The Irrationality Measure of Pi is at most 7.103205334137...

By Doron Zeilberger and Wadim Zudilin

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Appeared in Moscow Journal of Combinatorics and Number Theory 9 (2020), 407-419.

First Written: Dec. 12, 2019; This version: Jan. 7, 2020.

In memory of Naum Il'ich Feldman (1918-1994)

We use a variant of V. Kh. Salikhov's ingenious proof that the irrationality measure of Pi is at most 7.606308 to prove that, in fact, it is at most 7.103205334137...

Note: If you google "Irrationality measure of Pi" you would get this erroneous paper. Buyers beware!

# Maple packages

• SALIKHOVpi.txt, a Maple package that uses V. Kh. Salikhov's integral (in his paper "On the measure of irrationality of Pi" Math. Notes 88(4) (2010) 563-573), but with a different approach using recurrences obtained via the Almkvist-Zeilberger algorithm, to find diophantine approximations to Pi that lead to good irrationality measures.

Note that you also need to download the following data file SalikhovDataFile.txt, and put it in the same directory in your computer.

• SALIKHOVpiHuman.txt, a Maple package that empirically checks most of the claims in Part II (the human part).

# Sample Input and Output for SALIKHOVpi.txt

• If you want to see a computer-generated paper about the initial empirical exploration about how to tweak Salikhov's proof that uses exponents (3,5) to other exponents, and find other, possibly better, choices
the input file generates the following computer-generated article.
Note that indeed we found that, at least, empirically, (2,3) is better. This is confirmed via a fully rigorous proof in the (human) paper.

• If you want to see a computer-generated paper that does the (2,3)-analog of Salikhov's proof, with a seemingly better irrationality measure, that was confirmed rigorously in the paper
the input file generates the following computer-generated article.

• If you want to see a computer-generated paper that redoes, using our approach, the original (3,5) case
the input file generates the following computer-generated article.

• If you want to see a computer-generated paper that handles again the case (2,3), similar to the one above, but taking advantage of the fact (unlike the original (3,5) case where the odd n yield approximations of arctan(1/7)), that the corresponding Salikhov integral gives combinations of 1 and Pi for any n, hence the recurrence is simpler, but otherwise it is the same,
the input file generates the following computer-generated article.

Added Jan. 13, 2020: For those skeptics who doubt the identity at the bottom of page 7 of the paper
the input file generates the following output file with a fully rigorous proof (thanks to Shalosh B, Ekhad).