Tweaking the Beukers integrals in search of more miraculous irrationality proofs a la Apéry

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