A Combinatorial Approach to Sums of Two Squares and Related Problems

  • Christian ElsholtzEmail author


In this paper, we study elementary approaches to classical theorems on representations of primes of the form ax 2 + by 2, in particular the two squares theorem. While most approaches make use of quadratic residues, we study a route initiated by Liouville and simplified by Heath–Brown and Zagier.


Binary quadratic forms Fermat’s two squares theorem 



The author is grateful to B. Artmann and D. Spalt for introducing him to Zagier’s proof and for the challenge to understand how the proof could have been found. Further thanks goes to A.M. Decaillot for clarifying a question on Lucas’ work. Sections 2.1 and Sections 2.2 were found in 1990, Sect. 3 in 1996, and Sect. 1.6 in 2001, see also [11, 12, 13, 14].


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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Institut für Mathematik ATechnische Universität GrazGrazAustria

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