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

@article{DoughertyBliss2022TweakingTB,
  title={Tweaking the Beukers integrals in search of more miraculous irrationality proofs a la Ap{\'e}ry},
  author={Robert Dougherty-Bliss and Christoph Koutschan and Doron Zeilberger},
  journal={The Ramanujan Journal},
  year={2022}
}
As we all know, he was proven wrong by Gödel and Turing in general, but even for such concrete problems, like the irrationality of a specific, natural, constant, like the Euler-Mascheroni constant (that may be defined in terms of the definite integral − ∞ 0 e−x log x) , that is most probably decidable in the logical sense, (i.e. there probably exists a (rigorous) proof), we lowly humans did not yet find it, (and may never will!). 
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