## 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)

### MSC:

 11D25 Cubic and quartic Diophantine equations 11R16 Cubic and quartic extensions

### Citations:

Zbl 0657.10014; Zbl 0697.10011
Full Text:

### References:

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