Ooh, I have a new mathematical obsession, following, as the last several such have done, from what I have just brought to some state of (in)completion.
Consider the ceiling square of a positive integer m, the integer ceiling of the square root of m. Subtracting m from this number produces a remainder r. Applying this step now to r rather than m, and continuing in this fashion, I produce a set of numbers associated with m that I call its series of ceiling squares (SCS).
These have simple and at least seemingly profound properties, as I have shown in past posts and the Google Docs reports I have written, or at least started writing, concerning all of them. Series of Ceiling Squares appear to me to benefit from further study. I hope their analysis will indeed be fruitful.
Correspondingly, consider the floor square of m to be the integer floor of m’s square root. This time, subtract this number from m to produce the remainder, and repeat the process with r rather than m, and then with the resultant remainder, and so on. This provides terms in a representation of m as the sum of perfect squares, and I call it the series of floor squares (SFS) of m.
I have produced the beginnings of a spreadsheet showing SCS(m) and SFS(m) for the first few m. This table will, of course, grow bigger.
I look forward to finding general relationships between, and generating insights from, SCS(m) and SFS(m). And I will keep you posted, dear reader!