Yes, it appears mathematics is not done with giving me a flood of ideas in the early morning hours. This time, it was in the form of interesting observations, some of which I felt compelled to work out while I was trying to get back to sleep in preparation for my church choir job, for which I had set an 8 a.m. alarm. I truly had not expected to be an early riser working on math this morning, but here I am creating a new paper in the pre-dawn hours before I get ready to go sing. Daisy and Lucie, our two dogs, have shown enormous tolerance and patience for this mathematician’s new work schedule, so to speak.

The new ideas in question do necessitate a separate paper from the most recent one that I have at least temporarily finished. They concern the structure of, and questions about, series of ceiling roots for all the nonnegative integers, but particularly the odd positive ones.

Consider a positive integer m with a ceiling square s, so that s is the smallest perfect square greater than m and m=s^{2}-r for some positive integer remainder r. The number r has its own ceiling square and remainder, and that remainder has its own, and so on, forming a series of ceiling squares for the integer m associated with its representation made by expanding each number in the series into a ceiling square and a remainder. Since this ultimately becomes a series of perfect squares, taking the square root of each produces an integer **series of ceiling roots**. I call such a process **telescoping the integer m.**

Example: Telescoping m=105, we have 105=121-16=11^{2}-4^{2}, and the process is done.

It gets messy with even numbers. Since the ceiling square of 2 is 4, and the remainder is 2, I discovered that 2 telescopes infinitely:

2 = 2^{2}-2 = 2^{2}-(2^{2}-2) = 2^{2}-(2^{2}-(2^{2}-2)) = 2^{2}-(2^{2}-(2^{2}-(2^{2}-2))), and so on. This is another mathematical concept I’ve discovered that confining my attention to the odd positive integers makes much less quirky and messy. At the present time, however, I am not absolutely certain whether or not any of the terms in an odd m’s series of ceiling roots will be 2. My gut instinct tells me no, but we will see.

This came with all kinds of other thoughts at 2:15 a.m. after I walked the dogs, and although I did prevail for a bit and got back to sleep, those ideas did keep waking me up to formulate and refine them. One of those little ideas was related to that 2 quirk, and had to do with the fact that 2 is the positive integer root of the polynomial x^{2}-2x. I was struggling with that infinitely telescoping 2 until I realized that. Worked it all out in my head, in fact.

That’s the way those early morning ideas come.

The paper is underway, and I don’t think it will take long for it to have enough interesting information and observations that I will want to share it. Stay tuned.

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