Whenever I see prime visualisations, I go down a rabbit hole of investigating prime numbers in search of a better factoring algorithm. I’m no great mathematicia

Whenever I see prime visualisations, I go down a rabbit hole of investigating prime numbers in search of a better factoring algorithm. I’m no great mathematician, so I always come up better educated but ultimately fruitless. :) I have this nagging idea, though, that there must be a deep relationship among: 1. the polynomials on which the primes are clustered in the Ulam spiral, 2. algorithms to factor numbers, and 3. recovering information “destroyed” by addition. That is, if you ask me to find the factors of C = A × B, the information about which prime factors went into that product is preserved by multiplication, but if you ask me for the factors of C = A + B, I can’t say much about how the factors of A and B relate to those of C, can I? But if I could, I suspect I could factor numbers efficiently. Something something information theory something entropy… Anyone have any pointers to reading materials along these lines?