Legendre symbol

The Legendre symbol is a mathematical notation that indicates whether a number is a quadratic residue modulo a prime \(p\). Specifically, for an odd prime \(p\) and an integer \(a\), the Legendre symbol \(\left(\frac{a}{p}\right)\) equals 1 if \(a\) is a number that, when squared, equals some number modulo \(p\). It equals -1 if \(a\) is not a quadratic residue modulo \(p\), meaning no such square exists. If \(a\) is divisible by \(p\), the symbol is 0. This concept helps study solutions to quadratic equations within modular arithmetic systems.