On two-parametric quartic families of diophantine problems
The two-parametric Thue equation \[ x(x-y) (x-ay) (x-by)- y^4= \varepsilon, \quad \varepsilon\in \{+1,-1\} \tag{\(*\)} \] is studied for sufficiently large values of the integer parameters \(a,b\). The authors prove that if \[ 10^{2\times 10^{28}}< a+1< b\leq a\Biggl( 1+ \frac{1}{\log^4 a}\Biggr), \] then, the only solutions to \((*)\) are given by \[ (x,y,\varepsilon)= (\pm 1,0,1),\;(0,\pm 1,-1),\;(\pm 1,\pm 1,-1),\;(\pm a,\pm 1,-1),\;(\pm b,\pm 1,-1). \] The authors also prove that if \[ 10^{3\times 10^{18}}\leq a+1< b< a\Biggl(1+ \frac{1}{\log^2 a}\Biggr), \] then the only exceptional units in \(\mathbb{Z} [\alpha]\), where \(\alpha\) is a root of the polynomial \(x(x-1) (x-a)\cdot (x-b)-1\), are the trivial ones. We remind here that a unit \(\eta\) in an order of a number field is called exceptional if \(1-\eta\) is also a unit. In the present situation \(\alpha\) and \(1-\alpha\) are exceptional units. Also, if \(a=2\), then \(\alpha-2\) is also an exceptional unit and, finally, if \(a= b-1\), then \(\alpha-a\) and \(\alpha-a-1\) are also exceptional units. These are the trivial exceptional units mentioned above. As in all papers dealing with explicit solutions of parametric Thue equations, the method of proof is very technical and is based on the theory of linear forms in logarithms and on heavy symbolic computation, which, in the case of the present paper is done with MAPLE.
scientific article; zbMATH DE number 1204261