Simplifying the solution of Ljunggren's equation \(x^ 2+1=2y^ 4\)
type: Item
instance of: scholarly article
title: Simplifying the solution of Ljunggren's equation \(x^ 2+1=2y^ 4\)
zbMATH Open document ID: 0716.11016
DOI: 10.1016/S0022-314X(05)80029-0
published in: Journal of Number Theory
publication date: 1991-01-01T00:00:00Z
review text: \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.
Mathematics Subject Classification ID: 11D25
Mathematics Subject Classification ID: 11J86
Mathematics Subject Classification ID: 11J70
zbMATH DE Number: 4179443
zbMATH Keywords: Ljunggren's equation
zbMATH Keywords: quartic diophantine equation
zbMATH Keywords: continued fractions
zbMATH Keywords: linear forms in logarithms
author: Ray Phillip Steiner
author: Nikos Tzanakis
reviewed by: Roelof J. Stroeker
MaRDI profile type: Publication
cites work: Q3243346
cites work: A Continued Fraction Algorithm for Real Algebraic Numbers
cites work: On the practical solution of the Thue equation
cites work: Congruent numbers
full work available at URL: s0022-314x(05)80029-0
OpenAlex ID: W2011416821
description: scientific article; zbMATH DE number 4179443