Legendre

One of the most celebrated theorems in number theory is the law of quadratic reciprocity. We formulated it at the end of Chapter 3. The history of the discovery of this theorem is complicated and not quite clear, but we will shortly show that one is led to the theorem by the problem of deciding whether a given prime number divides a number of the form x2 + ay2. This was how Euler and later (around 1785), independently, Legendre discovered the theorem. Unlike Euler, Lagrange tried to prove the theorem, but his proof had serious gaps. We will discuss it below. Finally, it was rediscovered by Gauss, probably after numerical calculations and not in connection with the theory of binary forms. Gauss gave the first complete proof.