Tweaking the Beukers Integrals In Search of More Miraculous Irrationality Proofs A La Apery

TL;DR

This paper explores modifications of Beukers integrals inspired by Apery's proofs to discover new irrationality proofs for mathematical constants, using algorithmic proof theory to identify promising integral candidates.

Contribution

It introduces a systematic search for new irrationality proofs based on Beukers integrals, expanding the set of constants with potential irrationality proofs.

Findings

Identified several candidate integrals suggesting irrationality of specific constants.

Extended Beukers' approach to new classes of integrals involving Gamma functions.

Provided a framework for future proof-theoretic exploration of irrationality proofs.

Abstract

There are only aleph-zero rational numbers, while there are 2 to the power aleph-zero real numbers. Hence the probability that a randomly chosen real number would be rational is 0. Yet proving rigorously that any specific, natural, real constant, is irrational is usually very hard, witness that there are still no proofs of the irrationality of the Euler-Mascheroni constant, the Catalan constant, or Zeta(5). Inspired by Frits Beukers' elegant rendition of Apery's seminal proofs of the irrationality of Zeta(2) and Zeta(3), and heavily using algorithmic proof theory, we systematically searched for other similar integrals, that lead to irrationality proofs. We found quite a few candidates for such proofs, including the square-root of Pi times Gamma(7/3)/Gamma(-1/6) and Gamma(19/6)/Gamma(8/3) divided by the square-root of Pi.