On Representations of integers by the quadratic form x2 − Dy2

The representation of integers in binary quadratic forms has been a penchant for mathematicians throughout history including the well known Pierre de Fermat and Charles Hermite. The area has grown from simple representations as the sum of squares to representations of the form x²-Dy² where D\u3e1 and square-free. Based on congruence relations we will provide a classification criterion for the integers that can be represented in the form x²-Dy² for various values of D (specifically D=10 and 11). We will also discuss methods for constructing such representations using the theory of continued fractions, quadratic reciprocity and solutions to Pell\u27s equations