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.
Proof by infinite descent is a mathematical proof technique that is particularly effective in certain areas, such as number theory. It is based on the principle that a statement is true if assuming its negation leads to an infinite sequence of cases that cannot exist in practice. The idea can be summarized as follows: 1. **Assumption of Negation**: Start by assuming that there exists a solution (or an example) that contradicts the statement you are trying to prove.