Ljunggren--Jacobsthal and Bailey-Type Congruences for Rectangular Gaussian Binomial Coefficients

2026/07/31 by Kevin Calderon
Mathematics · #math.NT

paper · pdf

arxiv created 2026/07/31 · arxiv updated 2026/08/04

Abstract

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≡ 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≥ 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 ℤ_p[i] with factorizations of rectangular products into complete p^k× p^k blocks.

Citations

• A p-adic (p≡ 3\pmod 4) depth-5 supercongruence for Gaussian p-th power sums over a square
• Wolstenholme's theorem over Gaussian integers
• A q-analog of Ljunggren's binomial congruence