Given an integer n ≥ 3, let u1, …, un 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 ui− ε(I) for some I ∈ D, but it does not divide u1 · · · un? We answer this question in the positive when the ui are prime powers and " and D are subjected to certain restrictions. We use the result to prove that, if ε0 ∈ {±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−ε0 for which B is a finite nonempty proper subset of A, then A contains all the primes.

### On a system of equations with primes

#### Abstract

Given an integer n ≥ 3, let u1, …, un 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 ui− ε(I) for some I ∈ D, but it does not divide u1 · · · un? We answer this question in the positive when the ui are prime powers and " and D are subjected to certain restrictions. We use the result to prove that, if ε0 ∈ {±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−ε0 for which B is a finite nonempty proper subset of A, then A contains all the primes.
##### Scheda breve Scheda completa Scheda completa (DC) 2014
2014
http://jtnb.cedram.org/item?id=JTNB_2014__26_2_399_0
Leonetti, P; Tringali, S
File in questo prodotto:
File
Euclid.pdf

accesso aperto

Tipologia: Documento in Pre-print
Licenza: Creative commons
Dimensione 213.14 kB
Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11383/2142081`
• ND
• 0
• 0