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.


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

| cite | improve this answer | |
  • 1
    $\begingroup$ 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. $\endgroup$ – Franz Lemmermeyer Oct 8 '11 at 17:31
  • $\begingroup$ My interpretation of what I saw was that Weil agreed with Gauss on this point. $\endgroup$ – Igor Rivin Oct 8 '11 at 19:15
  • $\begingroup$ @Igor: I think it's Hammurapi and not Hammurabi $\endgroup$ – crskhr Jun 23 '12 at 8:20
  • 1
    $\begingroup$ 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"!] $\endgroup$ – Noam D. Elkies Mar 20 '13 at 16:20

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.