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

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.