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.