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On two congruence conjectures of Z.-W. Sun involving Franel numbers

Part of: Elementary classical functions Combinatorics Elementary number theory

Published online by Cambridge University Press:  16 May 2023

Guo-Shuai Mao  and
Yan Liu
Guo-Shuai Mao
Affiliation:
Department of Mathematics, Nanjing University of Information Science and Technology, Nanjing 210044, People's Republic of China (maogsmath@163.com, 1325507759@qq.com)
Yan Liu
Affiliation:
Department of Mathematics, Nanjing University of Information Science and Technology, Nanjing 210044, People's Republic of China (maogsmath@163.com, 1325507759@qq.com)
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Abstract

In this paper, we mainly prove the following conjectures of Z.-W. Sun (J. Number Theory 133 (2013), 2914–2928): let $p>2$ be a prime. If $p=x^2+3y^2$ with $x,y\in \mathbb {Z}$ and $x\equiv 1\ ({\rm {mod}}\ 3)$, then

\[ x\equiv\frac14\sum_{k=0}^{p-1}(3k+4)\frac{f_k}{2^k}\equiv\frac12\sum_{k=0}^{p-1}(3k+2)\frac{f_k}{({-}4)^k}\ ({\rm{mod}}\ p^2), \]
and if $p\equiv 1\pmod 3$, then
\[ \sum_{k=0}^{p-1}\frac{f_k}{2^k}\equiv\sum_{k=0}^{p-1}\frac{f_k}{({-}4)^k}\ ({\rm{mod}}\ p^3), \]
where $f_n=\sum _{k=0}^n\binom {n}k^3$ stands for the $n$th Franel number.

Keywords

CongruencesFranel numbersp-adic gamma functiongamma function

MSC classification

Primary: 11A07: Congruences; primitive roots; residue systems
Secondary: 05A10: Factorials, binomial coefficients, combinatorial functions 05A19: Combinatorial identities, bijective combinatorics 33B15: Gamma, beta and polygamma functions

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 154 , Issue 3 , June 2024 , pp. 887 - 905
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Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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