On the Diophantine equation \(x^{2}+p^{2k+1}=4y^{n}\)
Summary: It has been proved that if \(p\) is an odd prime, \(y>1\), \(k\geq 0\), \(n\) is an integer greater than or equal to \(4\), \((n,3h)=1\) where \(h\) is the class number of the field \(\mathbb{Q}(\sqrt{-p})\), then the equation \(x^{2}+p^{ 2k+1}=4y^{n}\) has exactly five families of solution in the positive integers \(x\), \(y\). It is further proved that when \(n=3\) and \(p=3a^{2}\pm 4\), then it has a unique solution \(k=0\), \(y=a^{2}\pm 1\).
scientific article; zbMATH DE number 1814808