Gauss's law for quadratic residues

Gauss's law for quadratic residues relates to the behavior of numbers when squared and divided by a certain number, often a prime. Specifically, it helps us understand which numbers, when squared, yield the same remainder when divided by that prime. In essence, it states that if you take pairs of residues, their squares will combine in a predictable way, allowing mathematicians to derive important properties about them. This principle has applications in number theory, cryptography, and understanding the symmetry and structure of numbers in modular arithmetic.