Supercongruences involving products of two binomial coefficients modulo $p^4$
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866908782496317440 |
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| author | Mao, Guo-Shuai |
| author_facet | Mao, Guo-Shuai |
| contents | In this paper, we mainly prove a congruence conjecture of Z.-W. Sun \cite{Sjnt}: Let $p>5$ be a prime. Then $$ \sum_{k=(p+1)/2}^{p-1}\frac{\binom{2k}k^2}{k16^k}\equiv-\frac{21}2H_{p-1}\pmod{p^4}, $$ where $H_n$ denotes the $n$-th harmonic number. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_06951 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Supercongruences involving products of two binomial coefficients modulo $p^4$ Mao, Guo-Shuai Number Theory Combinatorics In this paper, we mainly prove a congruence conjecture of Z.-W. Sun \cite{Sjnt}: Let $p>5$ be a prime. Then $$ \sum_{k=(p+1)/2}^{p-1}\frac{\binom{2k}k^2}{k16^k}\equiv-\frac{21}2H_{p-1}\pmod{p^4}, $$ where $H_n$ denotes the $n$-th harmonic number. |
| title | Supercongruences involving products of two binomial coefficients modulo $p^4$ |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2201.06951 |