Lucas congruences using modular forms

TL;DR

This paper proves that many Apéry-like sequences derived from modular forms satisfy Lucas congruences modulo primes, confirming several conjectures and providing new insights into their properties.

Contribution

It establishes that numerous sequences from modular forms obey Lucas congruences, confirming four recent conjectures and reinterpreting existing results.

Findings

Many Apéry-like sequences satisfy Lucas congruences modulo primes

Four conjectural Lucas congruences are fully proven

Reinterpretation of known results in the context of modular forms

Abstract

In this work, we prove that many Ap\'ery-like sequences arising from modular forms satisfy the Lucas congruences modulo any prime. As an implication, we completely affirm four conjectural Lucas congruences that were recently posed by S. Cooper and reinterpret a number of known results.