# A modular supercongruence for $_6F_5$: an Ap\'ery-like story

## TL;DR

This paper establishes a supercongruence relating a modular form's Fourier coefficient to a hypergeometric series, using novel harmonic sum identities and Apéry number comparisons, advancing understanding of supercongruences in number theory.

## Contribution

It introduces a new supercongruence for $_6F_5$ hypergeometric series linked to modular forms, employing innovative techniques involving harmonic sums and Apéry-like sequences.

## Key findings

- Proves a supercongruence modulo p^3 for a weight 6 modular form coefficient.
- Develops new harmonic sum identities through rational approximations of ζ(3).
- Reduces complex congruences via Apéry number comparisons.

## Abstract

We prove a supercongruence modulo $p^3$ between the $p$th Fourier coefficient of a weight 6 modular form and a truncated ${}_6F_5$-hypergeometric series. Novel ingredients in the proof are the comparison of two rational approximations to $\zeta (3)$ to produce non-trivial harmonic sum identities and the reduction of the resulting congruences between harmonic sums via a congruence between the Ap\'ery numbers and another Ap\'ery-like sequence.