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

2025/01/28 by Yip, Chi Hoi · 1 citation
#11B30 #11D45 #11P70 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2501.16620

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≥ 2, if one changes o(X^1/k) elements of 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≥ 3.

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