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arXiv · 2501.16620

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

Chi Hoi Yip
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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 $k\geq 3$.

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Chi Hoi Yip. 2025-01-28. Multiplicative irreducibility of small perturbations of the set of shifted $k$-th powers. https://doi.org/10.1007/s00493-025-00193-9

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