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

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arxiv 2101.08308 v1 pith:VWW4MK7O submitted 2021-01-20 math.NT math.CO

Robert Dougherty-Bliss , Christoph Koutschan , Doron Zeilberger This is my paper · ORCID
classification math.NTmath.CO
keywords proofsgammairrationalityconstantrealtherezetaaleph-zero
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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.

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  1. On Conservative Matrix Fields: Continuous Asymptotics and Arithmetic

    math.NT 2025-07 conditional novelty 6.0 of 10

    Conservative Matrix Fields generalize Apéry-type ratios of D-finite sequences to several dimensions and conjecturally have direction-continuous convergence and irrationality measures.

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