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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?

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@pedja! waiting for a solution –  gandhi Mar 26 '12 at 7:43
2  
Waiting for Godot –  pedja Mar 26 '12 at 7:51
    
related? –  draks ... Mar 26 '12 at 8:20
    
this too –  draks ... Mar 26 '12 at 8:21
    
@pedja! you have given a good material on "Waiting for Godot", really great.Thank you. –  gandhi Mar 26 '12 at 9:10

2 Answers 2

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).

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! I cam to know the third solution. Also, I understand that, there is no particular method to solve Diophantine equations. But, I had a question in my mind. Why we should learn about Diophantine equations? just for fun or any importance there in mathematics or other sciences? Thank you so much for your help. –  gandhi Mar 27 '12 at 3:51
    
@gandhi I have added a few references to check. I could not find any applications in Genetics (You can look up, and if you find share with us here). Keep up the good work! –  Kirthi Raman Mar 27 '12 at 11:55

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).

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! after your reply, I gone through the paper. I am not happy with the paper and lot of revision is required. Anyhow, thank you so much for your reply. Because of you, I came to know about other equations. –  gandhi Mar 27 '12 at 3:49

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