On $p$-adic congruences involving $\sqrt d$
Bo Jiang, Zhi-Wei Sun
TL;DR
This work generalizes Kalinin's p-adic congruences from Gaussian integers to the setting of $\sqrt{d}$ with $p\nmid d$. The authors evaluate the lattice-sum polynomial $P(x)=\prod_{1\le m,n\le p-1,\ p\nmid m^2-dn^2}(x-(m+n\sqrt{d}))$ modulo $p$, obtaining explicit forms depending on the Legendre symbol $\left(\frac{d}{p}\right)$: $(x^{p-1}-1)^{p-3}$ when $\left(\frac{d}{p}\right)=1$ and $(x^{p^2-1}-1)/(x^{2(p-1)}-1)=\sum_{k=0}^{(p-1)/2}x^{2k(p-1)}$ when $\left(\frac{d}{p}\right)=-1$, thereby extending Kalinin's conjecture. Additionally, a Wolstenholme-type congruence is established: $\displaystyle \sum_{\substack{1\le m,n\le p-1 \\ p\nmid m^2-dn^2}} \frac{1}{m+n\sqrt{d}} \equiv 0 \pmod{p^2}$. The proofs rely on classical congruences $x^{p-1}-1 \equiv \prod_{t=1}^{p-1}(x-t)$ and $(x-\alpha)^p \equiv x^p-\alpha^p \pmod p$, together with a case analysis guided by the quadratic character $\left(\frac{d}{p}\right)$. The results illuminate the p-adic structure of such products and their associated reciprocal sums, while indicating the optimality of the $p^2$ modulus in general.
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 $$\sum_{1\le m,n\le p-1\atop p\nmid m^2-dn^2}\ \ \frac1{m+n\sqrt d}\equiv0\pmod{p^2}.$$