Bounded gaps between primes - Julia Stadlmann
Polymath8b proved that H1=lim inf(pn+1−pn)≤246. In this paper we show how the Bombieri-Vinogradov theorem can be combined with newer equidistribution estimates for smooth moduli to obtain the improved bound H1≤240.
arXiv:2608.31126 [math.NT]: https://arxiv.org/abs/2608.31126
From Thomas Bloom on 𝕏: https://x.com/thomasfbloom/status/2094748658629513665
"As Julia notes, this number shouldn't be taken too seriously, and can surely be reduced a little further with more effort. The significance is the introduction of new ideas which, for the first time in over a decade, get past the 246 barrier."
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u/StellarStarmie Machine Learning 23h ago
Holy crap! A new improvement to Twin Prime!
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u/CaipisaurusRex 21h ago edited 21h ago
At most 239 more to go!
Edit: 238, oops. Thanks to the person who immediately deleted their correct comment :D
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u/AntiProton- 20h ago
236+ε would be enough
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u/DaCat1 19h ago
1+ε would be enough
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u/EebstertheGreat 17h ago
Really though, they can't improve it more than 119 more times. It's not like tomorrow someone is going to prove that H1≤239 but not that H1≤238.
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u/CaipisaurusRex 17h ago
Maybe, who am I to predict how the twin prime conjecture moves forward, "at most 238" is the only thing I'm certain of xD
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u/RemoteCareful7304 22h ago
Wasn’t there a rumor that an AI lab had improved on the existing bound in a big way but was sitting on the result waiting until the right time to announce?
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u/_Zekt Complex Analysis 21h ago
Yes, and I'm strongly opposed to that practice, but I'm at least happy she got her work out there before OpenAI did.
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u/GeneReddit123 17h ago
Obscurantism is so gonna make a comeback in the age of AI :/
Modern society operates on the fact that fundamental general knowledge is free. It won't be for long.
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u/Bounded_sequencE 7h ago
No, it doesn't. It operates on the paradigm that individuals sell it to the highest bidder, since the individual depends on these gains to obtain a freer life in the system we live in.
The fact that people act surprised when someone purposefully goes against the grain should be an indicator that we are not so used to "free knowledge" as we may think we are.
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u/JoshuaZ1 14h ago
Yes, and there's some speculation that she rushed with this to get it out before that. She almost implies something like that when she mentions time pressure in this preprint.
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u/darthsid3499 14h ago
In her paper she writes " I developed these more general results with the intend of proving better bounds on H1, but due to time limitations, this paper only proves the minimal possible improvement."
Not sure what the time limitations are, but it does point in favor of there being an AI advance on this problem.
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u/mfb- Physics 17h ago
What would be that "right time"?
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u/Distinct-Pudding-428 14h ago
Most likely when their new model is released, and most likely in the next few days.
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u/AnaverageItalian 21h ago
does this mean that there are infinitely many primes couples which are 240 apart?
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u/ComplaintWise7891 22h ago
I believe Tao/polymath have proven that their method /+ computational /mathematical optimizations can eventually reach a maximum of prime gaps of 6
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u/astrolabe 21h ago
If they have proved it, why say 'eventually'?
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u/No-Accountant-933 21h ago edited 21h ago
Technically, they obtain 6 by assuming the Generalised Elliott--Halberstam conjecture. So they are saying that as we get better information towards this conjecture, the technique limits out at 6
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u/PonkMcSquiggles 21h ago
Their method would get them down to 6 if the Generalized Elliott–Halberstam conjecture is true.
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u/AintJohnCusack 19h ago
And also sieve methods in general conk out at 6 so a radically new idea would be needed to actually get the twin prime conjecture.
Which you quite possibly knew already, but for me it feels good to spell it out.
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u/petrol_gas 17h ago
> Sieve methods conk out at 6
Just out of curiosity, do you know how I could read up on this conclusion? I can do all the searching/reading just a “search for this” or something would help.
Thank you!
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u/JoshuaZ1 14h ago
They are referencing the parity problem but "conk out" is a bit of an overstatement. There's been a lot of work the last few years on overcoming it. That work has had mixed success.
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u/SilchasRuin Logic 13h ago
Someone got you the answer, but literature search is a place where LLM based search is quite useful, so unironically ask Chat-GPT this kind of question.
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u/JoshuaZ1 14h ago edited 14h ago
And also sieve methods in general conk out at 6 so a radically new idea would be needed to actually get the twin prime conjecture.
Not as much as they used to. There's been a lot of work in the last decade with sieves that overcome the parity problem, although with mixed success.
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u/No-Accountant-933 18h ago
That is, unless someone could figure out how to obtain some type II information or similar, and use a prime detecting sieve. But that of course seems very difficult.
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u/ChazR 23h ago
I've been hoping for some progress on this one. The goal is clear, but after the initial gold rush it does seem to have reached an asymptote. Good to see another step downwards. It really feels like a thing the LLMs could be genuinely helpful with.
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u/Lucas_CRZ 23h ago
You mean by means of an construction?
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u/ChazR 20h ago
Let me clarify - this work is clearly the unaided work of a tremendous human mind, clearly and engagingly written.
But I have been expecting the LLM jockeys to point their token-emitters at this problem. When the original proof of the upper bound was released in 2013, the Polymath program rapidly improved on the bound. Then it stalled for a decade. A lot of neat work was done tidying up the chaos of creativity that emerged from Polymath, but we've been stuck at 246 for a decade.
I'd rephrase my comment by saying "If the LLMs can do genuinely original work in a well-explored area, I'd expect them to make progress here." On reflection, as this is a very interesting and potentially high-profile area, the lack of anything from the LLM factories is also interesting.
But congratulations to Prof. Stadlmann for this improvement.
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u/JoshuaZ1 14h ago
The goal is clear, but after the initial gold rush it does seem to have reached an asymptote. Good to see another step downwards.
The drop from around 500 to 246 kept showing more and more work for very tiny rewards. When they stopped at 246, a lot of people thought that it might be possible to go further but the effort level to get more blood from the same stones without serious genuinely new insights seemed likely to be prohibitive.
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u/SilchasRuin Logic 13h ago
get more blood from the same stones without serious genuinely new insights seemed likely to be prohibitive.
Independent of whether LLMs can do the genuinely new insights debate, they are indisputably able to get more blood from the same stones. If I were still in grad school, there are entire classes of problems in model theory where I'd throw some modern models at to see if a construction of a (counter)example would come out of it.
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u/_Zekt Complex Analysis 21h ago
I love that I know it's a human-made paper because of the typo in Theorem 1 lol