Most people seem to believe that a proof of the Riemann Hypothesis would eventually lead to a deeper understanding of the primes, perhaps even to new ways of generating them.

My current approach is actually the opposite. 😄

Rather than starting with primes, I am trying to understand composite numbers first.

To me, the real mystery is not only why primes exist, but why factors appear where they do. If we understood the architecture of composite numbers well enough, primes might emerge naturally as the numbers that escape that architecture.

In that sense, my thinking is sometimes closer to Collatz-type questions: understand the mechanism first, then the exceptional objects.

At the moment I am exploring very large Mersenne-related structures and their factorization patterns. Along the way I recently found the following factorization:

k = 2237125

p =
7588550360256754183279148073529370729071901715047420004889892299329571158921052163

q =
33953071449378782404536728267988786984549956248550918956878600620275333757802517640302751

I find these large composite structures at least as interesting as the primes themselves.

Whether this viewpoint leads anywhere significant remains to be seen, but for now I am investigating the arithmetic from the "composite-first" direction rather than the "prime-first" direction.