On the existence of S-Diophantine quadruples

2018/07/09 by Volker Ziegler, Ziegler, Volker
Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1807.02972

openalex publication_date 2018/07/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let S be a set of primes. We call an m-tuple (a₁,…,a_m) of distinct, positive integers S-Diophantine, if for all i≠ 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

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