# 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 .

Note 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, if D is a nonsquare then there are infinitely many distinct solutions to the pell equation. To prove this it must first be shown that there is a single solution to the pell equation.

Claim: If D is a positive integer that is not a perfect square, then the equation has a solution in positive integers.

Proof: Let be an integer greater than 1. We will show that there exists integers and such that with . Consider the sequence . By the pigeon hole principle it can be seen that there exists i, j, and p such that i < j, and

.

So we now have

.

We can now create a sequence of such that and which implies r and s. However we can see by the pigeon hole principle that there is another infinite sequence which will be denoted by such that . Once again, from the pigeon hole principle we can see that there exist integers f and g such that mod H, mod H, and . Define and notice that . Also note that mod H which means that Y = 0 mod H also. We can now see that is a nontrivial solution to pell's equation.

## Contents

## Family of solutions

Let be the minimal solution to the equation . Note that if are solutions to this equation then which means is another solution. From this we can guess that is obtained from . This does indeed generate all the solutions to this equation. Assume there was another solution . If is non-minimal, then there exists some integer such that

Next, multiply the inequality by to obtain:

.

However, it can be seen that

Meaning is a solution smaller than the minimal solution which is a contradiction.

Therefore, such cannot exist and so the method of composition generates every possible solution to Pell's equation.

For a Pell equation in form of , its roots are in the form of , in which and are the elementery roots of the Pell equation.

Q.E.D.

## 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.

## Introductory Problems

Show that if and are the solutions to the equation , then .

## Intermediate Problems

- Find the largest integer satisfying the following conditions:

- (i) can be expressed as the difference of two consecutive cubes;
- (ii) is a perfect square. (Source)