Discovering Quadratic Reciprocity Is there anything similar to this (page written by Field Medalist Timothy Gowers) for quadratic reciprocity ? 
I mean, the link there explains how you can figure out the solution of cubic equation by yourself without having a suddent flash of inspiration/ genius genes. Is there some similar guide for quadratic reciprocity ?
 A: There is a book by David Marshall, Edward Odell, and Michael Starbird called "Number Theory Through Inquiry". Chapter 7 is about quadratic reciprocity. It's sort of guided inquiry. The authors suggest questions for the reader, such as looking for patterns in certain tables. So the reader has some guidance and is not just completely on her own. But on the other hand there are a lot of exercises and theorems to be proven; the authors give the reader an opportunity to develop everything. Perhaps this book might fit your needs.
A: The first part of this paper http://www.math.ubc.ca/~belked/lecturenotes/620E/Frei%20-%20The%20Reciprocity%20Law%20from%20Euler%20to%20Eisenstein.pdf shows you how the QR law was discovered historically and shows the path all the way back to Diophantus. 
It seems that the first question explicitly stated that is equivalent to a case of QR is
p is an odd prime of the form $x^2+ y^2$ with x, y integers $\iff$ p is equal to 1 mod 4. 
This leads to the more general question:
Given $N\in \mathbb{Z}$, describe the primes $p \notin 2$ for which $p = x^2 + Ny^2$. 
This was considered by Fermat and studied carefully by Euler and leads to the full QR theorem whereas the question for sum of two squares leads to the evaluation of the quadratic character of -1 mod p only.
A: The first chapter of Cox's book, "Primes of the form $x^2+ny^2$", discusses the origins of quadratic reciprocity, and shows through the discussion and by extended examples how Euler was led to discover quadratic reciprocity from considerations of solutions to this equation for various $n$.
