By Scott Ahlgren,
Shalosh B. Ekhad,
Ken Ono, and
Doron Zeilberger
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.
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are no longer valid. The old version
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Doron Zeilberger's List of Papers