Multiplicative irreducibility of small perturbations of the set of shifted k-th powers
Chi Hoi Yip · COMBINATORICA · 2026
Abstract Motivated by a conjecture of Erdős on the additive irreducibility of small perturbations of the set of squares, recently Hajdu and Sárközy studied a multiplicative analogue of the conjecture for shifted k -th powers. They conjectured that for each $$k\ge 2$$ k ≥ 2 , if one changes $$o(X^{1/k})$$ o ( X 1 / k ) elements of $$M_k'=\{x^k+1: x \in \mathbb {N}\}$$ M k ′ = { x k + 1 : x ∈ 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 $$k\ge 3$$ k ≥ 3 .