I’ve been building a new table of the positive odd discrete semiprimes. My objective is to consolidate observations about all such among the integers less than 20,000, but right now I’m not even up to 2000, so there’s plenty of work to be done, and it’s all aimed at the objective of finding something that will help me make factorization as fast and formulaic as multiplication. (I see the two as inverse processes of each other, even though right now finding that inverse process is as tricky as making Time’s Arrow flow backward.)
For the discrete semiprimes between 0 and 432 (which is simply the range of them that I have charted so far), the following chart has a scatter plot with an interesting shape that invites further study:
It looks to me like a snake’s head – or perhaps that of one of Frank Herbert’s fictional sandworms, but with a horizontal line through the middle – but what I’ve done here is calculate an (x,y) value for a discrete semiprime m = c2-r, for c2 and r being the Ceiling Square and Remainder of m that I previously have discussed here. The (x,y) position on the chart for each m is the relative position of m between (c-2)2 and c2, plotted versus the relative position of r between the nearest even or odd squares to both sides of r on the number line, as r is even or odd correspondingly.
As I suggested, it shows an interesting shape, and I hope it may be an informational one.