For my latest mathematical explorations, the inspiration for which have come like the Ceiling Squares study in the form of ideas demanding me to pursue them in the early hours of morning, I have come up with a way to assign odd integer values to integer points on the Cartesian grid of x>0 and y>0, with the following restriction: x takes on the odd integer values from 1 to positive infinity, and y takes on the even integer values from 0 to positive infinity. The function that assigns an odd integer value to each point on this “number pegboard” is the function f(x,y)=|x2-y2|. The vertical bars indicate the absolute value function.
Each odd positive integer will appear on at least one point of the pegboard according to how many ways there are to express the integer as the difference of two perfect squares. An odd positive integer will be either below or above the line y=x corresponding to whether the integer is congruent to 1 or -1 modulo 4, respectively. Another way of expressing this is whether the integer has a remainder of 1 or 3 when divided by 4. This has to do with whether the odd term in the function, x2, is greater or less than the even y2 term, since all even perfect squares are congruent to 0 modulo 4, and all odd perfect squares are congruent to 1 modulo 4.
A composite number has more than one expression as the difference of two squares, each such expression corresponding to a distinct way of expressing the integer as the product of two not necessarily distinct integers.
This lends itself to some interesting observations, at least so far. The finite number of x,y points on the pegboard corresponding to a particular numeric value of f(x,y) form the integer points of the curve in real xyz space – the contour line, in mapping terms – where f(x,y) gives that value. Finding an equation for that line might be helpful in exploring any integer’s factorization structure.