Integer Factorization: An Overdue Update


New revelations in my study of factorization, Ceiling Squares, and the Zone Grid have not been coming lately, but I have indeed been looking at the data and trying out new ways to visualize their relationships, every day, and my work is still underway, despite no evidence of these facts in this blog. Right now, you will have to satisfy yourself with some recapitulation, which I have found to be useful, at the very least, in organizing and updating my thought processes.

Screenshot of my current study worksheet in Apple Numbers

The Ascent – the least number one must add to the Ceiling Root to produce a Ceiling Square that differs from odd integer m=ab by a perfect square amount, enabling Fermat factorization – is tantalizing but also continues to have the frustrating quality that knowing an answer ahead of time always has. Of course you would be able to do this and that and the other if you had that part of the answer!

So I am still examining the quantities c and r in the expression m=c2-r, and the related Remainder Series rn=(c+n)2-m which satisfies the quadratic function f(n)=rn=n2+2cn+r, for patterns that will lead to a formulaic expression for the Ascent (without, of course, any prior knowledge of it). This will be the least non-negative n for which f(n) is a perfect square, but that’s just a rearrangement of the problem. As mathematicians know who have looked in vain at Fermat factorization for any kind of algorithmic speed-up, rearranging a problem does not always shine new and useful light thereon.

The Zone Grid, a grid overlaid on the integer intersection points of the Cartesian Plane assigning integer values to those intersection points by the function Z(x,y)=x(x+1)+y, so that each line through integer x parallel to the y axis forms a shifted version of a standard number line, has been an interesting insight, though not entirely fruitful. Placing the values rn of the Remainder Series for a particular m=(c+n)2-rn on number lines for successive integer x values gives straight lines, just as the perfect squares form straight lines, on this Zone Grid. One can set formulaically where to start one of these straight lines, since the family of perfect square lines gives one infinite choices, in order to uniquely determine for m a Remainder Line on the Zone Grid. When that Remainder Line first strikes a point whose Z(x,y) value is a perfect square, the factorization of m becomes possible. However, this so far has been just another rearrangement of the problem.

When one considers a discrete semiprime m with prime factors p and q where p<q, and examines m within the finite set of all discrete semiprimes whose greater positive prime factor is q, one can quickly note that the Ascent of m decreases as the value of the lesser prime p increases while q is fixed. Again, is there something useful hiding in this fact? If there is, I have not yet found it.

This is me, J. Calvin “Bimp” Smith, at the Mountain River Chalet, signing off and getting to necessary household tasks before resuming my mathematics work this evening.

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