### A family of quartic Thue inequalities

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The famous problem of determining all perfect powers in the Fibonacci sequence ${\left({F}_{n}\right)}_{n\ge 0}$ and in the Lucas sequence ${\left({L}_{n}\right)}_{n\ge 0}$ has recently been resolved [10]. The proofs of those results combine modular techniques from Wiles’ proof of Fermat’s Last Theorem with classical techniques from Baker’s theory and Diophantine approximation. In this paper, we solve the Diophantine equations ${L}_{n}={q}^{a}{y}^{p}$, with $a\>0$ and $p\ge 2$, for all primes $q\<1087$ and indeed for all but $13$ primes $q\<{10}^{6}$. Here the strategy of [10] is not sufficient due to the sizes of...

In this paper, we use Baker’s method, based on linear forms of logarithms, to solve a family of Thue equations associated with a family of number fields of degree 3. We obtain all solutions to the Thue equation$$\begin{array}{cc}\hfill {\Phi}_{n}(x,y)& ={x}^{3}+({n}^{8}+2{n}^{6}-3{n}^{5}+3{n}^{4}-4{n}^{3}+5{n}^{2}-3n+3){x}^{2}y\hfill \\ & \phantom{\rule{1em}{0ex}}-({n}^{3}-2){n}^{2}x{y}^{2}-{y}^{3}=\pm 1,\hfill \end{array}$$for $n\ge 0$.

Let $M$ be a given real quadratic field. We give a fast algorithm for determining all dihedral quartic fields $K$ with mixed signature having power integral bases and containing $M$ as a subfield. We also determine all generators of power integral bases in $K$. Our algorithm combines a recent result of Kable [9] with the algorithm of Gaál, Pethö and Pohst [6], [7]. To illustrate the method we performed computations for $M=\mathbb{Q}\left(\sqrt{2}\right),\mathbb{Q}\left(\sqrt{3}\right),\mathbb{Q}\left(\sqrt{5}\right).$

Let A be an arbitrary integral domain of characteristic 0 that is finitely generated over ℤ. We consider Thue equations F(x,y) = δ in x,y ∈ A, where F is a binary form with coefficients from A, and δ is a non-zero element from A, and hyper- and superelliptic equations $f\left(x\right)=\delta {y}^{m}$ in x,y ∈ A, where f ∈ A[X], δ ∈ A∖0 and $m\in {\mathbb{Z}}_{\ge 2}$. Under the necessary finiteness conditions we give effective upper bounds for the sizes of the solutions of the equations in terms of appropriate representations for A, δ, F, f, m. These...

We study integral points and generators on cubic twists of the Fermat cubic curve. The main results assert that integral points can be in a system of generators in the case where the Mordell-Weil rank is at most two. As a corollary, we explicitly describe the integral points on the curve.