Fraser, Owen; Gordon, Basil On representing a square as the sum of three squares. (English) Zbl 0207.35204 Am. Math. Mon. 76, 922-923 (1969). 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. Reviewer: David A. Klarner (Lincoln) Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page MSC: 11E25 Sums of squares and representations by other particular quadratic forms Keywords:representation of squares of integers; sum of three squares Citations:Zbl 0007.19504; JFM 59.0035.01; Zbl 0042.26702 PDFBibTeX XMLCite \textit{O. Fraser} and \textit{B. Gordon}, Am. Math. Mon. 76, 922--923 (1969; Zbl 0207.35204) Full Text: DOI Online Encyclopedia of Integer Sequences: Solutions n to n^2 = a^2 + b^2 + c^2 (a,b,c > 0). Theta series of simple cubic lattice; also number of ways of writing a nonnegative integer n as a sum of 3 squares (zero being allowed). Squares of the form x^2+y^2+z^2 with x,y,z positive integers.