# Legendre and sums of three squares

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.

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## 1 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

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Weil says little if anything going beyond Gauss's remarks in his Disquisitiones. Apparently Weil did not think it worthwile studying this part of Legendre's work. –  Franz Lemmermeyer Oct 8 '11 at 17:31
My interpretation of what I saw was that Weil agreed with Gauss on this point. –  Igor Rivin Oct 8 '11 at 19:15
@Igor: I think it's Hammurapi and not Hammurabi –  S.C. Jun 23 '12 at 8:20
It seems that both transliterations are correct but Weil used "Hammurapi". [When I first learned the name in Hebrew it was חמורבי with a B(eth), which had the unfortunate effect of making it look like a portmanteau "ass(=donkey) rabbi"!] –  Noam D. Elkies Mar 20 '13 at 16:20