By Arthur T. Benjamin
and Doron Zeilberger
First Written: Aug. 2, 2004.
Revised version: Aug. 27, 2004.
A month ago, I met the legendary mathemagician
Arthur T. Benjamin
for the first time (on a one-to-one basis, I have watched his
amazing performances at AMS and MAA National meetings before).
It was at the 11th Fibonacci Conference that was held in
Braunschweig. During the traditional Wed. afternoon excursion,
I told him about Smith's proof, that he should have
included in his great book with Jenny Quinn `Proofs that Really Count:
The Art of the Combinatoiral Proof'.
Not surprisingly, he grasped the proof right away, without any
visual aids like paper or computer (or even hand-motions).
I also promised him a prize of
1 Euro, if he could perform the implied algorithm (in his head)
to express 41 as a sum of two perfect squares (it was the
honor system, he was not allowed to use paper (or computer), and
it was to be done with continued fractions). To my great
disappointment he didn't do it during the excursion (because
he is such a friendly guy, and he rather talk to people than do mental
math, even though he is so good at it). But the next morning,
over breakfast, sure enough he came with the answer, and I gave
him half a Euro.
To my even greater amazement, he wrote up this proof, and
made me co-author. Notice that this paper is
more than an exposition of Smith's proof. The
point of view is brand-new. None of these 19th century guys (or 18th century
, notably Euler)
ever thought of `proofs that count' in the style of
Benjamin and Quinn.
.pdf
.ps
Appeared in
INTEGERS,
v. 5(1) (2005), A30.
Added Aug. 23, 2004: Our dedicated librarian, Mei Ling Lo, kindly scanned
Henry Smith's 1855 original paper
, written in Latin. You don't
have to know much Latin to realize that his proof is not entirely
self-contained, and definitely not purely combinatorial,
since he used linear algebra.
Doron Zeilberger's List of Papers