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Will Jagy
Will Jagy
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Finding Generators of GLO( Z^3,x^2 + xy + y^2 - z^2) and integer solutions

All pythagorean triples can be generated from $(3,4,5)^T$ and $(5,4,3)^T$ by the matrices:

\[ A = \left(\begin{array}{ccc} -1 & 2 & 2 \\\\ -2 & 1 & 2 \\\\-2 & 2 & 3 \end{array} \right) \hspace{0.25in} B = \left(\begin{array}{ccc} 1 & 2 & 2 \\\\ 2 & 1 & 2 \\\\2 & 2 & 3 \end{array} \right) \hspace{0.25in} C = \left(\begin{array}{ccc} 1 & -2 & 2 \\\\ 2 & -1 & 2 \\\\2 & -2 & 3 \end{array} \right) \]

Giving pythagorean triples the structure of a ternary tree. These correspond to generators of $\mathrm{SL}(\mathbb{Z}^3, x^2+y^2-z^2)$$\mathrm{O}(\mathbb{Z}^3, x^2+y^2-z^2)$ and we're looking for integer points on the ``light cone" where the norm is zero.

I remember seeing there being 5 generators for the quadratic $x^2 + xy + y^2 = z^2$. Does anyone have the reference?

Finding Generators of GL( Z^3,x^2 + xy + y^2 - z^2) and integer solutions

All pythagorean triples can be generated from $(3,4,5)^T$ and $(5,4,3)^T$ by the matrices:

\[ A = \left(\begin{array}{ccc} -1 & 2 & 2 \\\\ -2 & 1 & 2 \\\\-2 & 2 & 3 \end{array} \right) \hspace{0.25in} B = \left(\begin{array}{ccc} 1 & 2 & 2 \\\\ 2 & 1 & 2 \\\\2 & 2 & 3 \end{array} \right) \hspace{0.25in} C = \left(\begin{array}{ccc} 1 & -2 & 2 \\\\ 2 & -1 & 2 \\\\2 & -2 & 3 \end{array} \right) \]

Giving pythagorean triples the structure of a ternary tree. These correspond to generators of $\mathrm{SL}(\mathbb{Z}^3, x^2+y^2-z^2)$ and we're looking for integer points on the ``light cone" where the norm is zero.

I remember seeing there being 5 generators for the quadratic $x^2 + xy + y^2 = z^2$. Does anyone have the reference?

Finding Generators of O( Z^3,x^2 + xy + y^2 - z^2) and integer solutions

All pythagorean triples can be generated from $(3,4,5)^T$ and $(5,4,3)^T$ by the matrices:

\[ A = \left(\begin{array}{ccc} -1 & 2 & 2 \\\\ -2 & 1 & 2 \\\\-2 & 2 & 3 \end{array} \right) \hspace{0.25in} B = \left(\begin{array}{ccc} 1 & 2 & 2 \\\\ 2 & 1 & 2 \\\\2 & 2 & 3 \end{array} \right) \hspace{0.25in} C = \left(\begin{array}{ccc} 1 & -2 & 2 \\\\ 2 & -1 & 2 \\\\2 & -2 & 3 \end{array} \right) \]

Giving pythagorean triples the structure of a ternary tree. These correspond to generators of $\mathrm{O}(\mathbb{Z}^3, x^2+y^2-z^2)$ and we're looking for integer points on the ``light cone" where the norm is zero.

I remember seeing there being 5 generators for the quadratic $x^2 + xy + y^2 = z^2$. Does anyone have the reference?

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john mangual
john mangual
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tree generating solutions to x^2 Finding Generators of GL( Z^3,x^2 + xy + y^2 =- z^2) and integer solutions

All pythagorean triples can be generated from $(3,4,5)^T$ and $(5,4,3)^T$ by the matrices:

\[ A = \left(\begin{array}{ccc} -1 & 2 & 2 \\\\ -2 & 1 & 2 \\\\-2 & 2 & 3 \end{array} \right) \hspace{0.25in} B = \left(\begin{array}{ccc} 1 & 2 & 2 \\\\ 2 & 1 & 2 \\\\2 & 2 & 3 \end{array} \right) \hspace{0.25in} C = \left(\begin{array}{ccc} 1 & -2 & 2 \\\\ 2 & -1 & 2 \\\\2 & -2 & 3 \end{array} \right) \]

Giving pythagorean triples the structure of a ternary tree. These correspond to generators of $\mathrm{SL}(\mathbb{Z}^3, x^2+y^2-z^2)$ and we're looking for integer points on the ``light cone" where the norm is zero.

