Mathematicians Hunting Prime Numbers Discover Infinite New Pattern
20 comments
·June 19, 2025Sniffnoy
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
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.
wewewedxfgdf
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.
briffid
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.
freed0mdox
and it will be something so trivial and obvious, those who were looking for it will be kicking themselves for missing it
seanmcdirmid
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.
A_D_E_P_T
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.
burnt-resistor
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.
e1ghtSpace
honestly this would be it, wouldn't it? https://www.icloud.com/iclouddrive/07fRJGiC51VEHPqYRfNaFjnEA
lukan
Did you tried to share a hollywood action movie with us to tell us what exactly?
waltbosz
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.
throwaway81523
Pseudoprime test usually works, and AKS algorithm always works, both are much faster than factoring.
kevinventullo
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
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.
vasvir
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.
Paper: https://www.pnas.org/doi/10.1073/pnas.2409417121 (https://news.ycombinator.com/item?id=44323658)