After reading the article in question, I was thinking about my own experience. I too have been working on prime factorization for the past 6 years in the same fashion that you have and what you just wrote here describe perfectly how I feel about my progress and journey. Which makes me wonder... How many people out there are like us, working on the same problem while feeling the same way?
It's like the problem is laughing at you. Baiting you back in with its extreme simplicity. It truly is a mesmerizing problem that you can't help but feel is solvable if only one single Eureka moment is reached. Boy do I want this one solved...
So far the journey has been really rewarding though. I’ve recently been focusing my efforts on remainders and decision paths.
The most invaluable lessons learned have been my understanding of the complexity of simplicity.
I’ve written some insanely complicated, convoluted and frankly weird ass code throughout all this.
For example: bignum math libraries, modified Euclidean geometry, imaginary number libs, fractal factoring, Russian peasant multiplication, 3D volumetric fluids to triangle area fitting. Multiple geometric shape fitting libraries, Gallois fields and lots and lots of weird what-if code that popped up in my head.
I’d always been good at algorithms, but straightforward head-first attacks of the problem always ends in failure.
Now I’ve learned to build analysis, statistics and proofs by necessity. It’s completely changed my mindset on design. (Obviously I studied far my comp-sci than mathematics.)
Other than sieving and brute force, I think the only other general solution to be a sort of ‘halving’ problem.
The relation between factors and q is essential sum or area. You can subdivide either into n buckets and remainders building a tree of nodes iteratively. At many nodes I have found some interesting patterns. I find nodes after some iterations that sum up to a factor. So now my question is finding the path every time.
Anyway, best of luck on your journey and enjoy the ride! Maybe in another few years somebody will finally slay this beast.
For every possible input 0 to X you need to map to a different 'compressed' format. So, if you want to handle every possibility then some of your compressed versions need to be as long if not longer than the original version. That's not a problem if their is some bias in how your generating the data. Just map the more common versions to shorter encoding and you get a useful compression even if sometimes it makes things worse on average their is an improvement. But, with unbiased random data you can't chose which subset is more likely because every version is equally likely. Which is why it does not work.
It reminds me of a sci-fi story where people who decided that they were tired of being emulated retired to a virtual mine where they try to discover new mathematical proofs.
They didn't have to do it, since it served no real purpose. But it opened up new avenues of knowledge.
Could you mention the the name?
More down to earth though, I'm a biologist and whenever I get this feeling related to work, scientifically I have learned to trust it and keep at it, it usually does mean there is something of value in my ideas.
For me, it's factoring large numbers in short times. I've spent 18 years on and off of the problem. Most of that time was coming up with novel ideas and then arriving at the same conclusions as everybody else.
I'm too obstinate and "stupid" to know that the problem is practically unsolvable - but fortunately I'm not obsessive enough to throw my life into it. It's basically a hobby I do during my downtime.
I like to call it mathematical finger painting, or nature walks through obscure numeric systems. Studying topics, geometries, ancient multiplication methods for fun, and generally scratching under the surface of "why" without knowing how.