[HN Gopher] Mathematicians hunting prime numbers discover infini...
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Mathematicians hunting prime numbers discover infinite new pattern
Author : georgecmu
Score : 126 points
Date : 2025-06-19 03:28 UTC (2 days ago)
(HTM) web link (www.scientificamerican.com)
(TXT) w3m dump (www.scientificamerican.com)
| wewewedxfgdf wrote:
| This sort of thing makes me feel there is some deep understanding
| of reality only inches away from us, we glimpse it through these
| patterns but the secret remains hidden.
| freed0mdox wrote:
| and it will be something so trivial and obvious, those who were
| looking for it will be kicking themselves for missing it
| seanmcdirmid wrote:
| It's a huge refrain that shows up again every 20 years or so.
| Wolfram wrote a huge book with this premise, but I don't
| think it's gone anywhere even though it's surely 25 years old
| by now.
| burnt-resistor wrote:
| GEB was similar in a cycle prior. It's cool to dream but
| the limits of accepted knowledge requires the hard work of
| assembling data, evidence, and reasoning.
| A_D_E_P_T wrote:
| It's arguably ~2500 years old, dating back to the
| Pythagoreans, who believed that "all is number" and had a
| very large and complex system of musical rituals.
|
| The modern manifestation is mostly the intellectual product
| of Konrad Zuse, who wrote "digital physics" in 1969.
|
| > https://en.wikipedia.org/wiki/Digital_physics
|
| Wolfram, Tegmark, Bostrom, etc. are mostly downstream of
| Zuse.
| Kaijo wrote:
| Wolfram came to our evolutionary biology department to
| preach that book about 20 years ago. We all got our heads
| into cellular automata for a while, but in the end they
| just don't have the claimed profound explanatory power in
| real biological systems.
| robin_reala wrote:
| You can read it for free at
| https://www.wolframscience.com/nks/ if you're interested.
| e1ghtSpace wrote:
| honestly this would be it, wouldn't it?
| https://www.icloud.com/iclouddrive/07fRJGiC51VEHPqYRfNaFjnEA
| lukan wrote:
| Did you tried to share a hollywood action movie with us to
| tell us what exactly?
| e1ghtSpace wrote:
| Did you listen? The audio is different yet it still
| works.
| lukan wrote:
| No, I did not download a big movie and likely won't to
| get a point on HN.
| e1ghtSpace wrote:
| All you want is points on HN when you comment? Anyway,
| try to only focus on the top middle screen in this video.
| https://www.youtube.com/watch?v=T_dLx_J2oVs
| lukan wrote:
| Your point of argument/information. Not karma points.
| amelius wrote:
| Or someone proves that there is no pattern and they will be
| kicking themselves for wasting their time searching.
| mensetmanusman wrote:
| The experience gained along the journey is more valuable
| than the result.
| waltbosz wrote:
| Wouldn't it be fun if someone out there already knows a simple
| way to determine if a number is prime without factoring, but to
| them it is so obvious that they didn't even consider others may
| be interested.
| vasvir wrote:
| Well I have a really elegant proof for this but I don't have
| enough space in the HN reply box to write it out -- but it is
| trivial, I am sure you will work it out.
| Fermat Reincarnation.
| kevinventullo wrote:
| As far as I know, the Lucas-Lehrer test used by GIMPS does
| not actually factor: https://en.m.wikipedia.org/wiki/Lucas%E2
| %80%93Lehmer_primali...
| Someone wrote:
| That works for very few numbers. From that Wikipedia
| article: _"In mathematics, the Lucas-Lehmer test (LLT) is a
| primality test for Mersenne numbers"_
|
| That's fine for GIMPS, which only searches for Mersenne
| primes, but doesn't work in general.
|
| https://en.wikipedia.org/wiki/Primality_test#Fast_determini
| s... mentions several tests that do not require
| factorization, though.
| throwaway81523 wrote:
| Pseudoprime test usually works, and AKS algorithm always
| works, both are much faster than factoring.
| adgjlsfhk1 wrote:
| Since 2002 this has been known, and it's one of the least
| intuitive things in modern math. (versions with probability
| of 1-\epsilon have existed since Miller-Rabin in 1976)
| briffid wrote:
| I had a similar feeling. But I think this is indeed a glimpse
| to the intrinsic structure of reality itself, not just a
| promise of seeing reality. Like we can have a blink of turning
| around in Plato's cave. I think the patterns of the Mandelbrot
| set is a similar thing. And there are only a handful of other
| things that shows the very basic structure of reality. And the
| encouraging thing is that it seems the core of reality is not
| an infinite void.
| dcow wrote:
| I don't think this understanding will be related to the
| structure of reality but instead the structure of discrete
| math. Math is not an observed property of reality it's a system
| of describing quantities and relations between them, often with
| plenty of practical application. Math is applied philosophy and
| physics is applied math.
| bmacho wrote:
| Discrete math is the single most "observed property of
| reality", and nothing else comes even close.
