Azarya’s Angels did not give me a realization as the result of a train of thought that seized me upon popping awake very early. However, I did pop awake early, and began studying data while enjoying coffee and a donut, and by about 7 a.m., the realization for which the Angels apparently awakened me arrived through the most curious of vehicles: an assertion I made and shared on Facebook and Twitter which turned out to be dead wrong. We learn from our mistakes, and this is an especially gratifying example of doing so.
Here is the correct realization I had: When an odd integer’s series of ceiling roots (see this report for the definition of “ceiling root”) telescopes that integer to infinity, by virtue of having a remainder of 2 at some point, that integer is itself 2 greater than an integer that is the difference between two perfect squares greater than 1.
DISCLAIMER: I have not proven this yet. I will endeavor to do so this evening. It is good to have a sense of purpose in this study. Thank you, Angels!