Mathematics
Quadratic Reciprocity: The Law of Modular Arithmetic
Quick fact
For any two odd primes p and q, the question of whether p is a square modulo q is answered by the sign of (-1)^((p-1)(q-1)/4), a single bit of information that depends only on the primes themselves.
Why this is interesting
You know that 1 is a perfect square, and that 4 is a perfect square. But what does it mean to be a square in the world of remainders? The answer holds a hidden symmetry that has fascinated mathematicians for centuries.