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pro vyhledávání: '"W., Sun"'
Autor:
Wei, Chuanan
Harmonic numbers are significant in various branches of number theory. With the help of the digamma function, we prove ten conjectural series of Z.-W. Sun involving harmonic numbers. Several ones of them are also series expansions of $\log2/\pi^2$.
Externí odkaz:
http://arxiv.org/abs/2304.09753
Autor:
Mao, Guo-Shuai
In this paper, we mainly prove two congruence conjecture of Z.-W. Sun. Let $p\equiv3\pmod 4$ be a prime. Then $$\sum_{k=0}^{p-1}\frac{\binom{2k}k^2}{8^k}\equiv-\sum_{k=0}^{p-1}\frac{\binom{2k}k^2}{(-16)^k}\pmod{p^3}.$$ And for any odd prime $p$, if $
Externí odkaz:
http://arxiv.org/abs/2304.04548
Akademický článek
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Autor:
Mao, Guo-Shuai, Liu, Yan
Publikováno v:
Proceedings of the Royal Society of Edinburgh Section A-Mathematics (2024)
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 $$x\equiv\frac14\sum_{k=0}^{p-1}(3k+4)\frac{f_k} {2^k}\equiv\frac12\sum_{k=0}^
Externí odkaz:
http://arxiv.org/abs/2111.08775
Autor:
Mao, Guo-Shuai
Publikováno v:
Chinese Annals of Mathematics, Series B (2022) 43(3), 2022, 417-424
In this paper, we partly prove a supercongruence conjectured by Z.-W. Sun in 2013. Let $p$ be an odd prime and let $a\in\mathbb{Z}^{+}$. Then if $p\equiv1\pmod3$, we have \begin{align*} \sum_{k=0}^{\lfloor\frac{5}6p^a\rfloor}\frac{\binom{2k}k}{16^k}\
Externí odkaz:
http://arxiv.org/abs/2003.14221
Autor:
Wang, Chen
In this paper, we mainly prove two conjectural supercongruences of Sun by using the following identity $$ \sum_{k=0}^n\binom{2k}{k}^2\binom{2n-2k}{n-k}^2=16^n\sum_{k=0}^n\frac{\binom{n+k}{k}\binom{n}{k}\binom{2k}{k}^2}{(-16)^k} $$ which arises from a
Externí odkaz:
http://arxiv.org/abs/2003.09888
Autor:
Mao, Guo-Shuai, Wen, Chen-Wei
Publikováno v:
The Ramanujan Journal(2021)
In this paper, we prove two supercongruences conjectured by Z.-W. Sun via the Wilf-Zeilberger method. One of them is, for any prime $p>3$, \begin{align*} \sum_{n=0}^{p-1}\frac{6n+1}{256^n}\binom{2n}n^3&\equiv p(-1)^{(p-1)/2}-p^3E_{p-3}\pmod{p^4}. \en
Externí odkaz:
http://arxiv.org/abs/1910.00779
Autor:
Mao, Guo-shuai1 (AUTHOR) maogsmath@163.com
Publikováno v:
Chinese Annals of Mathematics. May2022, Vol. 43 Issue 3, p417-424. 8p.
Autor:
Guo, Song, Guo, Victor J. W.
The polynomials $d_n(x)$ are defined by \begin{align*} d_n(x) &= \sum_{k=0}^n{n\choose k}{x\choose k}2^k. \end{align*} We prove that, for any prime $p$, the following congruences hold modulo $p$: \begin{align*} \sum_{k=0}^{p-1}\frac{{2k\choose k}}{4^
Externí odkaz:
http://arxiv.org/abs/1604.05019