Integer Factorization: An Important Graph


Let odd positive m be the product of two unequal prime numbers p and q.

Let the Ascent of m be the difference between (p+q)/2 and c, the Ceiling Root of m, which is the value of ceil(sqrt(m)) with 1 added, if necessary, to make c even or odd as (p+q)/2 is even or odd.

Integer factorization formulaically depends, I think, on the relationship between the Ceiling Root of m, its remainder c2-m, and its Ascent. But that formula in any closed form has eluded mathematicians so far in human history.

This figure below is the closest I come in my mind to a glimpse of what such a closed-form solution to factorization might be: It is a graph of m versus Ascent for almost all discrete semiprimes (products of two primes) less than m=20000. I stopped at largest prime less than or equal to 563 because, well, the graph is pretty straightforward. One can see the outlines of smallest prime and biggest prime in how the Ascents organize themselves. And it is that straightforwardness into which I am trying to see further, to find a hitherto undiscovered useful property or formula.

Here is the graph, a scatter plot of m vs. Ascent:

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