The Legendre symbol, denoted as \left(\fracap\right), is a mathematical notation used in number theory to determine whether an integer a is a quadratic residue modulo a prime p. Specifically, it indicates if there exists an integer x such that x^2 \equiv a \mod p. The symbol takes the value 1 if a is a quadratic residue, -1 if it is a non-residue, and 0 if a is divisible by p. Legendre symbols are closely related to the study of quadratic residues and modular arithmetic. They play a significant role in number theory, particularly in quadratic reciprocity, which is a fundamental theorem that describes the solvability of quadratic equations in modular arithmetic. The properties of Legendre symbols help mathematicians understand the distribution of prime numbers and their relationships.