Integer Factorization: Chasing Elusive Characteristics


I have computed a chart for all the odd discrete semiprimes less than 250 (I think the count is 45 of them), by way of teasing out characteristics and properties that I hope I can tease out formulaically in days/weeks/months to come. It’s the current tree up which I’m barking, so to speak. By way of a little update without much description, here’s an image of the chart as it stands now.

What’s going on here that’s interesting to me to see in this form, is that when the two primes whose product is a discrete semiprime are sufficiently close together, the remainder when that discrete semiprime subtracts from its ceiling square is always a perfect square, enabling Fermat factorization. The reasons are probably obvious after a bit more calculation, but this chart, grouping the discrete semiprimes together according their smallest prime factors, also shows the ascent climbing monotonically as the gap between the smaller prime and the greater prime widens. This is indeed obvious, since the ceiling square and the non-trivial value of s for which m=s2-t2 grow farther apart as the difference between smaller and larger prime grows.

But I’ve also put the characteristic function f(x) for m here, along with the slope of the line corresponding to m and to f(x) on the Zone Grid.

Will keep studying this data until I’ve found something, or I’ve determined more or less robustly that it won’t lead anywhere interesting.

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