原始欧拉砖生成器

Djamel Himane
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引用次数: 0

摘要

保罗-哈尔克(Paul Halcke)发现的最小欧拉砖的边长为 $(177,44,240)$,面对角线为 $(125, 267, 244 )$,由原始毕达哥拉斯三重 $ (3, 4, 5) $ 生成。让$(u,v,w)$原始勾股定理三重边,Sounderson 对边做了广义参数化:a = \vert u(4v^2 - w^2) \vert, \quad b = \vert v(4u^2 -w^2)\vert, \quad c = \vert 4uvw \vert \end{equation*} 给出面对角线(begin{equation*})。{\displaystyle d=w^{3},\quad e=u(4v^{2}+w^{2}),\quadf=v(4u^{2}+w^{2})}\end{equation*} 引出欧拉砖。除上述公式外,找到生成这些原始砖块的其他公式,或者做出可以在以后加以改进的初步猜测,是理解这些砖块是如何生成的关键。
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Primitive Euler brick generator
The smallest Euler brick, discovered by Paul Halcke, has edges $(177, 44, 240) $ and face diagonals $(125, 267, 244 ) $, generated by the primitive Pythagorean triple $ (3, 4, 5) $. Let $ (u,v,w) $ primitive Pythagorean triple, Sounderson made a generalization parameterization of the edges \begin{equation*} a = \vert u(4v^2 - w^2) \vert, \quad b = \vert v(4u^2 - w^2)\vert, \quad c = \vert 4uvw \vert \end{equation*} give face diagonals \begin{equation*} {\displaystyle d=w^{3},\quad e=u(4v^{2}+w^{2}),\quad f=v(4u^{2}+w^{2})} \end{equation*} leads to an Euler brick. Finding other formulas that generate these primitive bricks, other than formula above, or making initial guesses that can be improved later, is the key to understanding how they are generated.
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