Integer Factorization: Recent Work


Ideas kept me up when the dogs got me up around 3 a.m. this morning for a walk and some water. That’s good, even though the ideas did not pan out to much. I had church to attend in a few hours, after all. But it occurs to me to share where it’s all brought me, because despite the feeling of not having made any breakthroughs, it’s also definitely not back to the drawing board. A process I hope will result in eventual realization/revelation seems still to be at work in my mind, and I hope I am translating it effectively into the steps of my continuing integer factorization study.

First of all, I need to move past my last satisfactory attempt at programming a factorization algorithm,, to use the ceiling squares/Fermat-like factorization concept as I now frame it, but also to take advantage of the information that when m, the odd positive integer I wish to factor, produces a value of (m+1)/2 that is odd – i.e., when m is congruent to 1 modulo 4 – then all expressions of m as s2-t2 will have s odd and t even, just as (m-1)/2 will be even. And it flips neatly in the opposite case: (m+1)/2 is even -> m is congruent to 3 modulo 4, s will be even, and t will be odd. I do need to look again at the approach I’d taken in – I’ve nearly wholly forgotten it – and see where and whether this even-or-odd-s-only property will improve the approach I took in that code at all. An equivalent every-other-integer process may already be at work there.

Secondly, I believe I can get some value out of the “look at the data” task that has absorbed some of my time: I have constructed a table and graphs based on all of the discrete semiprimes whose prime factors are less than 100, their Ceiling Squares, Ascents, and characteristic polynomials for those whose Ascents are greater than 5 (5 or less means that the current characterization algorithm finds a difference-of-squares expression for m in the course of creating the polynomial).

“Look at the data” has been as enjoyable but also as frustrating as usual, in terms of finding patterns. Here, for example, are some of the plots that tantalize by being not wholly random, but lacking a genuinely suggestive structure – basically, by exhibiting the structural properties of the primes and semiprimes and their distributions. Each of these plots’ data is for the entire set of discrete semiprimes whose Ascent is 6 or greater, with prime factors less than 100.

This plot is of the Ascent vs. the Discriminant of the Characteristic Polynomial.
This chart graphically shows the coefficients a, b, and c, for characteristic function f(x)=ax2+bx+c corresponding to values of m=pq, sorted by p and q.
And, finally, I took a scatter plot of the “a” coefficients vs. the “b” coefficients in the characteristic polynomials for those same values of m.

Typical bugaboos in this work have been mainly emotional/energetic ones: my motivational irregularities that I usually call “laziness,” compounded with some somewhat urgent pet interaction problems, and their disruption of the normal process of housecleaning. My beloved wife will be home again in a little over a week, and the place needs to be ready for her to arrive so that my housekeeping habits do not cause her undue dismay. So I have to work with the energies I have, striving as always to make them the energies I need. The Integer Factorization Study is a priority, but it is most definitely not the only nor the highest one, not always. There were also some other practical life struggles involved, which definitely have affected my mood and energy level, but which I do not wish to share here, except to say that I am taking concrete steps to deal with them, and honestly expect to succeed in doing so.

As Timbuk3 once sang, “Life is good; knock on wood.”

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