Fermat's two-squares theorem -- a study for heuristics of proving
The author discusses in considerable detail a number of different proofs of the two squares theorem (a prime \(p=4n+1\) is a sum of two integer squares). In particular Euler's proof using infinite descent, Minkowski's proof, the Heath-Brown--Zagier proof, and some further lattice point counting proofs are studied. The author emphasizes the historical, heuristic and aesthetic aspects of these proofs. The paper is especially aimed at teacher training. \{Reviewer's comment: A similar, but more concise, discussion can also be found in the reviewer's article [Kombinatorische Beweise des Zweiquadratesatzes und Verallgemeinerungen, Math. Semesterber.~50, No. 1, 77--93 (2003; Zbl 1050.11045)]. At the end of the paper the author gives the impression that Euclid's method for proving there are infinitely many primes, cannot prove there are infinitely many of type \(4n+1\), whereas the lattice counting proof would establish this. This may be a bit misleading. The clarification the author gave in private communication pointed to the standard trick: \((p_1 p_2 \ldots p_r)^2+4\) necessarily has a new prime divisor of the type \(4n+1\). But this argument is of course in the spirit of Euclid's method, and is independent of the type of proof of the two squares theorem for primes.
scientific article; zbMATH DE number 2051551