Simplifying the solution of Ljunggren's equation \(x^ 2+1=2y^ 4\)
\textit{W. Ljunggren} solved the title equation in his 1942 paper [Zur Theorie der Gleichung \(x^ 2+1=Dy^ 4\), Avh. Norske Vid. Akad. Oslo No.5, 1-27 (1942; Zbl 0027.01103)] by a rather complicated argument depending on the structure of units of relative norm -1 in a quadratic extension of a quartic number field. The solutions are \((x,y)=(1,1)\), (239, 13) and no others. In the present paper the authors offer an entirely different proof, essentially depending on a deep result of \textit{M. Mignotte} and \textit{M. Waldschmidt} [Acta Arith. 53, 251-287 (1989; Zbl 0642.10034)] on lower bounds of linear forms in logarithms of algebraic numbers. It is remarkable that the high-precision calculations involved do not require more than 30 significant decimal digits.
scientific article; zbMATH DE number 4179443