ensemble de trois ou plusieurs nombres premiers qui contient les diviseurs premiers de tous les nombres ${\prod }_{p\in B}p-{\epsilon }_0$ pour lesquels $B$ est un sous-ensemble propre, fini et non vide de $A$, alors $A$ contient tous les nombres premiers.

Given an integer $n\ge 3$, let $u_1,...,u_n$ be pairwise coprime integers $\ge 2$, $𝒟$ a family of nonempty proper subsets of $\{1,...,n\}$ with “enough” elements, and $\epsilon$ a function $𝒟\to \{\pm 1\}$. Does there exist at least one prime $q$ such that $q$ divides ${\prod }_{i\in I}{u}_i-\epsilon \left(I\right)$ for some $I\in 𝒟$, but it does not divide $u_1\cdots u_n$? We answer this question in the positive when the $u_i$ are prime powers and $\epsilon$ and $𝒟$ are subjected to certain restrictions.

We use the result to prove that, if ${\epsilon }_0\in \{\pm 1\}$ and $A$ is a set of three or more primes that contains all prime divisors of any number of the form ${\prod }_{p\in B}p-{\epsilon }_0$ for which $B$ is a finite nonempty proper subset of $A$, then $A$ contains all the primes.

Classification : 11A05, 11A41, 11A51, 11D61, 11D79, 11R27
Mots clés : Agoh-Giuga conjecture, cyclic congruences, prime factorization, Pillai’s equation, Znam’s problem.

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