  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.
