Original scientific paper

https://doi.org/10.3336/gm.54.2.03

On the existence of S-Diophantine quadruples

Volker Ziegler ; Institute of Mathematics, University of Salzburg, Hellbrunnerstrasse 34/I, A-5020 Salzburg, Austria

Abstract

Let \(S\) be a set of primes. We call an \(m\)-tuple \((a_1,\ldots,a_m)\) of distinct, positive integers \(S\)-Diophantine, if for all \(i\neq j\) the integers \(s_{i,j}:=a_ia_j+1\) have only prime divisors coming from the set \(S\), i.e. if all \(s_{i,j}\) are \(S\)-units. In this paper, we show that no \(S\)-Diophantine quadruple (i.e.~\(m=4\)) exists if
\(S=\{3,q\}\). Furthermore we show that for all pairs of primes \((p,q)\) with \(p

Keywords

Diophantine equations; S-unit equations; Diophantine tuples; S-Diophantine quadruples

Hrčak ID:

229600

URI

https://hrcak.srce.hr/229600

Publication date:

11.12.2019.

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