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This is my another question on Diophantine equations. Prove the following great and special problem.

Let $D$ and $k$ be positive integers and $p$ be a prime number such that $gcd(D, kp) = 1$. Prove that there is an absolute constant $C$ such that the Diophantine equation $x^2 + D = kp^n$ has at most $C$ solutions $(x, n)$. Also prove that $x^2 + 119 = 15\cdot2^n$ has only six solutions.

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A little more information, please. How do you know there is an absolute constant $C$ such that $x^2+D=kp^n$ has at most $C$ solutions? How do you know $x^2+119=15\cdot2^n$ has only six solutions? And what is great and special about these problems? – Gerry Myerson Mar 23 '12 at 5:41

These two problems are proposed by Yann Bugeaud in the $2007$ paper :some open problems about diophatines equations as open. This equation is also known as the generalized Ramanujan–Nagell equation, and there is a lot of work in this problem recently and it was confirmed that the number of solution is finite and there are reasonable estimation of the constant $C$ for some particular cases.

"This equation has a rich history and it has attracted the attention of several great mathematicians" so there is a lot of papers which can be found easily in the internet and I don't point to anything in particular here.

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