H.J.S. Smith and the Fermat Two Squares Theorem

The history of this theorem of Fermat is given in detail by Dickson [7, 224-237]. Dickson names the theorem after Girard, who discussed the result in 1632; however the common practice now is to attribute the result to Fermat, who stated in 1659 that he possessed an irrefutable proof by the method of infinite descent; see [7, p. 228] and [2, p. 89]. The first recorded proof is due to Euler given in 1749 [7, pp. 230-231]; Bell writes, "It was first proved by the great Euler in 1749 after he had struggled, off and on, for seven years to find a proof" [2, p.89]. The first proof that such prime numbers can be uniquely represented as the sum of squares of two positive integers was given by Gauss in 1801 [7, p. 233]. See also the account of the two squares theorem of Fermat in the books by Burton [4, Chapter 12, Section 2], and Hardy and Wright [14, Chapter XX]. The last sentence in the quotation from Hardy is significant. Hardy had an interest in the classification of proof; see, in particular, [13, p. 6] in connection with the "elementary" proof of inequalities. In this context the word elementary must not be confused with the words obvious or easy; many of the elementary proofs in [13] are subtle, ingenious, and far from obvious. When Hardy wrote [12] he was, more than likely, not aware that an elementary proof of this theorem of Fermat had been given in 1855 by H. J. S. Smith, one of his predecessors in the Savilian Chair of Geometry in the University of Oxford. This simple but remarkable proof of Smith is within the comprehension of those with knowledge of elementary