Incoming links: Riemann hypothesis
bbchallenge.org/story#what-is-known-about-bb lists some (all?) cool examples,
- BB(15): Erdős' conjecture on powers of 2, which has some relation to Collatz conjecture
- BB(27): Goldbach's conjecture
- BB(744): Riemann hypothesis
- BB(748): independent from the Zermelo-Fraenkel axioms
- BB(7910): independent from the ZFC
wiki.bbchallenge.org/wiki/Cryptids contains a larger list. In June 2024 it was discovered that BB(6) is hard.
Ciro Santilli would like to fully understand the statements and motivations of each the problems!
Easy to understand the motivation:
- Navier-Stokes existence and smoothness is basically the only problem that is really easy to understand the statement and motivation :-)
- p versus NP problem
Hard to understand the motivation!
- Riemann hypothesis: a bunch of results on prime numbers, and therefore possible applications to cryptographyOf course, everything of interest has already been proved conditionally on it, and the likely "true" result will in itself not have any immediate applications.As is often the case, the only usefulness would be possible new ideas from the proof technique, and people being more willing to prove stuff based on it without the risk of the hypothesis being false.
- Yang-Mills existence and mass gap: this one has to do with finding/proving the existence of a more decent formalization of quantum field theory that does not resort to tricks like perturbation theory and effective field theory with a random cutoff valueThis is important because the best theory of light and electrons (and therefore chemistry and material science) that we have today, quantum electrodynamics, is a quantum field theory.
visualizing the Riemann hypothesis and analytic continuation by 3Blue1Brown (2016) is a good quick visual non-mathematical introduction is to it.
One of the Millennium Prize Problems and Hilbert's problems.
What is the Riemann hypothesis REALLY about? by HexagonVideos (2022)
Source. Ciro once commented that the best game is an infinitely hard one, where you can progress infinitely. To which his great friend J. replied:Or more broadly, one may argue that the perfect video game is life itself, or difficult life goals like making money, becoming famous or changing the world.
Fine, so the perfect game for you is mathematics. Stage one: prove the Riemann hypothesis!
Thinking about it, "infinitely hard" is perhaps not a very precise term, as it could be interpreted as impossible. And if you have mathematical proof that something is impossible, it would be "pointless" to try, trying would be equivalent to pure meditation.
Maybe a better way to put it would be in terms of a difficulty curve. Real life also involves a lot of waiting, either for some experiment to finish running, of for you mental energy to restore a bit.
But so be it, you get the idea.
But this is basically what Ciro feels on every video game. It happens too often on PVE games that things are is either:
- too slow and easy (Ciro would rather skip those with saves made by other)
- or too fast hard, Ciro would rather tool-assisted speedrun those parts
Not to mention the incredible breach of suspension of disbelief of most PvE games where enemies are unbelievably stupid. E.g., why doesn't Bowser just build one fucking wall 15 tiles high to prevent Mario from coming through to his castle? And then put a gate and a hundred guards in front of it? TODO there was a YouTube video of this, I think it was Toad pointing it out to Mario that it is quite weird that Bowser is so stupid, it almost feels like he wants to be beaten.