Q:

Pythagorean Theorem on Spiral of Theodorus Triangles

I have 1 right triangle of dimensions $\sqrt75$$, 11, 14$. I'd like to know how to quickly obtain the other right triangles with $\sqrt75$ as a leg, and two integers as the hypotenuse and the other leg (as per the Pythagorean theorem). It is to my understanding that these triangles are all connected somehow geometrically and, consequently, algebraically.  Are the necessary techniques for quickly obtaining them related to: https://en.wikipedia.org/wiki/Spiral_of_Theodorus and/or the proof using differential techniques shown here: https://en.wikipedia.org/wiki/Pythagorean_theorem ?

A:

Factor $75=3 \cdot 5^2$. This implies that $75$ has 6 divisors, namely $1,3,5,15,25,75$.
Your problem is equivalent on looking for all integer solution to
$$x^2+75=y^2$$
Now, write this equation as
$$(y+x)(y-x)=75$$
so that you have to solve 6 different linear systems
$$\left\{ \begin{matrix} y &+&x& =& a \\ y&-&x&=& 75/a\end{matrix} \right.$$
where $a$ is a divisor of $75$. For example, for $a=25$ you get the solution $x=11, y=14$.
The other solutions are $(37,38)$ and $(5,10)$ (and other 3 negative solutions, which must be discarded).

