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

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
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