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Im trying to connect his work on quadratic reciprocity with some simple question, like solution to certain diophantine equation or representing primes. Any ideas? I find it hard to imagine that he out of the blue just wanted to characterize the primes for which certain equations were solvable. I did read something about him wanting to write primes as sums of squares just like Fermat had done in $p=x^{2}+y^{2}$ for instance. But having $ax^{2}+bx+c=pk$ dont look like an attempt to achieve this, he doesn't seam to try getting $k=1$ either

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  • $\begingroup$ Are you trying to learn this directly from the Disquisitiones? $\endgroup$ – Hagen von Eitzen Dec 20 '14 at 9:31
  • $\begingroup$ @HagenvonEitzen yes, I looked into some of Fermat's work aswell. $\endgroup$ – user1 Dec 20 '14 at 9:45
  • $\begingroup$ Quadratic reciprocity was discovered by Euler (without proof) $\endgroup$ – user8268 Dec 20 '14 at 9:55
  • $\begingroup$ @user8268 any idea what his question was? $\endgroup$ – user1 Dec 20 '14 at 9:59
  • $\begingroup$ sorry, very unreliable info: Euler was trying to prove Fermat's claims about primes of the from $x^2+ny^2$; he did it for $n=1$, maybe for some other $n$'s (?). If $\mathbb Z(\sqrt{-n})$ is a UFD then knowing that $-n$ is a quadratic residue mod $p$ is enough to conclude that $p=x^2+ny^2$. Anyway, Euler's form of reciprocity is (as I read) different from Gauss's, namely: $(d/p)$ depends only on $p$ mod $4d$. $\endgroup$ – user8268 Dec 20 '14 at 10:18
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See Chapter 1 of Cox. Primes of the form $x^2+ny^2$ and Chapter 1 of Lemmermeyer. Reciprocity Laws: From Euler to Eisenstein for some info and further references.

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