bibkey: calderon2026rectangular authors: Kevin Calderon year: 2026 title: “Ljunggren–Jacobsthal and Bailey-Type Congruences for Rectangular Gaussian Binomial Coefficients” doi: 10.48550/arXiv.2608.00347 url: https://arxiv.org/abs/2608.00347v1 claim: “Conjecture 6.3 asks for reciprocal first and second moment congruences for inert primes in imaginary quadratic orders.” strata_touched:

• D5/S3/Arith/Congruence/QuadraticOrderMoments/CalderonReciprocalMoments license: citation-only triage: anchor

Reciprocal moments in imaginary quadratic orders

Verified locator

DOI: 10.48550/arXiv.2608.00347. Source: https://arxiv.org/abs/2608.00347v1. Conjecture 6.3 is on PDF page 19; equations (6.1) and (6.3) specify its setting.

Source statement

“Let ω satisfy (6.1), and let p satisfy (6.3). Then, for every k ≥ 1, H₁,ω(k) ∈ p²ᵏOω,p, H₂,ω(k) ∈ pᵏOω,p.”

The source TeX states:

Let $\omega$ satisfy \eqref{eq:omega-minimal-polynomial}, and let $p$ satisfy \eqref{eq:omega-inert-prime}. Then, for every $k\ge 1$,

$$H_{1,\omega }(k)\in p^{2k}{\mathcal{O}}_{\omega ,p},H_{2,\omega }(k)\in p^k{\mathcal{O}}_{\omega ,p}.$$

The setting is ω² − Tω + N = 0, T,N ∈ ℤ, Δ = T² − 4N < 0, p > 5, p ∤ Δ, and (Δ/p) = −1. The source defines

$$U_{\omega ,k}:=\{x+y\omega :1\le x,y\le p^k,p\nmid (x,y)\},H_{r,\omega }(k):=\sum _{z\in U_{\omega ,k}}{z}^{-r}.$$

Here p ∤ (x,y) means that the two coordinates are not simultaneously divisible by p. The local order is Oω,p = ℤp[ω]. Both coordinates use positive representatives, including pᵏ for the zero residue.

Scope

The paper states Conjecture 6.3 as a conjecture. The Lean result proves both congruences for all its admissible parameters, using the literal adjoin-root ring over the p-adic integers. Conjecture 6.4, the ω-Ljunggren analogue, uses the hypotheses of Conjecture 6.3, with k ≥ 1, A ≥ C ≥ 1 and B ≥ D ≥ 1. Its preceding derivation assumes the two bounds H₁,ω(k) ∈ p²ᵏOω,p and H₂,ω(k) ∈ pᵏOω,p; this result discharges that reciprocal-moment premise. Conjecture 6.5, the ω-Bailey analogue, instead states the inert-prime condition (6.3) and the digit ranges 1 ≤ γ ≤ α ≤ p − 1 and 1 ≤ δ ≤ β ≤ p − 1. The source precedes it with Frobenius strip factorizations and states no Conjecture 6.3 reciprocal-moment premise for it, so this result discharges no such premise for Conjecture 6.5. Both neighbouring conjectures’ conclusions and sharpness of the moment bounds remain open.

Definition fidelity

In the admissible setting every element of U is a unit. Mathlib’s Ring.inverse equals the underlying value of that unit’s inverse; its zero value on nonunits does not occur in the conjecture. The internal shifted coordinate subtype is related bijectively to the literal finite set inside the proof of result, so the sums count each ring element once.