Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Acta Arithmetica / All issues

Abstract

Let $p \gt 3$ be a prime, and let $a,b$ be two rational $p$-adic integers. We present general congruences for $\sum _{k=0}^{p-1}\binom ak\binom {-1-a}k\frac p{k+b}\pmod {p^2}$. Let $\{D_n\}$ be the Domb numbers given by $D_n=\sum _{k=0}^n\binom nk^2\binom {2k}k\binom {2n-2k}{n-k}$. We also prove that $$\sum _{n=0}^{p-1}\frac {D_n}{16^n}\equiv \sum _{n=0}^{p-1}\frac {D_n}{4^n}\equiv \begin {cases} 4x^2-2p\pmod {p^2} &\text {if $3\mid p-1$ and so $p=x^2+3y^2$,}\\ 0\pmod {p^2} &\text {if $p\equiv 2\pmod 3$,}\end {cases}$$ which was conjectured by Z. W. Sun.