[HN Gopher] Algorithm-assisted discovery of an intrinsic order a...
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Algorithm-assisted discovery of an intrinsic order among
mathematical constants
Author : BerislavLopac
Score : 76 points
Date : 2023-09-20 08:39 UTC (14 hours ago)
(HTM) web link (fermatslibrary.com)
(TXT) w3m dump (fermatslibrary.com)
| YeGoblynQueenne wrote:
| Original on arxiv:
|
| https://arxiv.org/abs/2308.11829
| miguelmurca wrote:
| This should be the link. As is, currently, you get a PDF viewer
| that does not allow zoom (without zooming surrounding UI), and
| with a weird comments integration.
| nonrandomstring wrote:
| This sounds big. Can some mathematicians please weigh in.
| kevinventullo wrote:
| Seems to be in the spirit of the "Ramanujan Machine" from a few
| years ago, which... well here is one prominent mathematician's
| take: https://www.galoisrepresentations.com/2019/07/17/the-
| ramanuj...
|
| Until they prove a new irrationality result I personally won't
| be paying much attention.
| nonrandomstring wrote:
| Thanks. This is a fascinating rabbit hole in itself
| regardless of the merit of the claims, because I was unaware
| of how rapidly advancing experimental mathematics is banging
| heads with traditional work.
|
| I see the problem, which goes deep into issues of ML/AI, that
| interesting results without "proofs of understanding" are
| "Idiotic" in the original Greek sense - that they stand alone
| and separate from the wider corpus - leaving someone else the
| work of "connecting them up", as it were.
|
| Am I even half right?
| bigbacaloa wrote:
| [dead]
| henrydark wrote:
| Still waiting to see the first new irrationality proof with these
| ideas, wishing you lots of good luck!
| ackbar03 wrote:
| Can someone tldr this please?
| pringk02 wrote:
| Some people involved in this have a YouTube explainer here:
| https://www.youtube.com/watch?v=Uk04gfIt8yM
| contravariant wrote:
| There are a lot of continued fraction formulas for mathematical
| constants of the form:
|
| c = a(0) + a(1) / (b(1) + a(2) / ( ....
|
| Here a and b are polynomials (this looks a bit more sensible in
| their paper, I don't really know how to denote continued
| fractions in ASCII).
|
| If you stop this fraction at a certain point you get a fraction
| like p_n / q_n. In general the value of p_n and q_n grow
| extremely rapidly. Their conjecture is that if you look at this
| fraction in its lowest terms then p_n and q_n grow _way less_
| rapidly (exponential vs. a power of a factorial), but only if
| the limit is a mathematical constant.
|
| Obviously this can't hold for all mathematical constants, but
| as a rule of thumb it seems to work well enough to identify
| when two polynomials a and b correspond to some mathematical
| constant or a combination of mathematical constants.
| Garlef wrote:
| And what is a "mathematical constant" in this context? Not in
| terms of examples but rather: Is this something that can be
| defined in a meaningful way? Remembering that one joke about
| inductive proofs ("there are no uninteresting natural
| numbers"), my first impression would be that this is a bit of
| a void concept.
|
| Regarding the joke: "there are no uninteresting natural
| numbers because if there were any, there would also be a
| smallest uninteresting number, making this number
| interesting, which is a contradiction"
| jjgreen wrote:
| I'd say not, mathematical constants are usually irrational
| (often transcendental) numbers which tend to crop-up
| consistently in different contexts, so they're culturally
| (and so ill-) defined. Such concepts are not uncommon in
| mathematics, "canonical" springs to mind.
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