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), 407419.
First Written: Dec. 12, 2019; This version: Jan. 7, 2020.
In memory of Naum Il'ich Feldman (19181994)
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
Sample Input and Output for SALIKHOVpi.txt

If you want to see a computergenerated 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
computergenerated 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 computergenerated 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
computergenerated article.

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

If you want to see a computergenerated 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
computergenerated 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).
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