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On representing a square as the sum of three squares. (English) Zbl 0207.35204

The authors give an ingenious proof of the following theorem stated without proof by A. Hurwitz [Mathematische Werke. Bd. 2. Basel: Birkhäuser (1933; Zbl 0007.19504; JFM 59.0035.01), p. 751]: For each natural number \(n\) there exist natural numbers \(x\), \(y\), \(z\) such that \(n^2 = x^2 + y^2 + z^2\) if and only if \(n\ne 2^k, \ 5\cdot 2^k\) for \(k = 0, 1,\ldots\).
This theorem clears up a statement made by T. Nagell [Introduction to number theory. Stockholm: Almqvist & Wiksell (1951; Zbl 0042.26702), p. 194] which might lead one to think that every natural square can be expressed as a sum of three natural squares. The proof of Hurwitz’s theorem given by the authors is completely elementary, although it depends on Lagrange’s theorem (every natural number can be expressed as a sum of four integer squares), and Fermat’s theorem that every prime \(p\equiv 1\pmod 4\) can be expressed as a sum of two squares.

MSC:

11E25 Sums of squares and representations by other particular quadratic forms
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