Integer Factorization: The Mystery Remains


Exhibit A: The “Fivers”

Yes, I am still barking up trees on this problem, even as particularly fruitful trees up which to bark become harder and harder to find.

Behold! The quintessential devil in these matters! These are most or all of the odd positive discrete semiprimes less than 20,000 whose difference from their Ceiling Squares is 5. I am calling them The Fivers. They appear to be a well-behaved set of numbers, simple and polite, until one considers the Ascent, the least even number a that one must add to the ceiling square c in order that (c+a)2 will differ from m, the discrete semiprime itself, by a perfect square quantity, thus permitting Fermat factorization. As students of Fermat factorization, and its lack of advantage over other factorization methods, well know, there is not yet a simple formula associating the Ascent with the Ceiling Square or its remainder, whether it be 5 or some greater number.

But I am still investigating. I still have a faint wisp of a belief that there’s an answer out there, a glimmer of hope that it is an answer that I could recognize if I could only see its shape. I also believe that that answer will have properties evident in cases as above where the numbers involved are small, and other cases where the numbers are larger, regardless in both cases of the size of the remainder. As above, so below. Such a general formula would work, I hope, equally well all along the set of odd positive discrete semiprimes.

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