论文标题
$ \ thera $ - 昂贵的数字椭圆曲线通过fermat-type定理
The $\thera$-congruent numbers elliptic curves via a Fermat-type theorem
论文作者
论文摘要
如果有一个$ \ ta $ -triangle $（a，b，c）$，一个正整数$ n $称为$θ$ -congruent号码，其角度为$ a $ a $ a $ a $ a $ b $等于$θ$，并且其面积为$ n \ sqrt {r^2-s^2} $，$ a $ a和$ b $之间$ \ cos（θ）= s/r $，$ 0 \ leq | s | <r $是Coprime Integers。它归因于fujiwara \ cite {fujw1}，$ n $是一个$ \ ta $ -congruent的数字，并且仅当椭圆曲线$ e_n^\ ta：y^2 = y^2 = x（x+（x+（r+s）n）（x-（x-（r+s）n）$）的订单更大比$ 2 $ 2 $ 2 $ 2 ratival $ 2 rative。此外，如果$ e_n^\ ta（\ q）$的等级大于零，则自然数量$ n \ neq 1,2,3,6 $是$ \ ta $ - congruent数字。 在本文中，我们可以积极回答有关存在新的理性$θ$三角形的问题，以$θ$ - 串联数字$ n $从给定的算法中概括为Fermat的算法，该算法可以从给定的角度$θ$的条件，从而为给定的一个条件，从而产生一个新的合理右三角形，从而为给定的一个符合符合人数。我们表明，这种概括类似于$ e_n^θ（{\ mathbb q}）$中的重复公式。然后，基于$ e_n^θ（{\ mathbb q}）$中添加两个不同的点，我们提供了一种使用给定的两个不同的方法来找到$θ$ - congruent $ n $的新的理性$ \ ta $ -triangles。最后，我们为藤拉（Fujiwara）的定理2.2和定理2.3的一侧提供了替代证明。特别是，我们提供了$ e_n^θ（{\ mathbb q}）$中的所有扭转点的列表，
A positive integer $N$ is called a $θ$-congruent number if there is a $\ta$-triangle $(a,b,c)$ with rational sides for which the angle between $a$ and $b$ is equal to $θ$ and its area is $N \sqrt{r^2-s^2}$, where $θ\in (0, π)$, $\cos(θ)=s/r$, and $0 \leq |s|<r$ are coprime integers. It is attributed to Fujiwara \cite{fujw1} that $N$ is a $\ta$-congruent number if and only if the elliptic curve $E_N^\ta: y^2=x (x+(r+s)N)(x-(r-s)N)$ has a point of order greater than $2$ in its group of rational points. Moreover, a natural number $N\neq 1,2,3,6$ is a $\ta$-congruent number if and only if rank of $E_N^\ta(\Q)$ is greater than zero. In this paper, we answer positively to a question concerning the existence of methods to create new rational $θ$-triangle for a $θ$-congruent number $N$ from given ones by generalizing the Fermat's algorithm, which produces new rational right triangles for congruent numbers from a given one, for any angle $θ$ satisfying the above conditions. We show that this generalization is analogous to the duplication formula in $E_N^θ({\mathbb Q})$. Then, based on the addition of two distinct points in $E_N^θ({\mathbb Q})$, we provide a way to find new rational $\ta$-triangles for the $θ$-congruent number $N$ using given two distinct ones. Finally, we give an alternative proof for Fujiwara's theorem 2.2 and one side of Theorem 2.3. In particular, we provide a list of all torsion points in $E_N^θ({\mathbb Q})$ with corresponding rational $θ$-triangles
