| INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 5(1) (2005), #A30 PYTHAGOREAN PRIMES AND PALINDROMIC CONTINUED FRACTIONS (2009) | |||||||||||||
Abstract | |||||||||||||
| 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. For n ≥ 1 and positive integers a0,..., an, let [a0,..., an] denote the finite continued fraction a0 + 1 a1 + | |||||||||||||
Publication details | |||||||||||||
| |||||||||||||