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Mathematics (HMC)

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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 ([3], [5], and [6]) most recently described in [4] (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.


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.

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© 2005 Arthur T. Benjamin

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