What question did this study set out to answer?

The research aims to explore integer sequences generated by Sym³(Gauss) and their supercongruence properties.

April 1, 20260 citationsOpen Access

Sym³(Gauss) produces Apéry-like sequences with Dwork supercongruences beyond mod p³

Key Points

The research aims to explore integer sequences generated by Sym³(Gauss) and their supercongruence properties.
Analyzed integer sequences derived from Sym³(Gauss) using series expansion techniques.
Tested sequences for supercongruences mod p⁴ with primes p up to 23.
Utilized computational tools including Python scripts for exact integer arithmetic.
The sequence A_n shows an order-2 recurrence and satisfies A(mp) ≡ A(m) (mod p⁴) for primes p ≤ 23.
Comparative benchmarks from classic sequences A036917 and A002895 only reach mod p³.
A second sequence exhibits CM splitting that holds mod p⁴ if p ≡ 1 (mod 6).

Abstract

Integer sequences from Sym³ (Gauss) with Dwork supercongruences beyond mod p³. The sequence Aₙ = 27ⁿ zⁿ ₂F₁ (1/3, 1/3;1;z) ³ satisfies an order-2 recurrence and A (mp) ≡ A (m) (mod p⁴) for all tested primes p ≤ 23 and mp ≤ 499; classical Sym² benchmarks A036917 and A002895 reach only mod p³. A second sequence from (1/6, 1/3;1) exhibits CM splitting: mod p⁴ iff p ≡ 1 (mod 6). Archive: proofs (PDF), computational report (PDF), Python script (exact integer arithmetic), sequence data (JSON).

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