Remark. Trivial cases of the Pell equation [004N]

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