The solutions to some Diophantine equations

Question

I have proved these two theorems (are they correct?) but I think most probably they have occurred somewhere else already. Would you please help me find references. Thank you.

Answer 1

Your first theorem is just some subcases of Pell's Equation, so it is possible that this characterization has been done before. You should definitely search the literature further.

Your second theorem deals with a Diophantine equation that can be reduced to an instance of Pell's Equation. Thus I would expect that you might be able to use first theorem to prove the second almost immediately.

As far as verifying your proofs, we can't do that unless you show us your work! But I suspect that what you have done is probably right anyway.