Multiplicatively reducible subsets of shifted perfect $k$-th powers and bipartite Diophantine tuples

TL;DR

This paper explores the structure of multiplicatively reducible subsets of shifted perfect powers and bipartite Diophantine tuples, establishing bounds and extending previous results in the field of polynomial sequence decompositions.

Contribution

It introduces bounds on the size of bipartite Diophantine tuples related to shifted perfect powers and improves existing results in the study of Diophantine tuples.

Findings

Bound on the minimum size of bipartite Diophantine tuples in terms of log |n|

Upper bounds on the product of the sizes of the sets A and B

Extension of previous results to broader cases with k ≥ 6

Abstract

Recently, Hajdu and S\'{a}rk\"{o}zy studied the multiplicative decompositions of polynomial sequences. In particular, they showed that when $k \geq 3$, each infinite subset of $\{x^k+1: x \in \mathbb{N}\}$ is multiplicatively irreducible. In this paper, we attempt to make their result effective by building a connection between this problem and the bipartite generalization of the well-studied Diophantine tuples. More precisely, given an integer $k \geq 3$ and a nonzero integer $n$, we call a pair of subsets of positive integers $(A,B)$ a bipartite Diophantine tuple with property $BD_k(n)$ if $|A|,|B| \geq 2$ and $AB+n \subset \{x^k: x \in \mathbb{N}\}$. We show that $\min \{|A|, |B|\} \ll \log |n|$, extending a celebrated work of Bugeaud and Dujella (where they considered the case $n=1$). We also provide an upper bound on $|A||B|$ in terms of $n$ and $k$ under the assumption $\min \{|A|,|B|\}\geq 4$ and $k \geq 6$. Specializing our techniques to Diophantine tuples, we significantly improve several results by B\'{e}rczes-Dujella-Hajdu-Luca, Bhattacharjee-Dixit-Saikia, and Dixit-Kim-Murty.