Bipartite Diophantine tuples and their applications

2025/12/03 by Tsang, Kin Ming, Yip, Chi Hoi
#11B30 #11D45 #11D72 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2512.03441

Abstract

This paper investigates bipartite variants of generalized Diophantine tuples and their applications. We generalize a result of Bugeaud--Dujella on a special family of bipartite Diophantine tuples and affirmatively resolve a related question posed by the second author. Additionally, we establish new connections between bipartite Diophantine tuples and several known variants of Diophantine tuples, including those introduced by Banks--Luca--Szalay and Kihel--Kihel.

Citations

• Diophantine tuples and product sets in shifted powers
• f-Diophantine sets over finite fields via quasi-random hypergraphs from multivariate polynomials
• Multiplicative irreducibility of small perturbations of the set of shifted k-th powers
• On Diophantine graphs
• There is no Diophantine quintuple
• Hilbert cubes in arithmetic sets
• Hilbert cubes in progression-free sets and in the set of squares II
• On a problem of Diophantus for higher powers

Related

• A Generalization of Diophantine Tuples
• Diophantine tuples and product sets in shifted powers
• There is no Diophantine quintuple
• k-Diophantine m-tuples in Finite Fields
• A note on the number of S-Diophantine quadruples