Quadratic reciprocity is the first theorem I encountered in number theory for which I never developed an intuition for why it's true, and it's heartening to rea

Quadratic reciprocity is the first theorem I encountered in number theory for which I never developed an intuition for why it's true, and it's heartening to read in the article that professional mathematicians still feel the same way about it. It can turn up unexpectedly. For instance, here's a problem/puzzle that a co-worker came up with: Find all positive integers b for which whenever (x^2 + xy + y^2) written in base b (for some integers x and y) ends with 0, it ends with two 0s. (Answer in rot13: gubfr gung ner n cebqhpg bs qvfgvapg cevzrf gung ner rnpu gjb zbqhyb guerr. This can be proved using quadratic reciprocity, at least the special case that (-3) is a square mod p iff p is a square mod 3.)