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:

such that and .

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?

Jonathan RaynerJanuary 19, 2018 at 7:59 amI’ll give it a shot:

To cheat a bit and simplify the argument, assume that we have verified the cases up to and including . By direct calculation

(this is true for and so true for all ). So the interval is of length >2, which means that that it contains two distinct natural numbers . Now:

where , . Then and satisfy the requirements of the question.

Jonathan ShockJanuary 19, 2018 at 8:22 pmLovely

So, the next question is, what other sufficient conditions for non-traceability might exist?