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| Main Author: | |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.17434 |
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| _version_ | 1866917351372357632 |
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| author | Cherukupally, Srikanth |
| author_facet | Cherukupally, Srikanth |
| contents | For number $n>1$, let $\mathcal{A}(n) = \{1\leq a<n: n|a^2-1, a|n^2-1 \}$.
We show that the size of $\mathcal{A}(n)$ is connected to
a property concerning integer evaluations of
Fibonacci-like polynomials. In the process, we prove that $|\mathcal{A}(n)|< \log_2 n$,
and establish the average value of $|\mathcal{A}(n)|$ to be a little above $2$, asymptotically.
But the empirical data up to
$n<10^7$ indicate that $|\mathcal{A}(n)|\leq 3$, proving which is left as an open issue. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_17434 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the size of $\{a: 1\leq a<n, n|a^2-1, a|n^2-1\}$ for number $n$ Cherukupally, Srikanth Number Theory For number $n>1$, let $\mathcal{A}(n) = \{1\leq a<n: n|a^2-1, a|n^2-1 \}$. We show that the size of $\mathcal{A}(n)$ is connected to a property concerning integer evaluations of Fibonacci-like polynomials. In the process, we prove that $|\mathcal{A}(n)|< \log_2 n$, and establish the average value of $|\mathcal{A}(n)|$ to be a little above $2$, asymptotically. But the empirical data up to $n<10^7$ indicate that $|\mathcal{A}(n)|\leq 3$, proving which is left as an open issue. |
| title | On the size of $\{a: 1\leq a<n, n|a^2-1, a|n^2-1\}$ for number $n$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2603.17434 |