Ceiling Squares: My New Mathematical Study

Welcome to my new mathematical study! My work on integer factorization led, not to any new discoveries in number theory, but to a way of organizing the odd positive integers that lends itself to visualization and can provide insight: the Ceiling Square.

Simply put – or, at least, as compactly phrased as I can imagine to put it – the Ceiling Square of size s-1 is an arrangement of the s-1 positive odd integers ranging from (s-1)2 minus either 1 or 2, depending on s being even or odd, respectively, to s minus either 1 or 2, depending on s being odd or even, in s-1 columns corresponding to their values (residue classes) modulo 1 to s-1, with s being a positive integer greater than 1.

I call these arrangements Ceiling Squares because they are ways of looking at the odd integers as values m=s2-r, where s is the integer ceiling of the square root of m, The rows correspond to values of r from either 2s-3 or 2s-2 down to 1 or 2, according to the value of s and thus s2 being even or odd, respectively. This produces all of the odd positive integers between (s-1)2 and s2.

I created the Ceiling Squares Tower by stacking Ceiling Squares, starting with the 1 by 1 square corresponding to the number 3, atop one another, aligned by the column for the modulus 1. In the picture of the Ceiling Squares Tower above, that column is the rightmost one. I made this choice, arranging moduli from s-1 down to 1, for the sake of one of the questions I wish to study: In row m of a Ceiling Square, what is the leftmost value of n that will make the entry for column s-n be zero? This corresponds to the largest positive integer s-n that divides m. Here is a note on coloration of the image above: In the Ceiling Squares Tower, the case where 1 is the largest such integer indicates a prime number, which is marked by a green box on the rightmost side of that row (storey) of the Tower. If the leftmost zero residue occurs at n less than s-1, the corresponding block gets a blue background instead.

Odd perfect squares become bright yellow intermediate storeys in the illustration above. Squares of even integers, being even themselves, do not appear as storeys.

Some properties of Ceiling Squares jump out immediately: Every number in the column for modulus 1 will always be 0, since 1 evenly divides every integer, and the column for modulus 2, right next to it, will always be 1 due to the restriction of rows to odd numbers only. Every row will have at least one zero, in the rightmost column, corresponding to the factorization m=m*1. Additional zeros in a row indicate additional possible factorizations of m, corresponding to its factors that are greater than 1 and less than s. If m is composite, there will be at least one additional such zero in the row. If there are more than one way to factor m besides m*1, there will be more than one zero in the row.

The number in the leftmost top box of a Ceiling Square will be 1 or 2, according to whether the size of the Square is even or odd, respectively. This sets a kind of starting position in the sequence of residues that increases by 2 modulo n going down column s-n, a property that holds not only for the rest of the residues in that Ceiling Square, but in all subsequent Squares, provided the value also advances by 2 when the row corresponds to an odd perfect square between two Ceiling Squares.

Other properties may not be so immediately apparent: A special factorization case arises when r is a perfect square: s-r will be a factor of m, as x-r is a term in the factorization of a quadratic polynomial in x, x2-r2, evaluated at x=s. In the Ceiling Squares Tower visualization, I indicate this special case with an orange line to the left of the row of numbers. Also, with the columns arranged in this order and colored according to the choices already explained, a Ceiling Square with odd side length will always have its bottom left box be blue, except for the 1 by 1 Ceiling Square corresponding to m=3. If the side length is even, a box one to the right and one upward is the box that will always be blue, except for in the 2 by 2 Ceiling Square.

What happened to Row 1? Since there are no positive odd integers between 0 and 1, imagine Row 1 being there, but of zero dimension, at the very top of Ceiling Squares Tower, Row 1 is therefore invisible, and it resists simple categorization by Ceiling Square rules, since 1 is neither prime nor composite.

Ceiling Squares, and the Ceiling Squares Tower I made by stacking them atop one another, may be little more than an enhanced visualization of principles of the Sieve of Eratosthenes and other elementary analyses of the positive integers. But, as with the special cases described above, a good visualization can sometimes lead to less obvious insights. I have found this concept, and how it emerged from my study of integer factorization, fascinating and exciting. The arithmetic structures involved are also, in my view, beautiful. I count them among the patterns of nature, the Nature of Number.