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# A modular supercongruence for 6F5: An Apéry-like story

A modular supercongruence for 6F5: An Apéry-like story
Robert Osburn, Armin Straub, Wadim Zudilin — Annales de l'Institut Fourier — 2017

## 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 Apéry numbers and another Apéry-like sequence.

@article{supercongruence-6f5-2017,
title = {A modular supercongruence for $_6F_5$: An Ap\'ery-like story},
}