If solve and is a solution then . Thus, and is a solution. We can use this to generate infinitely many solutions.
I need help understanding the following theorem.
Theorem. Suppose d is a positive integer. If there is one solution to the Fermt-Pell equation (1): , then there are infinitely many solutions. If there is one solution to the Fermat-Pell equation (2): , then there are infinitely many solutions to both (1) and (2).
Proof.
Suppose there are integers a and b,
,where c will shortly be chosen to be . Suppose further that for some there are integers and such that
.When n = 1 this is possible with and . Set
[This is where I'm confused, how do they get these equations? I'm lost from here on.]These values are legal since
,
.
and
,[...] The rest of the proof follows, but I think I posted the relavent parts.
.
Thanks.