Here is a review paper from 2007 that describes Dirac’s large number hypothesis and the many variations on it and additional apparent “coincidences” found by others over the years, experimental evidence and arguments for and against it, and the status of investigation of changes in the gravitational constant G over cosmological time, “Large Number Hypothesis: A Review”. Full text [PDF] is available at the link.
The coincidences involving the radius and mass of the proton seem odd to consider fundamental to me since the proton is not a fundamental particle and most of its mass comes from the binding energy of the massless gluons that hold its three constituent quarks together.
Another numerological observation that’s always intrigued me is the universe’s inordinate fondness for the number three:
- Three extended dimensions of space
- Three colour charges for quarks
- Three generations of fundamental particles (e.g. electron, muon, tau)
- Three fundamental non-gravitational forces, mediated by gauge bosons
Wouldn’t it be neat if all of these were related somehow?