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

Abstract

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Given an integer n ≥ 3 , 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 ∏ i ∈ I u i - ε ( I ) for some I ∈ 𝒟 , 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 ∏ p ∈ B p - ε 0 for which B is a finite nonempty proper subset of A , then A contains all the primes.

How to cite

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Leonetti, 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 -

References

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  9. F. Luca, On the diophantine equation p x 1 - p x 2 = q y 1 - q y 2 . Indag. Mathem. (N.S.) 14, 2 (2003), 207–222. Zbl1080.11031MR2026815
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