The diophantine equation \(x^3-3xy^2-y^3=1\) and related equations
type: Item
instance of: scholarly article
title: The diophantine equation \(x^3-3xy^2-y^3=1\) and related equations
zbMATH Open document ID: 0552.10008
DOI: 10.1016/0022-314X(84)90053-2
author: Nikos Tzanakis
published in: Journal of Number Theory
publication date: 1984-01-01T00:00:00Z
review text: The equation of the title is an example of an equation to which Skolem's p-adic method cannot be directly applied since the equation \(\theta^ 3- 3\theta -1=0\) has three real roots. Indeed, it was solved by \textit{W. Ljunggren} [Acta Math. 75, 1-21 (1942; Zbl 0027.15702)] by the use of intermediate extension fields of degree 6. In the present paper the author gives a more direct solution of the title equation. This is done as follows: First, the author solves the equation \(x^ 2+3=4y^ 3\) by reducing it to solving an equation of the form \(f(x,y)=1\), where f is a binary quartic form with two real roots. This reduction allows Skolem's method to be applied directly, and, in fact, implements a basic idea of \textit{W. Ljunggren} [Acta Arith. 17, 379-387 (1971; Zbl 0216.041)] concerning such problems. A further simple reduction then leads to the solution of the equation of the title. Finally, the author points out that the method of the paper has a general character in that an equation \(F(x,y)=1\), where F is a binary cubic form with positive discriminant can be reduced to solving an equation of the form \(Ay^ 2+B=x^ 3\), which can then be reduced to solving a finite number of equations \(f(x,y)=1\), where f is a binary quartic form with two real roots. He gives 3 examples to illustrate this point.
Mathematics Subject Classification ID: 11D25
zbMATH DE Number: 3879007
zbMATH Keywords: binary cubic equations
zbMATH Keywords: binary quartic equations
zbMATH Keywords: Skolem's p-adic method
zbMATH Keywords: \(x^ 2+3=4y^ 3\)
zbMATH Keywords: binary quartic form
zbMATH Keywords: binary cubic form
zbMATH Keywords: \(Ay^ 2+B=x^ 3\)
reviewed by: Ray Phillip Steiner
MaRDI profile type: Publication
cites work: Q5582375
cites work: Q5849879
cites work: Q4760981
cites work: On the diophantine equation y²-k=x³
cites work: On Mordell's equation \(y^2-k=x^3\): an interesting case of Sierpiński
cites work: Algebraic Number-Fields with two Independent Units
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cites work: On the representation of integers by certain binary cubic and biquadratic forms
description: scientific article; zbMATH DE number 3879007