On Le's and Bugeaud's papers about the equation \(ax^2 + b^{2m-1} = 4c^p\) (Q1849561)

This paper corrects an inacurracy which appeared in papers of M. H. Le, M. Mignotte and D. Bugeaud on the generalized Ramanujan-Nagell equation. In a paper published in Monatsh. Math. 120, 121--125 (1995; Zbl 0877.11020), \textit{M. H. Le} claimed that the equation \(ax^2+b^{2m-1}=4c^p\) has only finitely many solutions in positive integers with \(\gcd(ax,b)=1\) when \(p>5\) is a prime which does not divide the class number of \({\mathbb Q}(\sqrt{-ab})\). The author gives a counter-example with \(p=7\), and he proves that the results of the previous papers can be obtained under the extra assumption that \(b\) is a square-free integer.