On a family of cubics. (English) Zbl 0734.11025

This is one more step towards the “practical solution” of Thue equations invented by the second author and B. M. M. de Weger [J. Number Theory 31, 99-132 (1989; Zbl 0657.10014)]. After solving single equations, now the method is able to solve families of equations. The first one was the “simplest” cubic family \[ X^ 3-(n-1)X^ 2Y- (n+2)XY^ 2-Y^ 3=\pm 1, \] which was completely solved [see E. Thomas, ibid 34, 235-250 (1990; Zbl 0697.10011)] and the first author [Verification of a conjecture of Thomas, Preprint, Strasbourg].
The second example, investigated in this paper, is the family of cubic equations \[ (*)\quad f(X,Y):\;X^ 3-nX^ 2Y-(n+1)XY^ 2-Y^ 3=1. \] A new difficulty arose now, because the cubic extension given by one root of \(f(X,1)=0\) isn’t Galois any more. Nevertheless, the authors are able to show that the only solutions of (*) for \(n>3.67\times 10^{32}\) are \[ (1,0),\quad (0,-1),\quad (1,-1),\quad (-n-1,-1),\quad (1,-n). \]
Reviewer: B.Richter (Berlin)


11D25 Cubic and quartic Diophantine equations
11R16 Cubic and quartic extensions
Full Text: DOI


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