Elliptic curves and the residue-counts of $x^2+bx+c/x$ modulo $p$
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866909264057991168 |
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| author | Sun, Zhi-Hong |
| author_facet | Sun, Zhi-Hong |
| contents | For any prime $p>3$ and rational $p$-integers $b,c$ with $c(b^3-27c)\not\equiv 0\pmod p$ let $V_p(x^2+bx+\frac cx)$ be the residue-counts of $x^2+bx+\frac cx$ modulo $p$ as $x$ runs over $1,2,\ldots,p-1$. In this paper, we reveal the connection between $V_p(x^2+bx+\frac cx)$ and the number of points on certain elliptic curve over the field $\Bbb F_p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_15432 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Elliptic curves and the residue-counts of $x^2+bx+c/x$ modulo $p$ Sun, Zhi-Hong Number Theory Primary 11A15, Secondary 11A07, 11E25, 11L10, 11G20 For any prime $p>3$ and rational $p$-integers $b,c$ with $c(b^3-27c)\not\equiv 0\pmod p$ let $V_p(x^2+bx+\frac cx)$ be the residue-counts of $x^2+bx+\frac cx$ modulo $p$ as $x$ runs over $1,2,\ldots,p-1$. In this paper, we reveal the connection between $V_p(x^2+bx+\frac cx)$ and the number of points on certain elliptic curve over the field $\Bbb F_p$. |
| title | Elliptic curves and the residue-counts of $x^2+bx+c/x$ modulo $p$ |
| topic | Number Theory Primary 11A15, Secondary 11A07, 11E25, 11L10, 11G20 |
| url | https://arxiv.org/abs/2407.15432 |