Of course the hard problem of integer factorization remains! Should I succeed where so many others, with so many more well developed analytical tools have failed?

And yet, I press on.

In short, this is . . .

## The Hard Problem, in a(n Impenetrable) Nutshell

Ceiling squares have gotten me a nice little efficient integer factorization algorithm, whose Perl code I have shared with my dear readers elsewhere. But that algoritm still must walk the ceiling squares upward – raise the ceiling, as it were, until the odd integer m shows itself as the difference of two squares that gives a non-trivial factorization. **I have not, as yet, found a formula that, given m, spits out the number of times I have to “raise the ceiling,” as it were, to yield a good value for m=s ^{2}-t^{2}.** A graph for the factorization of m=856981362557 illustrates this nicely. Below I have plotted the values of s versus the difference between r’s ceiling square and itself. Welcome to chaos! 😀

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