Long strings of composite values of polynomials and a basis of order 2
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914350516666368 |
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| author | Radomskii, Artyom |
| author_facet | Radomskii, Artyom |
| contents | We show that for any polynomial $f: \mathbb{Z}\to \mathbb{Z}$ with positive leading coefficient and irreducible over $\mathbb{Q}$, if $N$ is large enough then there are two strings of consecutive positive integers $I_{1}=\{n_1-m,\ldots, n_1+m\}$ and $I_{2}=\{n_2-m, \ldots, n_2+m\}$, such that $m = [(\log N) (\log \log N)^{1/325525}]$, $I_{1}\cup I_{2} \subset [1, N]$, $N = n_1 + n_2$, and $f(n)$ is composite for any $n\in I_{1}\cup I_{2}$. This extends the result in [5] which showed the same result but with $f(n)=n$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_15641 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Long strings of composite values of polynomials and a basis of order 2 Radomskii, Artyom Number Theory We show that for any polynomial $f: \mathbb{Z}\to \mathbb{Z}$ with positive leading coefficient and irreducible over $\mathbb{Q}$, if $N$ is large enough then there are two strings of consecutive positive integers $I_{1}=\{n_1-m,\ldots, n_1+m\}$ and $I_{2}=\{n_2-m, \ldots, n_2+m\}$, such that $m = [(\log N) (\log \log N)^{1/325525}]$, $I_{1}\cup I_{2} \subset [1, N]$, $N = n_1 + n_2$, and $f(n)$ is composite for any $n\in I_{1}\cup I_{2}$. This extends the result in [5] which showed the same result but with $f(n)=n$. |
| title | Long strings of composite values of polynomials and a basis of order 2 |
| topic | Number Theory |
| url | https://arxiv.org/abs/2506.15641 |