S-Diophantine quadruples with \mathbfS=\2,q\

2014/03/24 by László Szalay, Volker Ziegler, Szalay, László +1
Mathematics · Physics and Astronomy · #11D61 #Advanced Mathematical Theories and Applications #Analytic Number Theory Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1403.5876

openalex publication_date 2014/03/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let S denote a set of primes and let a₁,…,a_m be positive distinct integers. We call the m-tuple (a₁,…,a_m) an S-Diophantine tuple if a_ia_j+1=s_i,j are S-integers for all i\not=j. In this paper, we show that no S-Diophantine quadruple (i.e~m=4) exists if S=\2,q\ with q≡ 3 (\bmod 4) or q<10⁹. For two arbitrary primes p,q<10⁵ we gain the same result.

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