Constructive proof need to know the solutions of the equations

Question

Observe the following equations:

$2x^2 + 1 = 3^n$ has two solutions $(1, 1) ~\text{and}~ (2, 2)$

$x^2 + 1 = 2 \cdot 5^n$ has two solutions $(3, 1) ~\text{and}~ (7, 2)$

$7x^2 + 11= 2 \cdot 3^n$ has two solutions $(1, 2) ~\text{and}~ (1169, 14)$

$x^2 + 3 = 4 \cdot 7^n$ has two solutions $(5, 1) ~\text{and}~ (37, 3)$

How one can determine the only number of solutions are two or three or four...depends up on the equation. especially, the above equations has only two solutions. How can we prove there is no other solutions? Or how can we get solutions by any particular method or approach?

Answer 1

The proof for solutions of $\displaystyle{2x^2+1=3^n}$ can be read from the paper at American Mathematical Society Volume 131, Number 12

According to that three solutions are $(1,1), (2,2)$ and $(11,5)$

NOTE: I believe one cannot attempt with one approach to solve all of those equations.

ADDING THESE NOTE (Since it was requested in the comment here) A few papers that explains applications of Diaphontine Equations 1. Application of Linear Diaphontine Equations in Teaching Mathematical Thinking 2. Applications of Diaphontine Equations to Combinatorial Problems

I believe it can also be applied in Genetic Algorithms (I am not a specialist in that area, but I believe it is true).

Answer 2

All four of your equations (and many more) are mentioned in Saradha and Srinivasan, Generalized Lebesgue-Ramanujan-Nagell equations, available at http://www.math.tifr.res.in/~saradha/saradharev.pdf. The solutions are attributed to Bugeaud and Shorey, On the number of solutions of the generalized Ramanujan-Nagell equation, J Reine Angew. Math. 539 (2001) 55-74, MR1863854 (2002k:11041).