Solutions to certain families of Thue equations. (English) Zbl 0774.11013

Thomas, Emery

J. Number Theory 43, No. 3, 319-369 (1993).

Let \(k\geq 3\) and let \(a_ 1(t),\dots,a_{k-1}(t)\in\mathbb{Z}[t]\) be monic polynomials with \(0<\deg a_ 1(t)<\cdots<\deg a_{k-1}(t)\). For \(n\in\mathbb{Z}\) set \(\Phi_ n(x,y)= x(x-a_ 1(n)y) \dots (x-a_{k- 1}(n)y)+uy^ k\) with \(u=\pm 1\). The author first formulates the general conjecture, that there exists an \(N\), such that for every \(n\geq N\) all solutions of the equation \[ \Phi_ n(x,y)=1 \qquad\text{in }x,y\in\mathbb{Z} \] are \(\varepsilon\{(1,0),(0,u),(a_ 1(n)u,u), \dots,(a_{k-1}(n)u,u)\}\) where \(\varepsilon=1\) if \(k\) odd, \(\varepsilon=\pm 1\) if \(k\) even.
The paper is devoted to the verification of this conjecture in case \(k=3\), under some very weak additional regularity conditions. This main result is illustrated by Theorem 3 by a doubly infinite family of cubic Thue equations:
Let \(0<a<b\) be integers. If \(n\) is an integer with \(n\geq [2\cdot 10^ 6\cdot (a+2b)]^{4.85/(b-a)}\) then the equation \[ x(x-n^ a y)(x-n^ b y)+uy^ 3=1 \qquad\text{in } x,y\in\mathbb{Z} \] (where \(u=\pm 1\)) has only the four solutions \((1,0)\), \((0,u)\), \((n^ a u,u)\), \((n^ b u,u)\).
The deep tools developed by the author in this paper will certainly have further applications.

Reviewer: I.Gaál (Debrecen)

MSC:

11D25

Cubic and quartic Diophantine equations

Keywords:

linear forms in logarithms; cubic Thue equations