Explicit congruences for the class equations $H_{-7p}(X)$ and $H_{-28p}(X)$

TL;DR

This paper provides computational proofs of congruences modulo prime p for specific class equations related to quadratic forms, expanding understanding of their arithmetic properties under certain prime congruences.

Contribution

It introduces explicit congruences for class equations H_{-7p}(X) and H_{-28p}(X) under specific prime residue conditions, with computational verification.

Findings

Proves congruences for H_{-28p}(X) when p ≡ 3 (mod 4)

Establishes congruences for the product H_{-7p}(X) H_{-28p}(X) when p ≡ 1 (mod 4)

Uses computational methods to verify class equation congruences

Abstract

A computational proof is given for congruences modulo $p$ for the class equation $H_{-28p}(X)$, when the prime $p$ satisfies $p \equiv 3$ (mod $4$), and for the product $H_{-7p}(X) H_{-28p}(X)$, when $p \equiv 1$ (mod $4$).