Pythagorean Primes and Palindromic Continued Fractions

By Arthur T. Benjamin and Doron Zeilberger

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Appeared in INTEGERS, v. 5(1) (2005), A30.

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.

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.
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