# What motiveted Gauss to formulate his theorem on quadratic reprocity?

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

• Are you trying to learn this directly from the Disquisitiones? – Hagen von Eitzen Dec 20 '14 at 9:31
• @HagenvonEitzen yes, I looked into some of Fermat's work aswell. – user1 Dec 20 '14 at 9:45
• Quadratic reciprocity was discovered by Euler (without proof) – user8268 Dec 20 '14 at 9:55
• @user8268 any idea what his question was? – user1 Dec 20 '14 at 9:59
• 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 $4d$. – user8268 Dec 20 '14 at 10:18

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.