A new solution of the diophantine equation \(X^ 2+1=2Y^ 4\)
The equation (1) \(x^ 2 + 1 = 2y^ 4\) was first solved by \textit{W. Ljunggren} [Avh. Norske Vid. Akad. Oslo 1942, No. 5, 1-27 (1942; Zbl 0027.01103)] by using Skolem's method. Answering a question of Mordell, \textit{R. P. Steiner} and \textit{N. Tzanakis} [J. Number Theory 37, 123-132 (1991; Zbl 0716.11016)] gave a simpler proof by using Baker's method and numerical reduction algorithms. In the present paper the author uses the Thue-Siegel method and Padé approximations to algebraic functions to deduce the surprisingly sharp upper bound \(| y| < 4.233 \cdot 10^ 7 \cdot | N|^{1/0.3676}\) for the solutions of the Thue equation \[ x^ 4 - 12x^ 2 y^ 2 + 16xy^ 3 - 4y^ 4 = N.\tag{2} \] This bound allows equation (2) to be solved for \(N = 1\) from which one obtains the solutions of (1).
Padé approximations to algebraic functions
scientific article; zbMATH DE number 638545