On two congruence conjectures of Z.-W. Sun involving Franel numbers
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866910611354419200 |
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| author | Mao, Guo-Shuai Liu, Yan |
| author_facet | Mao, Guo-Shuai Liu, Yan |
| contents | In this paper, we mainly prove the following conjectures of Z.-W. Sun \cite{S13}: Let $p>2$ be a prime. If $p=x^2+3y^2$ with $x,y\in\mathbb{Z}$ and $x\equiv1\pmod 3$, then $$x\equiv\frac14\sum_{k=0}^{p-1}(3k+4)\frac{f_k} {2^k}\equiv\frac12\sum_{k=0}^{p-1}(3k+2)\frac{f_k}{(-4)^k}\pmod{p^2},$$ and if $p\equiv1\pmod3$, then $$\sum_{k=0}^{p-1}\frac{f_k}{2^k}\equiv\sum_{k=0}^{p-1}\frac{f_k}{(-4)^k}\pmod{p^3},$$ where $f_n=\sum_{k=0}^n\binom{n}k^3$ stands for the $n$th Franel number. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2111_08775 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On two congruence conjectures of Z.-W. Sun involving Franel numbers Mao, Guo-Shuai Liu, Yan Number Theory Combinatorics In this paper, we mainly prove the following conjectures of Z.-W. Sun \cite{S13}: Let $p>2$ be a prime. If $p=x^2+3y^2$ with $x,y\in\mathbb{Z}$ and $x\equiv1\pmod 3$, then $$x\equiv\frac14\sum_{k=0}^{p-1}(3k+4)\frac{f_k} {2^k}\equiv\frac12\sum_{k=0}^{p-1}(3k+2)\frac{f_k}{(-4)^k}\pmod{p^2},$$ and if $p\equiv1\pmod3$, then $$\sum_{k=0}^{p-1}\frac{f_k}{2^k}\equiv\sum_{k=0}^{p-1}\frac{f_k}{(-4)^k}\pmod{p^3},$$ where $f_n=\sum_{k=0}^n\binom{n}k^3$ stands for the $n$th Franel number. |
| title | On two congruence conjectures of Z.-W. Sun involving Franel numbers |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2111.08775 |