Solutions of $x^2 + 119 = 15 \cdot 2^n$ without trial and error

Question

I seen this equation at math.stack exchange

The equation $x^2 + 119 = 15 \cdot 2^n$ has only six solutions. Those are $(1,3) ,(11, 4), (19, 5), (29, 6), (61, 8)$ and other one is I don't know. This question I have seen in this site. There they given that, it has six solutions and they did not list these all solutions. I got $5$ of the solutions by my trail method. But, may be there is one more solution. Now my question is, how to find these solutions without using computer or calculator to determine these solutions and how one can conclude the number of solutions are six or some n?

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Thank you for providing the last solution (-1, 3). But, again by computation in trail and error I got the sixth solution. (701, 15). Now, I understand that, the above equation has 6 solutions in positive integers. So, we can consider this equation as Diophantine equation. Now my question, how to generalize this equations for finding the solutions?

Answer 1

look because $x^2$ then it could be also -1,so last solution is (-1,3)

Answer 2

Gandhi, there is a proof and therefore I am not going to reproduce that and claim it is mine. You may read it at here from the end of page 1 follow page 4. It shows there are only $6$ solutions.