A Binomial Coefficient Identity Associated To A Conjecture of Beukers

By Scott Ahlgren, Shalosh B. Ekhad, Ken Ono, and Doron Zeilberger

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Written: Jan. 29, 1998.

[Appeared in Electronic Journal of Combinatorics, v. 5(1998), R10.]

Like frugal-cheap, brave-reckless, smart-nerdy and countless other such pairs, where the same attribute becomes laudatory or derogratory according to human sentiment, so is the pair deep-complicated.

Using deep number-theoretic methods, Scott Ahlgren and Ken Ono reduced the proof of a 12-year-old conjecture of Frits Beukers to the proof of a binomial coefficient identity. Since its form was not immediately shaloshable, they could not use EKHAD. So they asked me, and I wrote a new Maple program, zeilWZP, implementing the extension that can handle their conjecture identity. Executing this program with the input file, Shalosh B. Ekhad (as your own computer can check for itself) got the `complicated' proof in the output file.

The present extension was already described in my article Closed Form(pun intended!), that was published in 1993 in Contemporary Math. 143. At the time, I was hoping that this extension will give irrationality proofs of new interesting numbers. This hope is still pending. It was a great surprise that it also proved useful in proving Frits Beukers's congruence conjecture.

The central NOTION here is that of a WZ pair, which I believe to be very deep, yet not at all complicated.

Added May 1, 2016: The urls in the original version in the Electronic Journal of Combinatorics are no longer valid. The old version has been replaced by the current version.
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