论文标题
$θ$ - 昂贵的数字，平铺编号和相关椭圆曲线的Selmer等级：奇数n
$θ$-Congruent Numbers, Tiling Numbers and the Selmer Rank of Related Elliptic Curves: odd n
论文作者
论文摘要
几个离散的几何问题与在理性字段$ \ mathbb {q} $上定义的椭圆曲线的算术理论密切相关。在本文中，我们考虑$θ$ -CONCRUENT编号，$θ= \fracπ{3} $和$ \ frac {2π} {3} {3} $和瓷砖编号n。对于$ n \ geqslant 2 $是无平方的整数y^2 = x（x+n）（x-3n）$为零。由此，我们为$θ= \fracπ{3} $和$ \ frac {2π} {3} $提供几个非$θ$ - 综合数字，以及具有任意许多prime分隔线的非平铺数字n。
Several discrete geometry problems are closely related to the arithmetic theory of elliptic curves defined on the rational fields $\mathbb{Q}$. In this paper we consider the $θ$-congruent number for $θ=\fracπ{3}$ and $\frac{2π}{3}$ and tiling number n. For the case that $n\geqslant 2$ is square-free odd integer, we determine all $n$ such that the Selmer rank of elliptic curve $E_{n,\fracπ{3}}:\ y^2=x(x-n)(x+3n)$ or/and $E_{n,\frac{2π}{3}}:\ y^2=x(x+n)(x-3n)$ is zero. From this, we provide several series of non $θ$-congruent numbers for $θ=\fracπ{3}$ and $\frac{2π}{3}$, and non tiling numbers n with arbitrary many of prime divisors.
