Searching for Apéry-Style Miracles
[Using, Inter-Alia, the Amazing Almkvist-Zeilberger Algorithm]

Shalosh B. Ekhad and Doron Zeilberger

.pdf   .ps   .tex  

First Written: May 17, 2014

[Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger and]

Dedicated to Gert Almkvist (b. April 17, 1934), on becoming (at least, chronologically), an octogenerian (plus one month).
[Added Dec. 19, 2019: Sadly Gert Almkvist died on Nov. 24, 2018. He was one of the most original mathematicians that I have known]

Older mathematicians make beautiful provers! One of my great heroes is Roger Apéry who at the age of 64 astounded the mathematical world with a proof that

1/13+1/23+1/33+ 1/43+ ...

is not a ratio of two integers. Since very soon I will turn 64 myself, I was hoping to do the analogous thing for

1/15+1/25+1/35+ 1/45+ ...        .

Alas, since I am not nearly as smart as Apéry, my only hope was to teach to Shalosh Apéry's neat tricks, but to our disappointment we could not do it.

Nevertheless, once I taught Shalosh Roger's tricks, it reproduced in one second the original proof (see below), and found lots and lots of new irrationality proofs of many new constants, but none of them happened to be ζ(5), or any of the other `famous', yet-to-be-proven-irrational, constants.

But maybe YOU would be able to use the present Maple package, or an extension of it, to do it. Good luck!

This article is dedicated to yet another hero of mine, GERT ALMKVIST, who has just turned 80. Happy Birthday Gert, and may you continue to do great experimental math!

Maple Package

Input and Output files

Personal Journal of Shalosh B. Ekhad and Doron Zeilberger

Doron Zeilberger's Home Page