Quadratic residues have significant applications in number theory and cryptography, particularly in the context of Dirichlet characters. A Dirichlet character is a function that focuses on integers relatively prime to a modulus, with examples illustrating their construction using the Legendre symbol. Primitive characters, which cannot be induced from non-trivial divisors, are uniquely described by quadratic symbols derived from quadratic reciprocity. Furthermore, modern integer factorization algorithms leverage small quadratic residues to find nontrivial factors of composite numbers. These applications highlight the relevance of quadratic residues beyond theoretical mathematics.