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On two congruence conjectures of Z.-W. Sun involving Franel numbers
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 154 / Issue 3 / June 2024
- Published online by Cambridge University Press:
- 16 May 2023, pp. 887-905
- Print publication:
- June 2024
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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.
