In this note, we prove that every prime of the form 4m + 1 is the sum of the squares of two positive integers in a unique way. Our proof is based on elementary combinatorial properties of continued fractions. It uses an idea by Henry J. S. Smith (, , and ) most recently described in  (which provides a new proof of uniqueness and reprints Smith's paper in the original Latin). Smith's proof makes heavy use of nontrivial properties of determinants. Our purely combinatorial proof is self-contained and elementary.
© 2005 Arthur T. Benjamin
Benjamin, A.T., & Zeilberger, D. (2005). Pythagorean primes and palindromic continued fractions. Integers: The Electronic Journal of Combinatory Number Theory, 5(1): 1-5.
First published in INTEGERS: The Electronic Journal of Combinatorial Number Theory, vol. 5, no. 1 (December 2005), by the State University of West Georgia, Charles University, and DIMATIA.
This article is also available at http://www.integers-ejcnt.org/vol5.html.