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 $$ \prod_{1\le m,n\le p-1\atop p\nmid m^2-dn^2}\ (x-(m+n\sqrt{d})) \equiv \begin{cases}\sum_{k=1}^{p-2}\frac{k(k+1)}2x^{(k-1)(p-1)}\pmod p &\text{if}\ (\frac dp)=1,\\\sum_{k=0}^{(p-1)/2}x^{2k(p-1)} \pmod p&\text {if}\ (\frac dp)=-1, \end{cases}$$ where $(\frac dp)$ denotes the Legendre symbol. This extends a recent conjecture of N. Kalinin. We also obtain the Wolstenholme-type congruence $$\sum_{1\le m,n\le p-1\atop p\nmid m^2-dn^2}\ \ \frac1{m+n\sqrt d}\equiv0\pmod{p^2}.$$