On the diophantine equation \(x^ 2+D=4p^ n\) (Q1187816)

The author considers the diophantine equation \(x^ 2+D=4p^ n\), \(x\), \(D\), \(n\) positive integers, \(p\) a prime, \((2p,D)=1\), where \(D\) and \(p\) are given. If there is a solution with \(x=1\), it is easy to construct another solution: \(x'=2p^ n-1\), \(n'=2n\). As the (historically famous) case \(D=7\), \(p=2\) shows, it is possible that there are 5 solutions. The author shows that if \(p\neq 2\) and \(D\neq 4p^ n-1\) there are at most two solutions. By a careful combination of elementary arithmetic lemmata with the known theorems on linear forms in logarithms by Baker, Mignotte, and Waldschmidt, he is able to prove that there is only one solution --- if there is any, if \(\max(D,p^ n)>\exp\exp\exp 105\).