| swayvil wrote:
| I for one never saw a "number 2" in the wild. But I'm a
| homebody.
| curtisblaine wrote:
| I guess you saw two things in the wild though.
| alphazard wrote:
| If I handed you 1 apple, and then handed you another
| apple, you wouldn't be surprised to find that you had 2
| apples. The same trick works with oranges and pears.
| verzali wrote:
| But not necessarily with rabbits. Can easily end up with
| dozens of 'em when you ony started out with two.
| brookst wrote:
| Whoa
| FredPret wrote:
| > If I handed you 1 apple,
|
| At this point I hold one object that we agree to label
| "apple". Note that even seeing it as a single object is a
| layer of abstraction. In reality it's a clump of
| fundamental particles temporarily banding together
|
| > and then handed you another apple,
|
| What's "another apple"? What does it have in common with
| the thing I'm already holding? We label this thing to be
| also an apple, but it's a totally different set of atoms,
| from a different tree, perhaps from the other side of the
| planet. Perhaps the atoms formed in stellar processes
| light years away from that of the other apple.
|
| Calling both of these things "apple" is a required first
| step to having two of them, but that is an of
| abstraction, a mental trick we use to simplify the world
| so we can represent it in our minds.
|
| I'm not a particle physicist but I hear electrons *can*
| be counted without any unwitting help from our lower-
| level neural circuitry.
| swayvil wrote:
| I wouldn't even go with particles. I'd call it a stream
| of sensations.
| swayvil wrote:
| There are a dozen leaps of abstraction occuring before
| you arrive at "2 apples".
|
| You are differentiating, classifying, etc.
| pixl97 wrote:
| Zero, one, infinity.
| datameta wrote:
| Infinity, aka 2 or more. I agree that those are truly
| three distinct classes of quantity/identity
| konfusinomicon wrote:
| bears shit in the woods so they're out there if you look
| in the right place
| ndsipa_pomu wrote:
| I've never seen gravity, but here I am, stuck to the
| ground
| hausrat wrote:
| The very notion of discreteness depends on subjective
| definitions of "objects". We take concepts of objects for
| granted because they make interacting with the world
| tractable, but it's really hard to define them outside of
| minds.
| yunwal wrote:
| As far as we know, the universe is made up of discrete
| units and any other type of math is an abstraction over
| discrete math.
| giardini wrote:
| As far as we know, the universe is a single unity, and
| any discrete units and any other type of math are human
| distinctions overlaid upon that unity.
| random3 wrote:
| Can you explain what you mean here? I mean yes there's a
| universe so it can be see as a unit. There's also quantum
| mechanics, telling us we can only distinguish discrete
| objects at the bottom of the scale. Can you give an
| example of a non-human distinction, or explain what you
| mean by that concept?
| Keyframe wrote:
| I thought it was a smooth continuous manifold
| datameta wrote:
| To what extent are the Planck length and Planck second
| confirmed smallest discrete units?
| Keyframe wrote:
| I was referring to spacetime in GR is modeled as smooth
| continuous manifold. In case you're serious though,
| planck length are not some fine-grained pixels/voxels in
| the cartesian 3d world, at least not confirmed; in-fact
| planck units are derived scales.
| feoren wrote:
| No, discrete math is exactly the same regardless of your
| definition of "object". It is completely _independent_ of
| that. Discrete math is important to any theoretical
| beings that have any concept of "objects" whatsoever. It
| would be mostly irrelevant to entities that have no such
| conception, but those entities are not writing math
| papers.
| random3 wrote:
| No, the discreetness comes from physical experiments. I
| do see a problem defining something outside of one's mind
| or outside the universe though :)
| swayvil wrote:
| Even counting and measurement are contrived abstractions. If
| any big ultimate truth is delivered it will probably be
| referring to our psychology.
| m3kw9 wrote:
| Math defines all that we do. Why do we want more? Because of
| addition.
| andsoitis wrote:
| https://archive.is/2025.06.17-194128/https://www.scientifica...
| gnabgib wrote:
| Paper: https://www.pnas.org/doi/10.1073/pnas.2409417121
| (https://news.ycombinator.com/item?id=44323658)
| Sniffnoy wrote:
| I'm a little confused at the significance here. Before I read the
| definition of the M_a, this seemed crazy, but on actually reading
| it, M_1 is just the sum-of-divisors function (usually denoted
| sigma).
|
| So, n is prime iff M_1(n)=n+1. That's much simpler than the first
| equation listed there!
|
| Indeed, looking things up, it seems that in general the functions
| M_a can be written as a linear combination (note: with polynomial
| coefficients, not constant) of the sigma_k (sigma_k is the sum of
| the k'th power of the divisors). So this result becomes a lot
| less surprising once you know that...
| bubblyworld wrote:
| Can you elaborate? How does this result become less surprising
| if you know that? Personally I would not have guessed that
| there are infinitely many characterisations of P involving
| sums-of-powers-of-divisors either.
| Sniffnoy wrote:
| I mean, if you can do something a simple way, it's not that
| surprising that you can also do it a complicated way, I'd
| say.
