Pell's equation is a type of Diophantine equation in mathematics that is expressed as x^2 - Dy^2 = 1, where D is a positive nonsquare integer.
There are several methods for solving Pell's equation, including the Chakravala method, continued fractions, and the Lagrange method. Each method involves finding the fundamental solution (smallest positive integer solution) and using it to generate infinitely many other solutions.
Pell's equation has applications in various fields such as cryptography, number theory, and physics. It is also used in solving certain types of recurrence relations and determining the shortest path between two points on a hyperbolic plane.
One common challenge is finding the fundamental solution, as it involves finding the square root of D. Another challenge is determining when to stop generating solutions, as there can be infinitely many. Additionally, the solutions can become very large, making it difficult to work with them.
Yes, there are infinitely many unsolved instances of Pell's equation, known as unsolvable or impossible cases. These occur when there are no integer solutions for a given D. The smallest unsolvable case is D = 61.