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.
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um, this is usually called stereographic projection. You take a fixed solution, $(1,0)$ take coprime integers $(u,v),$ write $(x,y) = (1+tu,tv)$ and solve for the nonzero value of $t$ that causes $(x,y)$ to solve your equation – Will Jagy Jan 05 '19 at 02:55
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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.
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@sai Ah, I assumed you were only caring about the positive $m$'s. Then a more complete solution is found here on page 92, remark 2. – ItsJustTranscendenceBro Jan 05 '19 at 03:37
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@sai So far, none of the references I have looked at since your comment deal with your cases this succinctly, but I think it is fair to say that these are generally known though specification from other theorems or general solution techniques. – ItsJustTranscendenceBro Jan 05 '19 at 03:40
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