| isaacfrond wrote:
| The M functions are the MacMahon's partition functions (see the
| paper [1]). They were not known to relate to the sum of
| divisors. The M_a function counts partitions in a parts but
| weighing multiplicities in the partion.
|
| [1]: https://arxiv.org/abs/2405.06451
| boothby wrote:
| Sorry, but M_1 is simply the sum of divisors, and I don't
| think that was ever a mystery. Specializing the notation from
| the paper for M_a, to a=1, and writing pythonic with finite
| bounds for clarity... M_1(n) = sum( m
| for m in range(1, n+1) for s in range(1, n+1)
| if m*s = n )
| Sniffnoy wrote:
| M_1 is obviously just sigma. That's straight from the
| definition, you can't tell me that wasn't known.
|
| As for the higher ones, I'm having trouble finding a proper
| citation saying that this was known earlier, but this
| math.stackexchange answer asserts that MacMahon himself
| worked some of this out:
| https://math.stackexchange.com/a/4922496/2884 No proper
| citation though, annoying.
|
| When you say "this wasn't known", on what basis is that? It's
| very hard to be sure that something wasn't known unless
| you're an expert on that particular thing!
| boothby wrote:
| I agree that the observation "M_1(n) = n+1 iff n is prime" is
| elementary. It certainly motivates some intuition behind the
| investigation in this paper, but I'd loathe to call it obvious.
|
| Note that the paper studies equations with polynomial
| coefficients on McMahon series. That is, the _n+1_ in our
| trivial observation is "stray" in a sense.
|
| For an at-a-glance indication of nontriviality, look no further
| than the conjecture associated with Theorem 1.2 -- that there
| are exactly five equations of this sort which are prime
| indicators. That seems spooky, to me; I can't help but wonder
| what structure underlies such a small number of relations.
| anthk wrote:
| Prime generating functions in polynomials? That's almost Lisp
| domain.
|
| Mathematicians should play with Scheme and SICP.
| swayvil wrote:
| Yeah but can we get a pretty picture out of it? A cool fractal is
| worth a thousand words.
| nprateem wrote:
| Oh. (3n3 - 13n2 + 18n - 8)M1(n) + (12n2 - 120n + 212)M2(n) -
| 960M3(n) = 0.
|
| I'd have thought that was obvious.
| drdunce wrote:
| This have implications for public key cryptography?
| datameta wrote:
| My naive notion on this is yes, iff the new method is
| computationally or memory-wise of lower complexity
| boothby wrote:
| Computing M_a(n) appears to be at least as hard as factoring n
| for a=1, so I think you're safe here.
| ysofunny wrote:
| I hope the twin prime conjecture will become a theorem during the
| remainder of my lifetime
|
| that's why I already got the double twin prime conjecture ready:
|
| there exists an infinite number of consecutive twin primes. 3
| examples: 11,13; 17,19. 101,103;107,109, AND 191,193;197,199... I
| know of another example near the 800s
|
| there's also the dubious, or trivial, or dunno (gotta generalize
| this pattern as well) of the first "consecutive" twin prime but
| they overlap which is 3,5 and 5,7.... which reminds me of how
| only 2 and 3 are both primes off by one; again, I need to
| generalize this pattern of "last time ever primes did that"
| D-Coder wrote:
| BTW the phone number in Jenny's song, 867-5309, is a twin prime
| (867-5311).
| jsisto wrote:
| Twin towers prime 9/11
| adgjlsfhk1 wrote:
| The broader generalization you're headed towards is
| https://en.wikipedia.org/wiki/Dickson%27s_conjecture (or the
| even more general
| https://en.wikipedia.org/wiki/Schinzel%27s_hypothesis_H) which
| basically say that the twin prime conjecture is true for every
| linear/polynomial generalization of twin primes.
| zck wrote:
| > there's also the dubious, or trivial, or dunno (gotta
| generalize this pattern as well) of the first "consecutive"
| twin prime but they overlap which is 3,5 and 5,7.... which
| reminds me of how only 2 and 3 are both primes off by one;
| again, I need to generalize this pattern of "last time ever
| primes did that"
|
| For the triplet n, n+2, n+4, exactly one of those numbers is
| divisible by 3. So the only triplet n, n+2, n+4 where all
| numbers are prime contains 3: 3, 5, 7.
| munichpavel wrote:
| Ken Ono, one of the authors, is the mathematician behind the
| University of Virginia women's swimming team's dominance in
| recent years, including world records and gold medals.
|
| https://news.virginia.edu/content/faculty-spotlight-math-pro...
| noqc wrote:
| Because the article doesn't actually say so (presumably because
| the author doesn't know the difference between "if" and "if and
| only if") the statement:
|
| (3n^3 - 13n^2 + 18n - 8)M_1(n) + (12n^2 - 120n + 212)M_2(n) -
| 960M_3(n) = 0
|
| is _equivalent_ to the statement that n is prime. The result is
| that there are infinitely many such characterizing equations.
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