Multiplicative irreducibility of small perturbations of the set of shifted $k$-th powers
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908751544451072 |
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| author | Yip, Chi Hoi |
| author_facet | Yip, Chi Hoi |
| contents | 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 $k\geq 2$, 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 $k\geq 3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_16620 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multiplicative irreducibility of small perturbations of the set of shifted $k$-th powers Yip, Chi Hoi Number Theory Combinatorics 11P70, 11B30, 11D45 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 $k\geq 2$, 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 $k\geq 3$. |
| title | Multiplicative irreducibility of small perturbations of the set of shifted $k$-th powers |
| topic | Number Theory Combinatorics 11P70, 11B30, 11D45 |
| url | https://arxiv.org/abs/2501.16620 |