User:Alexander V. B./sandbox

Pythagorean Triple Parabola Set
When Pythagorean triples are arranged in a certain order, they will form a set of infinite parabolas.

Function for the Equation of a Parabola
To have the equation of a parabola in the set, the function for the equation is:
 * $$P(m)$$ → $$y = \frac{1}{2m^2}x^2 - \frac{m^2}{2}$$

where $$m$$ is the order of the parabola in the set.

Points in a Parabola that Forms a Pythagorean Triple
Not all parts of the parabola form a Pythagorean triple. To have the specific points where a Pythagorean triple is formed, either the function for the $$x$$ coordinate or the $$y$$ coordinate of the point can be calculated. The formula for the $$x$$ coordinate of the point is:
 * $$X(m;n) = m^2 + 2mn$$

While the formula for the $$y$$ coordinate of the point is:
 * $$Y(m;n) = 2n^2 + 2mn$$

where $$m$$ is the order of the point's parent parabola, and $$n$$ is the order of the point itself.

Length of the Hypotenuse
Assuming that:
 * $$(0,0)$$ to $$(x,0)$$ is side $$p$$,
 * $$(x,0)$$ to $$(x,y)$$ is side $$q$$,
 * and $$(0,0)$$ to $$(x,y)$$ is side $$r$$, which is the hypotenuse,

the length of the hypotenuse is:
 * $$r = p + 2n^2 = q + m^2$$

where $$m$$ is the order of the parent parabola and $$n$$ is the order of the point in the parabola.

Primitive or Non-Primitive
A primitive Pythagorean triple is a Pythagorean triple in which the greatest common divisor of all of its three sides is 1.

A non-primitive Pythagorean triple, on the other hand, is a Pythagorean triple in which the greatest common divisor of all of its three sides is more than 1.

There are several rules that can be used to easily determine whether a Pythagorean triple is primitive or not using the value of $$m$$ and $$n$$, without knowing the length of any of its three sides. These rules are:
 * 1) If the value of $$m$$ of a parabola is equal to 1, all Pythagorean triples in the parabola are primitive.
 * 2) If the value of $$m$$ of a parabola is an odd number other than 1:
 * 3) * If the value of $$n$$ of a point in the parabola is coprime to $$m$$, the Pythagorean triple is primitive.
 * 4) * If the value of $$n$$ of a point in the parabola is NOT coprime to $$m$$, the Pythagorean triple is non-primitive.
 * 5) If the value of $$m$$ of a parabola is an even number, all Pythagorean triples in the parabola are non-primitive.