On a system of equations with primes
Journal de Théorie des Nombres de Bordeaux, Tome 26 (2014) no. 2, pp. 399-413.

Étant donné un entier n3, soient u 1 ,...,u n des entiers 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 ε une fonction 𝒟{±1}. Est-ce qu’il existe au moins un nombre premier q tel que q divise le nombre iI u i -ε(I) pour un certain I𝒟, mais q ne divise pas u 1 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 ε et 𝒟.

Nous utilisons ce résultat pour prouver que, si ε 0 {±1} et A est un ensemble de trois ou plusieurs nombres premiers qui contient les diviseurs premiers de tous les nombres pB p-ε 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 n3, let u 1 ,...,u n be pairwise coprime integers 2, 𝒟 a family of nonempty proper subsets of {1,...,n} with “enough” elements, and ε a function 𝒟{±1}. Does there exist at least one prime q such that q divides iI u i -ε(I) for some I𝒟, but it does not divide u 1 u n ? We answer this question in the positive when the u i are prime powers and ε and 𝒟 are subjected to certain restrictions.

We use the result to prove that, if ε 0 {±1} and A is a set of three or more primes that contains all prime divisors of any number of the form pB p-ε 0 for which B is a finite nonempty proper subset of A, then A contains all the primes.

Reçu le : 2013-03-02
Accepté le : 2013-07-08
Publié le : 2015-03-09
DOI : https://doi.org/10.5802/jtnb.873
Classification : 11A05,  11A41,  11A51,  11D61,  11D79,  11R27
Mots clés: Agoh-Giuga conjecture, cyclic congruences, prime factorization, Pillai’s equation, Znam’s problem.
@article{JTNB_2014__26_2_399_0,
     author = {Paolo Leonetti and Salvatore Tringali},
     title = {On a system of equations with primes},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {399--413},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {26},
     number = {2},
     year = {2014},
     doi = {10.5802/jtnb.873},
     mrnumber = {3320486},
     language = {en},
     url = {jtnb.centre-mersenne.org/item/JTNB_2014__26_2_399_0/}
}
Paolo Leonetti; Salvatore Tringali. On a system of equations with primes. Journal de Théorie des Nombres de Bordeaux, Tome 26 (2014) no. 2, pp. 399-413. doi : 10.5802/jtnb.873. https://jtnb.centre-mersenne.org/item/JTNB_2014__26_2_399_0/

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