Pell equation
A Pell equation is a type of diophantine equation in the form for a natural number . Generally, is taken to be square-free, since otherwise we can "absorb" the largest square factor into by setting .
Notice that if is a perfect square, then this problem can be solved using difference of squares. We would have , from which we can use casework to quickly determine the solutions.
Alternatively, we would like to find the set of solutions to such that the norm of , .
Family of solutions
Given a smallest solution , then all solutions are of the form for natural numbers .
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Continued fractions
The solutions to the Pell equation when is not a perfect square are connected to the continued fraction expansion of . If is the period of the continued fraction and is the th convergent, all solutions to the Pell equation are in the form for positive integer .
Generalization
A Pell-like equation is a diophantine equation of the form , where is a natural number and is an integer.