### 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.

Let $k\ge 5$ be an odd integer and $\eta $ be any given real number. We prove that if ${\lambda}_{1}$, ${\lambda}_{2}$, ${\lambda}_{3}$, ${\lambda}_{4}$, $\mu $ are nonzero real numbers, not all of the same sign, and ${\lambda}_{1}/{\lambda}_{2}$ is irrational, then for any real number $\sigma $ with $0<\sigma <1/\left(8\vartheta \right(k\left)\right)$, the inequality $$|{\lambda}_{1}{p}_{1}^{2}+{\lambda}_{2}{p}_{2}^{2}+{\lambda}_{3}{p}_{3}^{2}+{\lambda}_{4}{p}_{4}^{2}+\mu {p}_{5}^{k}+\eta |<{\left(\underset{1\le j\le 5}{max}{p}_{j}\right)}^{-\sigma}$$ has infinitely many solutions in prime variables ${p}_{1},{p}_{2},\cdots ,{p}_{5}$, where $\vartheta \left(k\right)=3\times {2}^{(k-5)/2}$ for $k=5,7,9$ and $\vartheta \left(k\right)=[({k}^{2}+2k+5)/8]$ for odd integer $k$ with $k\ge 11$. This improves a recent result in W. Ge, T. Wang (2018).