Heath-Brown [6] suggested a short proof of the two squares theorem, thereby simplifying ideas of Liouville. Zagier [15] suggested a particularly neat form of this, a "One sentence proof". It consists of two suitable involutions on the finite set of the solutions of p = x 2 +4yz in positive integers. A parity argument ensures the existence of a solution with y = z. The proof can be stated in one sentence since elementary calculations (namely to check that the mappings are well defined and are indeed involutory) can be left to the reader. The proof remained somewhat mysterious, since it is not obvious, where these mappings come from. In this paper we reveal this mystery and systematically explore similar proofs that can be given for related problems. We show that the very same method proves results on p = x 2 +2y 2 (see also Jackson [7]), p = x 2 2y 2 , p = 3x 2 + 4y 2 , and p = 3x 2 4y 2 .

@misc{Elsholtz2002,
title = {A combinatorial approach to sums of two squares and related problems},
author = {Elsholtz, Christian},
url = {https://link.springer.com/chapter/10.1007/978-0-387-68361-4{\_}8 http://www.math.tugraz.at/{\~{}}elsholtz/WWW/papers/papers.html http://www.math.tugraz.at/{\~{}}elsholtz/WWW/papers/zagierenglish9thjuly2002.ps},
urldate = {2010-07-16},
year = 2002,
abstract = {Heath-Brown [6] suggested a short proof of the two squares theorem, thereby simplifying ideas of Liouville. Zagier [15] suggested a particularly neat form of this, a "One sentence proof". It consists of two suitable involutions on the finite set of the solutions of p = x 2 +4yz in positive integers. A parity argument ensures the existence of a solution with y = z. The proof can be stated in one sentence since elementary calculations (namely to check that the mappings are well defined and are indeed involutory) can be left to the reader. The proof remained somewhat mysterious, since it is not obvious, where these mappings come from. In this paper we reveal this mystery and systematically explore similar proofs that can be given for related problems. We show that the very same method proves results on p = x 2 +2y 2 (see also Jackson [7]), p = x 2 2y 2 , p = 3x 2 + 4y 2 , and p = 3x 2 4y 2 .},
comment = {},
collections = {About proof,Fun maths facts}
}