I would really like to get some constructive feed back on this prime-seeking algorithm. Computationally it's no better than the rest. However, it does offer some unique insight. I have partitioned the set of naturals between prime and composites using a rigorous structural schema that I prove in the following thesis: https://sites.google.com/site/primenumbertheory/home/the-prime-thesis Let me know what you think. I appreciate any further insight the community can offer me. Shalom.
The article is quite long, so I skipped a few parts. If I understand correctly, your observation is that any odd composite number N is a sum of a series of consequtive integers of length < √N. This is a nice property, I for one didn't know it, and it wasn't covered in my number theory course. But I don't understand what the algorithm is. From what I could gather, you test all the bases for the sequence, from 1 up to N/3, and check if they start a sequence that sums up to N. This doesn't seem very efficient. Is this what you meant?
You have understood the primitive algorithm. You're right, it's not efficient. However, it is generalized as the thesis develops and removes all impossible values in the set of test subjects (you have to read the whole paper to understand this). Still, the fully developed algorithm is not all that efficient as a prime tester. The idea, however, is not to render a computationally efficient prime test so much as to stimulate and promote the idea that if natural number theory could be placed on some geometric palette, the key to primes might unfold.
I believe number theory involves a great deal of algebraic geometry nowadays. It's not at all like the approach in your paper, but if this leads to something, it'll be wonderful. Even if you don't prove new theorems, elementary proofs of existing theorems are ofter enlightening. PS. I liked your use of Hebrew variables in the paper. Nice touch.