For an integer we call a diophantine equation of the form Pell's equation.

1. If , or , then is the only solution.
2. If all solutions are and an aritrary .
3. If all solutions are .

Proof.

1. In the first case we can write the equation as , then . If , write . This forces and thus . If is of the form , then we have , forcing and .
2. This is clear.
3. We have . Our only options are and

Let be a natural number and not a square. We denote the set by , read as adjoint with .

For an element of we define its conjugate by .

We can identify a solution of the Pell equation with the pair . In this case is the inverse of , since . From this point of iew, the solutions of the Pell equation are precisely the units of .

We obtain new solutions by multiplying solutions we have already found.

Proof. Let and in be invertible; then is also invertible since . Therfore, also corresponds to a solution.

The minimum element of the set is well-defined. We call this element the fundamental solution of the Pell equation and denote it by .

Proof. TODO

Sei be non-square. Then the solutions of the Pell equation are exactly . where is the fundamental Solution.

Proof. TODO