On $p$-adic congruences involving $\sqrt{d}$

Abstract

Let $p$ be an odd prime and let $d$ be an integer not divisible by $p$. We prove that $$\prod _{\genfrac{}{}{0ex}{}{1\le m,n\le p-1}{p\nmid m^2-d{n}^2}}(x-(m+n\sqrt{d}))\equiv \{\begin{array}{ll}{\sum }_{k=1}^{p-2}\frac{k(k+1)}{2}{x}^{(k-1)(p-1)}(\mathrm{mod}p)& \text{if}(\frac{d}{p})=1,\\ {\sum }_{k=0}^{(p-1)/2}{x}^{2k(p-1)}(\mathrm{mod}p)& \text{if}(\frac{d}{p})=-1,\end{array}$$ where $(\frac{d}{p})$ denotes the Legendre symbol. This extends a recent conjecture of N. Kalinin. We also obtain the Wolstenholme-type congruence $$\sum _{\genfrac{}{}{0ex}{}{1\le m,n\le p-1}{p\nmid m^2-d{n}^2}}\frac{1}{m+n\sqrt{d}}\equiv 0(\mathrm{mod}{p}^2).$$

6 pages