Research on Rational Quadratic Residues

This paper introduces the concept of rational quadratic residues, namely fractional quadratic residues. It also presents a method for determining quadratic residues with fractions as moduli, as well as computational symbols for rational quadratic residues and formulas for converting rational Legendre symbols to general Legendre symbols. Formulas for calculating the number of rational quadratic residues and rational quadratic non-residues are derived. Additionally, several practical application examples of rational quadratic residues are highlighted. The paper further examines the discrimination and applications of rational quadratic residues. Finally, several conjectures related to rational quadratic residues are proposed.