Posted: Jan. 13, 2021

In honor of Wadim Zudilin, on the occaison of his 51-th birthday

Maple packages

• GenBeukersZeta2.txt, a Maple package to suggest irrationality proofs of natural constants defined in terms of Generalized versions of Beukers' integral that occurred in his elegant rendition of Apery's miraculous proofs that Zeta(2) is irrational.

• GenBeukersZeta3.txt, a Maple package to suggest irrationality proofs of natural constants defined in terms of Generalized versions of Beukers' integral that occurred in his elegant rendition of Apery's miraculous proofs that Zeta(3) is irrational.

Sample Input and Output for GenBeukersZeta2.txt

• If you want to see 123 promising consants that probably have an irrationality proof similar to Beukers' irrationality proof of Zeta(2)
  The input file yields the output file

• If you want to see 35 sketeches of proof of irrationality similar to Beukers' irrationality proof of Zeta(2) for potentially constants not yet proved to be irrational (at least Maple can't identify them)
  The input file yields the output file

• If you want to see 88 sketeches of proof of irrationality similar to Beukers' irrationality proof of Zeta(2) for identified constants, all of whom already have classical and standard irrationality proof, but the novelty is that they were computer-generated,
  The input file yields the output file

• The next input and output file gives 3 sketeches of proof of irrationality similar to Beukers' irrationality proof of Zeta(2) but using the generalized version. The first is potentially the first proof of irrationality of that constant. The two other ones suggest yet another proof of the irrationality of log(2)
  The input file yields the output file

• The next input and output file gives 4 sketeches of proof of fast convergent sequences that ALMOST made it to the Irrationality Proofs Hall of Fame, alas their estimated deltas are slightly negative. It also gives some other examples with more negative deltas, that Maple evaluates in terms of Pi^2
  The input file yields the output file

Sample Input and Output for GenBeukersZeta3.txt

• If you want to see many candidates for irrationality proofs similar to Beukers' irrationality proof of Zeta(3), with all positive paramers (up to denominator 7)
  The input file yields the output file
• If you want to see many candidates for irrationality proofs similar to Beukers' irrationality proof of Zeta(3), with paramers positive and negative (up to denominator 5)
  The input file yields the output file

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