The diophantine equation \(x^2+5^{2k+1}= y^n\)
The authors start with \textit{J. H. E. Cohn} [Acta Arith. 65, 367--381 (1993; Zbl 0795.11016)], who has solved the title equation in the case \(k=0\). Using \textit{T. Nagell}'s result [Nova Acta Soc. Sci. Upsal., IV. Ser. 16, No. 2, 1--38 (1955; Zbl 0057.28304)], that the equation \(5x^2+ 1=y^n\), where \(n\) is an odd integer \(\ge 3\) has no solution in integers \(x\) and \(y\) for \(y\) odd (except \(x=0\)), they show by a nice exercise that the title equation has no solution in integers \(x,y\) for all \(k\ge 0\) (except \(x=0\)).
scientific article; zbMATH DE number 1275976