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The aim is to establish a supercongruence modulo p⁴ for symmetric cube hypergeometric coefficients.

April 21, 2026Open Access

Order drop, Hecke descent, and an unconditional mod p⁴ supercongruence for Sym³ hypergeometric coefficients

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The aim is to establish a supercongruence modulo p⁴ for symmetric cube hypergeometric coefficients.
Used modular identification on X_0(3)
Applied an Eisenstein-type congruence for Hauptmodul coefficients
Employed Lagrange–Bürmann coefficient extraction and Hecke descent methods
Establish a supercongruence modulo p⁴ for specific hypergeometric coefficients
Demonstrate uniform vanishing of all defects through descent arguments

Abstract

We prove a supercongruence modulo p to the fourth for the coefficients of the symmetric cube of the hypergeometric function at parameters one third, one third, one. The proof combines a modular identification on X₀ (3), an Eisenstein-type congruence for the coefficients of the Hauptmodul derivative, Lagrange–Bürmann coefficient extraction, and a Fricke–Hecke descent argument that yields uniform vanishing of all defects.

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Alex Shvets (Sun,) studied this question.

www.synapsesocial.com/papers/69e713fdcb99343efc98d71a — DOI: https://doi.org/10.5281/zenodo.19648644

Order drop, Hecke descent, and an unconditional mod p⁴ supercongruence for Sym³ hypergeometric coefficients

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