### 14-term arithmetic progressions on quartic elliptic curves.

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Generalizing a result of Bombieri, Masser, and Zannier we show that on a curve in the algebraic torus which is not contained in any proper coset only finitely many points are close to an algebraic subgroup of codimension at least $2$. The notion of close is defined using the Weil height. We also deduce some cardinality bounds and further finiteness statements.

In this paper, we study equations of the form $C{y}^{d}=F(x,z)$, where $F\in \mathbb{Z}[x,z]$ is a binary form, homogeneous of degree $n$, which is supposed to be primitive and irreducible, and $d$ is any fixed integer. Using classical tools in algebraic number theory, we prove that the existence of a proper solution for this equation implies the existence of an integral ideal of given norm in some order in a number field, and also the existence of a specific relation in the class group involving this ideal. In some cases, this result...

T. Cochrane and R. E. Dressler [CD] proved that the abc-conjecture implies that, for every > 0, the gap between two consecutive numbers A ${A}^{0.4}$ with two exceptions given in Table 2.