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.
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.
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.
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.
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.
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?
I'm not a physicist, but I think those are the smallest units in the sense that they are the smallest units we could theoretically interact with/measure, not some hard limit. It's just that it's moot to consider anything smaller because there's no way for us to ever know.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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)
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.
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"
> 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.
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...
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.
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!
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
)
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.
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.
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:
A iff B doesn't mean "this is the only way for this to be true", it means A implies B and B implies A. B being the statement that a number is prime, but you can have any arbitrary A that is actually true.
Paper: https://www.pnas.org/doi/10.1073/pnas.2409417121 (https://news.ycombinator.com/item?id=44323658)
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...
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.
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.
Discrete math is the single most "observed property of reality", and nothing else comes even close.
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.
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.
No, the discreetness comes from physical experiments. I do see a problem defining something outside of one’s mind or outside the universe though :)
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.
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.
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?
I thought it was a smooth continuous manifold
To what extent are the Planck length and Planck second confirmed smallest discrete units?
I'm not a physicist, but I think those are the smallest units in the sense that they are the smallest units we could theoretically interact with/measure, not some hard limit. It's just that it's moot to consider anything smaller because there's no way for us to ever know.
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.
I for one never saw a "number 2" in the wild. But I'm a homebody.
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.
> 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.
I’m not sure I am intended to understand what your problem is.
I wouldn't even go with particles. I'd call it a stream of sensations.
But not necessarily with rabbits. Can easily end up with dozens of 'em when you ony started out with two.
There are a dozen leaps of abstraction occuring before you arrive at "2 apples".
You are differentiating, classifying, etc.
Zero, one, infinity.
Infinity, aka 2 or more. I agree that those are truly three distinct classes of quantity/identity
Whoa
I've never seen gravity, but here I am, stuck to the ground
bears shit in the woods so they're out there if you look in the right place
I guess you saw two things in the wild though.
Math defines all that we do. Why do we want more? Because of addition.
Even counting and measurement are contrived abstractions. If any big ultimate truth is delivered it will probably be referring to our psychology.
and it will be something so trivial and obvious, those who were looking for it will be kicking themselves for missing it
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.
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.
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.
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.
You can read it for free at https://www.wolframscience.com/nks/ if you’re interested.
Or someone proves that there is no pattern and they will be kicking themselves for wasting their time searching.
The experience gained along the journey is more valuable than the result.
honestly this would be it, wouldn't it? https://www.icloud.com/iclouddrive/07fRJGiC51VEHPqYRfNaFjnEA
Did you tried to share a hollywood action movie with us to tell us what exactly?
Did you listen? The audio is different yet it still works.
No, I did not download a big movie and likely won't to get a point on HN.
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
Your point of argument/information. Not karma points.
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.
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...
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_determinis... mentions several tests that do not require factorization, though.
Pseudoprime test usually works, and AKS algorithm always works, both are much faster than factoring.
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.
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)
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.
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"
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.
BTW the phone number in Jenny's song, 867-5309, is a twin prime (867-5311).
Twin towers prime 9/11
> 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.
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...
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
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!
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...
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.
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.
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.
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.
A iff B doesn't mean "this is the only way for this to be true", it means A implies B and B implies A. B being the statement that a number is prime, but you can have any arbitrary A that is actually true.
https://archive.is/2025.06.17-194128/https://www.scientifica...
This have implications for public key cryptography?
Computing M_a(n) appears to be at least as hard as factoring n for a=1, so I think you're safe here.
My naive notion on this is yes, iff the new method is computationally or memory-wise of lower complexity
Prime generating functions in polynomials? That's almost Lisp domain.
Mathematicians should play with Scheme and SICP.
Oh. (3n3 − 13n2 + 18n − 8)M1(n) + (12n2 − 120n + 212)M2(n) − 960M3(n) = 0.
I'd have thought that was obvious.
Yeah but can we get a pretty picture out of it? A cool fractal is worth a thousand words.