On a family of quintic Thue equations
Starting with a paper of \textit{E. Thomas} [J. Number Theory 34, 235-250 (1990; Zbl 0697.10011)], several parametrized families of Thue equations have been solved -- either completely or for parameter values exceeding some explicit bound. Now for the first time a family of degree \(5\) is investigated. The main result of the paper under review is: for integers \(a\) with \(| a| \geq 3.6 \cdot 10^{19}\) or \(| a| \leq 100\) the only solutions of the Thue equation \[ X(X^2-Y^2)(X^2-a^2Y^2)-Y^5 = \pm 1 \] are \((\pm 1,0)\), \((0,\pm 1)\), \((\pm 1, \pm 1)\) and \((\pm a,\pm 1)\). Supposing the existence of a further solution, one obtains a unit in a quintic algebraic number field. Let \(U\) be the -- in absolute value -- largest exponent when expressing that unit by fundamental units. Applying some standard techniques for Thue equations and Baker's method, the author derives an upper bound for \(U\). Calculating the asymptotic expansions of the exponents with respect to the parameter \(a\), it turns out that they all have the same order of magnitude. Therefore the author constructs small non-vanishing linear combinations of these exponents, from which he derives a lower bound for \(U\), exceeding the upper bound for \(| a| \geq 3.6 \cdot 10^{19}\).
scientific article; zbMATH DE number 1204262