I remember seeing there being 5 generators for the equationquadratic $x^2 + xy + y^2 = z^2$. Does anyone have the reference?

tree generating solutions to x^2 + xy + y^2 = z^2

All pythagorean triples can be generated from $(3,4,5)^T$ and $(5,4,3)^T$ by the matrices:

\[ A = \left(\begin{array}{ccc} -1 & 2 & 2 \\\\ -2 & 1 & 2 \\\\-2 & 2 & 3 \end{array} \right) \hspace{0.25in} B = \left(\begin{array}{ccc} 1 & 2 & 2 \\\\ 2 & 1 & 2 \\\\2 & 2 & 3 \end{array} \right) \hspace{0.25in} C = \left(\begin{array}{ccc} 1 & -2 & 2 \\\\ 2 & -1 & 2 \\\\2 & -2 & 3 \end{array} \right) \]

Giving pythagorean triples the structure of a ternary tree.

I remember seeing there being 5 generators for the equation $x^2 + xy + y^2 = z^2$. Does anyone have the reference?

Finding Generators of GL( Z^3,x^2 + xy + y^2 - z^2) and integer solutions

All pythagorean triples can be generated from $(3,4,5)^T$ and $(5,4,3)^T$ by the matrices:

\[ A = \left(\begin{array}{ccc} -1 & 2 & 2 \\\\ -2 & 1 & 2 \\\\-2 & 2 & 3 \end{array} \right) \hspace{0.25in} B = \left(\begin{array}{ccc} 1 & 2 & 2 \\\\ 2 & 1 & 2 \\\\2 & 2 & 3 \end{array} \right) \hspace{0.25in} C = \left(\begin{array}{ccc} 1 & -2 & 2 \\\\ 2 & -1 & 2 \\\\2 & -2 & 3 \end{array} \right) \]

Giving pythagorean triples the structure of a ternary tree. These correspond to generators of $\mathrm{SL}(\mathbb{Z}^3, x^2+y^2-z^2)$ and we're looking for integer points on the ``light cone" where the norm is zero.

I remember seeing there being 5 generators for the quadratic $x^2 + xy + y^2 = z^2$. Does anyone have the reference?

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john mangual
john mangual
  • 22.8k
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All pythagorean triples can be generated from $(3,4,5)^T$ and $(5,4,3)^T$ by the matrices:

\[ A = \left(\begin{array}{ccc} -1 & 2 & 2 \\\\ -2 & 1 & 2 \\\\-2 & 2 & 3 \end{array} \right) \hspace{0.25in} B = \left(\begin{array}{ccc} 1 & 2 & 2 \\\\ 2 & 1 & 2 \\\\2 & 2 & 3 \end{array} \right) \hspace{0.25in} C = \left(\begin{array}{ccc} 1 & -2 & 2 \\\\ 2 & -1 & 2 \\\\2 & -2 & 3 \end{array} \right) \]

Giving pythagorean triples the structure of a ternary tree.

I remember seeing there being 5 generators for the equation $x^2 + xy + y^2 = z^2$ or something similar. Does anyone have the reference?

All pythagorean triples can be generated from $(3,4,5)^T$ and $(5,4,3)^T$ by the matrices:

\[ A = \left(\begin{array}{ccc} -1 & 2 & 2 \\\\ -2 & 1 & 2 \\\\-2 & 2 & 3 \end{array} \right) \hspace{0.25in} B = \left(\begin{array}{ccc} 1 & 2 & 2 \\\\ 2 & 1 & 2 \\\\2 & 2 & 3 \end{array} \right) \hspace{0.25in} C = \left(\begin{array}{ccc} 1 & -2 & 2 \\\\ 2 & -1 & 2 \\\\2 & -2 & 3 \end{array} \right) \]

Giving pythagorean triples the structure of a ternary tree.

I remember seeing there being 5 generators for the equation $x^2 + xy + y^2 = z^2$ or something similar. Does anyone have the reference?

All pythagorean triples can be generated from $(3,4,5)^T$ and $(5,4,3)^T$ by the matrices:

\[ A = \left(\begin{array}{ccc} -1 & 2 & 2 \\\\ -2 & 1 & 2 \\\\-2 & 2 & 3 \end{array} \right) \hspace{0.25in} B = \left(\begin{array}{ccc} 1 & 2 & 2 \\\\ 2 & 1 & 2 \\\\2 & 2 & 3 \end{array} \right) \hspace{0.25in} C = \left(\begin{array}{ccc} 1 & -2 & 2 \\\\ 2 & -1 & 2 \\\\2 & -2 & 3 \end{array} \right) \]

Giving pythagorean triples the structure of a ternary tree.

I remember seeing there being 5 generators for the equation $x^2 + xy + y^2 = z^2$. Does anyone have the reference?

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john mangual
john mangual
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