The diophantine equation \(x^3-3xy^2-y^3=1\) and related equations
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.
scientific article; zbMATH DE number 3879007