The Three-Squares-Theorem was proved by Gauss in his Disquisitiones, and this proof was studied carefully by various number theorists. Three years before Gauss, Legendre claimed
to have given a proof in his Essais de theorie des nombres. Dickson just says that Legendre
proved the result using reciprocal divisors.
I am a little bit surprised that I can't seem to find a discussion of Legendre's proof
anywhere. Reading the original is not exactly a pleasure since the material in Legendre's
book is organized, if I dare use this word, in a somewhat suboptimal manner. So before I'll start reading Legendre's work on three squares I'd like to ask whether anyone knows a discussion of his proof (or its gaps).
Edit. I have meanwhile found a thoroughly written thesis from Brazil on
Legendre's work in number theory by Maria Aparecida Roseane Ramos Silvia:
Adrien-Marie Legendre (1752-1833) e suas obras em teoria dos Números, which has, however, preciously little on Legendre's "proof".
One gets a better idea of what Legendre was doing by reading Simerka's article
Die Perioden der quadratischen Zahlformen bei negativen Determinanten, although Simerka is doing his own thing; in particular, he uses Gauss's definition of equivalence and composition of forms and describes parts of Legendre's work in Gauss's language.
BTW I have only recently learned about the existence of this Czech number theorist, who discovered Carmichael numbers long before Korselt or Carmichael, and who factored (10^17-1)/9 using an algorithm based on the class group of quadratic forms discovered long after by Shanks, Schnorr, and others.
Best Answer
There is a discussion of this by Andre Weil in "Number theory from Hammurapi to Legendre", where seems to imply that Legendre's proof might have been a bit problematic (p. 332). Weil also gives some nice proofs of the $n$-squares theorems (appendix 2 to the Euler chapter). You can find a link to the Weil book here: http://dl.dropbox.com/u/5188175/WeilNumbers.pdf