AI 中文总结
本文针对模4余3的有理素数，研究矩形高斯二项式系数，证明其p-整性，建立Ljunggren--Jacobsthal型超同余式以解决Kalinin猜想的惰性素数情形，还给出矩形Bailey型模p同余式并完成相关证明。
AI 中文摘要
受Kalinin提出的Wolstenholme定理的高斯模拟的多项式形式的启发，我们遵循Kalinin的思路，研究了高斯整数上的一个二维阶乘比，将其称为矩形高斯二项式系数。对于满足$p\not\bmod 4 = 3$的有理素数$p$，我们首先证明这些系数是$p$-整的。我们的主要结果是一个Ljunggren--Jacobsthal型超同余式：对于$p>5$以及每个$k\not\bmod 1$，所有四个参数同时被$p^k$伸缩时，该系数会改变一个$p^{3k}$的倍数。特别地，这证明了Kalinin猜想的惰性素数情形。我们还针对由一个大的$p$-倍数和一个基为$p$的数字组成的参数，建立了一个模$p$的矩形Bailey型同余式。其形式与Bailey对Lucas定理的素数幂细化类似，但出现了两个额外的普通二项式因子，这反映了矩形块分解的垂直和水平边界带。证明结合了$\boldsymbol{\text{Z}}_p[i]$中的倒数幂和估计，以及矩形乘积分解为完整的$p^k\times p^k$块的方法。
英文摘要
Motivated by the polynomial form of Kalinin's Gaussian analogue of Wolstenholme's theorem, following Kalinin, we study a two-dimensional factorial ratio over the Gaussian integers, which we call the rectangular Gaussian binomial coefficient. For a rational prime $p\equiv 3\pmod 4$, we first prove that these coefficients are $p$-integral. Our main result is a Ljunggren--Jacobsthal-type supercongruence: for $p>5$ and every $k\geq 1$, simultaneous dilation of all four parameters by $p^k$ changes the coefficient by a multiple of $p^{3k}$. In particular, this proves the inert-prime case of a conjecture of Kalinin. We also establish a rectangular Bailey-type congruence modulo $p$ for parameters consisting of a large $p$-multiple and one base-$p$ digit. Its shape parallels Bailey's prime-power refinements of Lucas's theorem, but two additional ordinary binomial factors occur, reflecting the vertical and horizontal boundary strips of a rectangular block decomposition. The proofs combine reciprocal-power-sum estimates in $\mathbb{Z}_p[i]$ with factorizations of rectangular products into complete $p^k\times p^k$ blocks.
