Multiplicative irreducibility of small perturbations of the set of shifted $k$-th powers

TL;DR

This paper proves a conjecture about the multiplicative irreducibility of small perturbations of shifted k-th powers for k≥3, extending previous work on additive irreducibility and multiplicative analogues.

Contribution

It confirms a more general version of Hajdu and Sárközy's conjecture for k≥3, showing such small perturbations cannot be expressed as a nontrivial product set.

Findings

Small perturbations of shifted k-th powers are multiplicatively irreducible for k≥3.

The result extends previous additive irreducibility conjectures to a multiplicative setting.

The paper verifies the conjecture for a broader class of perturbations.

Abstract

Motivated by a conjecture of Erd\H{o}s on the additive irreducibility of small perturbations of the set of squares, recently Hajdu and S\'{a}rk\"{o}zy studied a multiplicative analogue of the conjecture for shifted $k$-th powers. They conjectured that for each $k\geq 2$, if one changes $o(X^{1/k})$ elements of $M_k'=\{x^k+1: x \in \mathbb{N}\}$ up to $X$, then the resulting set cannot be written as a product set $AB$ nontrivially. In this paper, we confirm a more general version of their conjecture for $k\geq 3$.