The Latest Math Idea to Get Me Up Early


All possible factorizations into two factors of the positive integers correspond to all integer points on the family of curves y=m/x, for m an element of the infinite set {1,2,3,4,5,…}, and we can restrict our attention, thanks to mirror symmetry, to the points where x and y are greater than zero (symmetric by Cartesian quadrant), and x is greater than y (symmetric on the main diagonal).

I believe this is, essentially, a geometric transformation of points on a structure relating to the Sieve of Eratosthenes, but I want to look at the graph of the data anyway.

I decided to plot these points for values of m up to 255, because that is the value of m for the storey of the Ceiling Squares Tower where I stopped building.

EDITED TO ADD: When I consider that each integer point on the x/y plane corresponds to such a factorization, and thus to an integer point of a curve in that family, my interest in seeing the entire set of integer points below a certain value of m just kind of dissipated. Sorry. It would have been a boring picture.

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