I thought more about the last question I added into the addendum of the Numberphile, Graph theory and Mathematica post

It can be succinctly stated as:

$(\forall m\in\mathbb{Z}, m\ge 19) (\exists p,q\in\mathbb{Z}, 1\le p,q such that $\sqrt{p+m}\in\mathbb{Z}$ and $\sqrt{q+m}\in\mathbb{Z}$.

In words:

For all integers m, greater than 19, there are two other distinct positive integers less than m such that the sum of each with m, when square rooted is an integer.

What is the shortest proof you can find for this statement?

 How clear is this post?