Not sure if I'm getting the math bug again or merely reminiscing, but here's my semi-prime factoring method (took 3 weeks from start to finish 3 years ago) which I've never shared. Instead of a formal paper, it's my messy hand written notes and finally the python code🧵👇
6
19
8,835
1. Wielding a High School education, Excel and the willingness to play with numbers, I hypothesized the remainder of the square root of a semi-prime could be useful in determining how far apart the prime factors are. For instance, 101 x 103 = 10403. 10403 ^0.5 = 101.995..
1
1
2,022
2. This isn't news, but I love to tackle a problem like this from scratch as much as possible and then find out how far I can get before hitting a dead end, writing equations and branching/building along the way.
1
4
1,015
3. Page numbers on the top left. I found an early equation with a sloppy correction factor which I later overcame. The messiness was just my method of not worrying and allowing time for playing with numbers to bring them into better focus.
1
747
5. I was working 50 hours a week back then and spending almost 10 hours a week on the road, but took what time I could to make progress as seen below.
1
629
11. Started writing python code to test things out. worked great, though my code was slow and I used late 2023 AI to try to improve it.
1
515
13. Cleaned up notes and equations as well as finding the ratio convergence toward 8/pi which was found by Lehman in 1974. This is one of the components which bounds the maximum efficiency of this method, though I took it beyond Lehman.
1
505
14. What did I rediscover?
1643 - Fermat discovered any product can be written as the difference of squares which then improves the efficiency of factorization to find that difference of squares vs trial division up to the square root of the number. A^2 - B^2 = (A - B)(A + B)
1
1
484
15. 1974 - Lehman found that by using a multiplication factor before searching for the difference of squares, optimization towards O ^1/3 could be reached. I discovered the same intuitively.
1
479
16. 1999 - McKee analyzed how the fractional remainders distribute continuously, utilizing the analytic constant 8/pi (2.54648) to shrink Lehman's required search bound. I found the same convergence when trying to optimize/understand "a" though didn't see it was 8/pi precisely
1
474
17. 2012 - Hart found that searching multiple values of "a" is redundant and instead focused on the c factor. I did exactly the same as it proved more efficient.
1
464
18. I'm certain it would require a tremendous amount of work for me to understand the published papers of any of those mentioned here. I've long held the belief that math was most beautiful when kept as simple as possible (as did Erdos).
1
457
19. Though my notes may not be easy to understand, the missteps, temporary walls and excitement is shared while keeping all of it at high school level. I've worked on half a dozen unsolved problems, always making progress (in my own way) and writing sub-equations to build on.
1
449
21. I'll never be some quack sending non-solutions of the Riemann Hypothesis to professional mathematicians. Instead, I'm the skeptical optimist willing to enjoy numbers and find my own way. I'm a big fan of Erdos. I also appreciate how Euler showed his methods in raw form.
1
1
924