Mersenne primes

Problem

This branch studies prime exponents \(q\) for which \(2^q-1\) is prime, using exact divisor traces instead of two-prime correlations.

Geometric formulation

For a prime exponent \(q\), a proper prime divisor of \(2^q-1\) must have the form \(\ell=2kq+1\). This exact divisor structure becomes the terminal trace used by the geometric atlas.

Main manuscript claim

The cumulative theorem in the paper claims a constant \(c_{\mathrm{Mer}}>0\) such that the number of prime exponents \(q\le X\) with \(2^q-1\) prime is at least \(c_{\mathrm{Mer}}\log X\) for all sufficiently large \(X\).

Proof architecture / transport mechanism

The comparison is stopped at the first active terminal row. The past is an observed no-hit prefix and the future is promoted. Dense positive excess is reduced to double-return structure; sparse positive excess is interpreted as first-hit wall flux.

Why this branch is different

Here the target is not a two-prime correlation. The geometry operates on the exact divisor trace of an exponential sequence, and the terminal conclusion is a survivor statement.

@misc{gotanegra2026mersenne,
  author       = {Jaume Gotanegra},
  title        = {On the Mersenne Prime Problem in a Prime-Specialized Multiplicative Geometric Model},
  year         = {2026},
  month        = {Jun},
  doi          = {10.5281/zenodo.20546633},
  url          = {https://doi.org/10.5281/zenodo.20546633},
  note         = {Preprint}
}