Pell's equation

Pell's equation is a specific type of Diophantine equation, which is an equation that seeks integer solutions. It is typically expressed in the form: \[ x^2 - Dy^2 = 1 \] Here, \( x \) and \( y \) are integers, and \( D \) is a positive integer that is not a perfect square. The main objective is to find integer pairs \((x, y)\) that satisfy this equation.