On the diophantine equation \(x^ 2-D=4p^ n\) (Q1199982)

Let \(D\) be an odd positive integer, and \(p\) an odd prime not dividing \(D\). Further let \(N(D,p)\) denote the number of positive solutions \((x,n)\) of the diophantine equation \(x^ 2-D=4p^ n\). This equation has three solutions if \(D\) is of type I, that is, \(D=p^{2m_ 2}+p^{2m_ 1} - 2p^{m_ 2+m_ 1} -2p^{m_ 2} -2p^{m_ 1}+1\), \(m_ 1,m_ 2\in\mathbb{N}\), \(m_ 2>m_ 1\), or \(D\) is of type II, where \(D=(p^{4m}- 14p^{2m}+1)/4\), \(p=3\), \(m\in\mathbb{N}\). By a careful combination of elementary arithmetic lemmata with the known theorems on linear forms in logarithms by Baker, Mignotte, and Waldschmidt, the author is able to prove: (i) If \(D\) is of type I, then \(N(D,p)\leq 5\). (ii) If \(D\) is of type II, then \(N(D,p)\leq 4\). For a suitable \(Z_ 0\in\mathbb{N}\) (related to the least positive solution \(Z\) of \(X^ 2-DY^ 2=4p^ Z)\): (iii) If \(p^{Z_ 0}>D\), then \(N(D,p)\leq 3\), if \(p^{Z_ 0}\geq 200\), otherwise \(N(D,p)\leq 4\). (iv) If \(\max(D,p^{Z_ 0})\geq 10^{60}\), and \(D\) is not of type I or II, then \(N(D,p)\leq 3\).