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 $\theta$-congruent number for $\theta=\frac{ \pi}{ 3}$ and $\frac{ 2\pi}{ 3}$ 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 $E_{ n,\frac{ \pi}{ 3}}:\ y^2=x(x-n)(x+3n)$ or/and $E_{ n,\frac{ 2\pi}{ 3}}:\ y^2=x(x+n)(x-3n)$ is zero. From this, we provide several series of non $\theta$-congruent numbers for $\theta=\frac{ \pi}{ 3}$ and $\frac{ 2\pi}{ 3}$, and non tiling numbers n with arbitrary many of prime divisors.
Categories: math.NT math.GR