It would seem, after reviewing factor_it_2.pl, that that code and the mathematics behind it are not due for an update – because they are beyond hope! 😀 No, really, I’ve moved far beyond those ideas, and they no longer seem fruitful enough to upgrade them. It would be like grafting a car onto a different car, to improve its performance.

What I hope to do in the next little while is to trim down to a leaner, simpler characterize.pl. I think my revelations last night led me to see, most of all, how unnecessarily complicated my scheme for the characteristic polynomial has been. What I am trying to do sees its essence in the difference between the Ceiling Square of m and m, and the quadratic sequence obtained by increasing that Ceiling Square’s Root by 1 and taking the difference again. This will go back to a polynomial I had played with back when this study was a bulging envelope of paper notes, but I think it will give me a more reliable, less jumpy set of numbers with which to work: a characteristic polynomial that will, I hope, be more evident in its characteristics.

I’ll start work right now on trimming down the code. See how far I get before the hour gets too late. Then again, I’ve nothing going on tomorrow that requires an early wake-up, and the coffee is yummy.

**EDITED TO ADD:** HAH! It simplified it too much. The characteristic function thus simplified is f(x)=ax^{2}+bx+c where a=1, b=the Ceiling Square of m times 2, and c= the remainder! Again, no, this isn’t back to the drawing board, but I once again must make an effort to determine my next steps; no new ones are apparent from this exercise.

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