Pell's equation (simple solutions)
Pell's equation is any Diophantine equation of the form where
is a given positive nonsquare integer, and integer solutions are sought for
and
Denote the sequence of solutions
It is clear that
During the solution we need:
a) to construct a recurrent sequence or two sequences
b) to prove that the equation has no other integer solutions.
Equation of the form ![$x^2 – 2y^2 = 1$](//latex.artofproblemsolving.com/2/7/9/27916919e274846741e72cf16f2b9b58113feea8.png)
Let integers
are the solution,
then
therefore integers are the solution of the given equation.