Bipartite Diophantine tuples and their applications

TL;DR

This paper explores bipartite Diophantine tuples, generalizes existing results, resolves an open question, and connects these structures to other variants, expanding understanding of their properties and applications.

Contribution

It generalizes a key result on bipartite Diophantine tuples, resolves an open question, and links these tuples to other known variants, broadening the theoretical framework.

Findings

Generalized a result of Bugeaud--Dujella

Resolved a question posed by the second author

Established new connections with known Diophantine tuple variants

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.