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On $θ$-congruent numbers on real quadratic number fields

Let ${\mathbb K}={\mathbb Q}(\sqrt{m})$ be a real quadratic number field, where $m>1$ is a squarefree integer. Suppose that $0 < θ< π$ has rational cosine, say $\cos (θ)=s/r$ with $0< |s|<r$ and $\gcd(r,s)=1$. A positive integer $n$ is called a $(\mathbb K,θ)$-congruent number if there is a triangle, called the $(\mathbb K,θ, n)$-triangles, with sides in $\mathbb K$ having $θ$ as an angle and $nα_θ$ as area, where ${α_θ}=\sqrt{r^2-s^2}$. Consider the $(\mathbb K,θ)$-congruent number elliptic curve $E_{n,θ}: y^2=x(x+(r+s)n)(x-(r-s)n)$ defined over $\mathbb K$. Denote the squarefree part of positive integer $t$ by ${\rm sqf}(t)$. In this work, it is proved that if $m\neq {\rm sqf}(2r(r-s))$ and $mn\neq 2, 3, 6$, then $n$ is a $(\mathbb K,θ)$-congruent number if and only if the Mordell-Weil group $E_{n,θ}(\mathbb K)$ has positive rank, and all of the $(\mathbb K,θ, n)$-triangles are classified in four types.