# On a system of equations with primes

Paolo Leonetti^{[1]}; Salvatore Tringali^{[2]}

- [1] Università Bocconi via Sarfatti 25 20100 Milan, Italy
- [2] Texas A&M University at Qatar PO Box 23874 Education City DOHA, 5825 QATAR

Journal de Théorie des Nombres de Bordeaux (2014)

- Volume: 26, Issue: 2, page 399-413
- ISSN: 1246-7405

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topLeonetti, Paolo, and Tringali, Salvatore. "On a system of equations with primes." Journal de Théorie des Nombres de Bordeaux 26.2 (2014): 399-413. <http://eudml.org/doc/275688>.

@article{Leonetti2014,

abstract = {Given an integer $n \ge 3$, let $u_1, \ldots , u_n$ be pairwise coprime integers $\ge 2$, $\mathcal\{D\}$ a family of nonempty proper subsets of $\lbrace 1, \ldots , n\rbrace $ with “enough” elements, and $\varepsilon $ a function $ \mathcal\{D\} \rightarrow \lbrace \pm 1\rbrace $. Does there exist at least one prime $q$ such that $q$ divides $\prod _\{i \in I\} u_i - \varepsilon (I)$ for some $I \in \mathcal\{D\}$, 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 $\varepsilon $ and $\mathcal\{D\}$ are subjected to certain restrictions.We use the result to prove that, if $\varepsilon _0 \in \lbrace \pm 1\rbrace $ and $A$ is a set of three or more primes that contains all prime divisors of any number of the form $\prod _\{p \in B\} p - \varepsilon _0$ for which $B$ is a finite nonempty proper subset of $A$, then $A$ contains all the primes.},

affiliation = {Università Bocconi via Sarfatti 25 20100 Milan, Italy; Texas A&M University at Qatar PO Box 23874 Education City DOHA, 5825 QATAR},

author = {Leonetti, Paolo, Tringali, Salvatore},

journal = {Journal de Théorie des Nombres de Bordeaux},

keywords = {Agoh-Giuga conjecture; cyclic congruences; prime factorization; Pillai’s equation; Znam’s problem; Pillai's equation; Znam's problem},

language = {eng},

month = {10},

number = {2},

pages = {399-413},

publisher = {Société Arithmétique de Bordeaux},

title = {On a system of equations with primes},

url = {http://eudml.org/doc/275688},

volume = {26},

year = {2014},

}

TY - JOUR

AU - Leonetti, Paolo

AU - Tringali, Salvatore

TI - On a system of equations with primes

JO - Journal de Théorie des Nombres de Bordeaux

DA - 2014/10//

PB - Société Arithmétique de Bordeaux

VL - 26

IS - 2

SP - 399

EP - 413

AB - Given an integer $n \ge 3$, let $u_1, \ldots , u_n$ be pairwise coprime integers $\ge 2$, $\mathcal{D}$ a family of nonempty proper subsets of $\lbrace 1, \ldots , n\rbrace $ with “enough” elements, and $\varepsilon $ a function $ \mathcal{D} \rightarrow \lbrace \pm 1\rbrace $. Does there exist at least one prime $q$ such that $q$ divides $\prod _{i \in I} u_i - \varepsilon (I)$ for some $I \in \mathcal{D}$, 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 $\varepsilon $ and $\mathcal{D}$ are subjected to certain restrictions.We use the result to prove that, if $\varepsilon _0 \in \lbrace \pm 1\rbrace $ and $A$ is a set of three or more primes that contains all prime divisors of any number of the form $\prod _{p \in B} p - \varepsilon _0$ for which $B$ is a finite nonempty proper subset of $A$, then $A$ contains all the primes.

LA - eng

KW - Agoh-Giuga conjecture; cyclic congruences; prime factorization; Pillai’s equation; Znam’s problem; Pillai's equation; Znam's problem

UR - http://eudml.org/doc/275688

ER -

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