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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
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[1] Baker, A., A sharpening of the bounds for linear forms in logarithms, III, Acta arith., 27, 247-252, (1975) · Zbl 0301.10030
[2] {\scJ. Blass, A. M. W. Glass, D. Manski, D. B. Meronk, and R. Steiner}, Constants for lower bounds for linear forms in the logarithms of algebraic numbers. I. The general case, Acta Arith., to appear. · Zbl 0709.11037
[3] Blass, J.; Glass, A.M.W.; Manski, D.; Meronk, D.B.; Steiner, R.P., Practical solution to thue equations over the rational integers, (1987), Bowling Green State University, preprint
[4] Ellison, W.J.; Ellison, J.F.; Pesek, J.; Stahl, C.E.; Stall, D.S., The Diophantine equation y2 + k = x3, J. number theory, 4, 107-117, (1972) · Zbl 0236.10010
[5] {\scM. Mignotte}, On a conjecture of E. Thomas, in preparation. · Zbl 0780.11013
[6] Mignotte, M.; Waldschmidt, M., Linear forms in two logarithms and Schneider’s method, III, Ann. fac. sci. Toulouse, 43-75, (1990) · Zbl 0702.11044
[7] Pethö, A., On the representation of 1 by binary cubic forms with positive discriminant, (), 185-196
[8] Pethö, A.; Schulenberg, R., Effektives Lösen von thue gleichungen, Publ. math. debrecen, 34, 189-196, (1987) · Zbl 0657.10015
[9] Steiner, R.P., On Mordell’s equation: A problem of Stolarsky, Math. comp., 46, 703-714, (1986) · Zbl 0601.10011
[10] Thomas, E., Complete solutions to a family of cubic Diophantine equations, J. number theory, 34, 235-250, (1990) · Zbl 0697.10011
[11] {\scE. Thomas}, Solutions to families of cubic Thue equations, I, preprint. · Zbl 0780.11014
[12] Thomas, E., Fundamental units for orders in certain cubic field, J. reine angew. math., 310, 33-55, (1979) · Zbl 0427.12005
[13] Tzanakis, N.; de Weger, B.M.M., On the practical solution of the thue equation, J. number theory, 31, 99-132, (1989) · Zbl 0657.10014
[14] Waldschmidt, M., A lower bound for linear forms in logarithms, Acta math., 37, 257-283, (1980) · Zbl 0357.10017
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