# Conjectures on Twin Euler Bricks

After this answer euler bricks: way to calculate them? and this question Cuboid nearest to a cube I had some three hundred primitive Euler Brick solutions, complete with their Twin solutions (see below), and wanted to sort them into some logical order.

The equations, with $x,y,z$ as the edges and $u,v,w$ as the face diagonals, are: $$x^2+y^2=u^2$$ $$y^2+z^2=v^2$$ $$x^2+z^2=w^2$$

I used the (IMHO) well known method:

“If $(x,y,z)$ is a solution, then (xy,xz,yz) is also a solution, but applying this to the new solution returns to the original shape.”

I don’t know when or where this method of producing Twins originated, or if it has been proven, but it works for every example I’ve tried. However, the calculations soon increase in magnitude.

Conjecture 1

If $(x_0,y_0,z_0)$ is a primitive Euler Brick solution then the following is also a solution,

$$(X_1,Y_1,Z_1)=(\frac{x_0y_0}{12},\frac{x_0z_0}{12},\frac{y_0z_0}{12})$$

Put $G_1=gcd(X_1,Y_1,Z_1)$ then a new primitive Euler Brick solution is given by,

$$(x_1,y_1,z_1)=(\frac{X_1}{G_1},\frac{Y_1}{G_1},\frac{Z_1}{G_1})$$

Now repeat using $(x_1,y_1,z_1)$ as the generator,

$$(X_2,Y_2,Z_2)=(\frac{x_1y_1}{12},\frac{x_1z_1}{12},\frac{y_1z_1}{12})$$

Put $G_2=gcd(X_2,Y_2,Z_2)$ to produce,

$$(x_2,y_2,z_2)=(\frac{X_2}{G_2},\frac{Y_2}{G_2},\frac{Z_2}{G_2})$$

If all went well, $x_2=x_0$, $y_2=y_0$ and $y_2=y_0$

Definition If $\frac{G_1}{G_2}<1$ then $(x_1,y_1,z_1)$ is the Good twin and $(x_2,y_2,z_2)$ is the Evil Twin.

If $\frac{G_1}{G_2}>1$ then $(x_1,y_1,z_1)$ is the Evil Twin and $(x_2,y_2,z_2)$ is the Good Twin.

Example $$(x_0,y_0,z_0)=(117,44,240)$$ $$(X_1,Y_1,Z_1)=(429,2340,880)$$ $$G_1=gcd(429,2340,880)=1$$ $$(x_1,y_1,z_1)=(429,2340,880)$$ $$(X_2,Y_2,Z_2)=(83655,31460,171600)$$ $$G_2=gcd(83655,31460,171600)=715$$ $$(x_2,y_2,z_2)=(117,44,240)$$

Hence, as $\frac{G_1}{G_2}=\frac{1}{715}$, we find $(117,44,240)$ is the Good Twin and $(429,2340,880)$ is the Evil Twin.

The product $G_1G_2=715$, whichever Twin we start from, and is the lowest I’ve found.

Conjecture 2 The value of $G_1G_2$ is equal for each of a pair of twins.

Conjecture 3 Except for twins, the value of $G_1G_2$ is unique for primitive solutions.

Conjecture 4 There are no Identical Twins, where the original solution is returned as $(x_1,y_1,z_1)$ in any order. The nearest I’ve found is $(72611,9180,206448)$ and its Evil Twin $(12915,290444,36720)$

My questions I would welcome any help proving or disproving any of these conjectures. Also, if there is an easier way to identify Twins, I’d love to know about it.

Please let me know if I’ve failed to make anything clear.

Update 26 May 2017 I generated $82$ solutions using Saunderson’s parametric formula, http://mathworld.wolfram.com/EulerBrick.html and found they were all Good Twins.

I would appreciate any other Euler Brick parametric formula.