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- W4376644865 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 $x,yin mathbb {Z}$ and $xequiv 1 ({rm {mod}} 3)$ , then [ xequivfrac14sum_{k=0}^{p-1}(3k+4)frac{f_k}{2^k}equivfrac12sum_{k=0}^{p-1}(3k+2)frac{f_k}{({-}4)^k} ({rm{mod}} p^2), ] and if $pequiv 1pmod 3$ , then [ sum_{k=0}^{p-1}frac{f_k}{2^k}equivsum_{k=0}^{p-1}frac{f_k}{({-}4)^k} ({rm{mod}} p^3), ] where $f_n=sum _{k=0}^nbinom {n}k^3$ stands for the $n$ th Franel number." @default.
- W4376644865 created "2023-05-17" @default.
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- W4376644865 date "2023-05-16" @default.
- W4376644865 modified "2023-09-28" @default.
- W4376644865 title "On two congruence conjectures of Z.-W. Sun involving Franel numbers" @default.
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- W4376644865 doi "https://doi.org/10.1017/prm.2023.41" @default.
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