Mathematicians hunting prime numbers discover infinite new pattern

https://www.scientificamerican.com/article/mathematicians-hunting-prime-numbers-discover-infinite-new-pattern-for/

by georgecmu • 1 year ago

163 102 1 year ago

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.

_feus• 1 year ago

and it will be something so trivial and obvious, those who were looking for it will be kicking themselves for missing it

seanmcdirmid• 1 year ago

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.

e1ghtSpace3• 1 year ago

honestly this would be it, wouldn't it? https://www.icloud.com/iclouddrive/07fRJGiC51VEHPqYRfNaFjnEA

amelius• 1 year ago

Or someone proves that there is no pattern and they will be kicking themselves for wasting their time searching.

waltbosz• 1 year ago

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• 1 year ago

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• 1 year ago

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...

throwaway81523• 1 year ago

Pseudoprime test usually works, and AKS algorithm always works, both are much faster than factoring.

adgjlsfhk12• 1 year ago

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• 1 year ago

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.

dcow2• 1 year ago

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• 1 year ago

Discrete math is the single most "observed property of reality", and nothing else comes even close.

swayvil5• 1 year ago

Even counting and measurement are contrived abstractions. If any big ultimate truth is delivered it will probably be referring to our psychology.

m3kw9• 1 year ago

Math defines all that we do. Why do we want more? Because of addition.

cyanydeez• 1 year ago

Makes me feel like math is mostly an arbitrary realm that after you've got 90% of the way in, the remaining 10% is just pointless coincidences.

andsoitis• 1 year ago

https://archive.is/2025.06.17-194128/https://www.scientifica...

gnabgib• 1 year ago

Paper: https://www.pnas.org/doi/10.1073/pnas.2409417121 (https://news.ycombinator.com/item?id=44323658)

Sniffnoy3• 1 year ago

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...

bubblyworld2• 1 year ago

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.

Sniffnoy3• 1 year ago

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• 1 year ago

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

boothby3• 1 year ago

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
  )

Sniffnoy3• 1 year ago

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!

boothby3• 1 year ago

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• 1 year ago

Prime generating functions in polynomials? That's almost Lisp domain.

Mathematicians should play with Scheme and SICP.

swayvil5• 1 year ago

Yeah but can we get a pretty picture out of it? A cool fractal is worth a thousand words.

nprateem• 1 year ago

Oh. (3n3 − 13n2 + 18n − 8)M1(n) + (12n2 − 120n + 212)M2(n) − 960M3(n) = 0.

I'd have thought that was obvious.

drdunce• 1 year ago

This have implications for public key cryptography?

datameta7• 1 year ago

My naive notion on this is yes, iff the new method is computationally or memory-wise of lower complexity

boothby3• 1 year ago

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• 1 year ago

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• 1 year ago

BTW the phone number in Jenny's song, 867-5309, is a twin prime (867-5311).

jsisto• 1 year ago

Twin towers prime 9/11

adgjlsfhk12• 1 year ago

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• 1 year ago

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• 1 year ago

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...

noqc3• 1 year ago

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.

ijustlovemath3• 1 year ago

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.

noqc3• 1 year ago

You could not be more wrong, and I am in a very bad mood unrelatedly, so that is all I will say.

lupire• 1 year ago

I have no idea what you are trying to say. What is "this" and what is the other "this"? A and B?

What is an arbitrary A?

coderatlarge3• 1 year ago

just this quote from the article shocked me:

“In 1976, Jones, Sato, Wada, and Wiens (2) (…) produced a degree 25 polynomial in 26 variables whose positive values, as the variables vary over nonnegative integers, is the set of primes.“

coderatlarge3• 1 year ago

here’s an explicit formula:

https://mathoverflow.net/questions/132954/the-jones-sato-wad...

coderatlarge3• 1 year ago

just to clarify this is well beyond my ability to verify or even situate in larger context.

Mathematicians hunting prime numbers discover infinite new pattern

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