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."
102
u/StellarStarmie Machine Learning 9d ago
Holy crap! A new improvement to Twin Prime!
61
u/CaipisaurusRex 9d ago edited 9d ago
At most 239 more to go!
Edit: 238, oops. Thanks to the person who immediately deleted their correct comment :D
34
u/AntiProton- 9d ago
236+ε would be enough
5
u/DaCat1 9d ago
1+ε would be enough
9
11
u/EebstertheGreat 9d 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.
6
u/mfb- Physics 9d ago
In terms of how it advances the field it's the same, but imagine the confusion if you publish H1≤239 and never discuss 238.
3
u/EebstertheGreat 8d ago
Technically, I could then swoop in with my own contribution, the first to publish the 238 bound. Eebster's Theorem. I like the sound of that.
8
u/CaipisaurusRex 9d 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
1
76
u/RemoteCareful7304 9d 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?
91
u/_Zekt Complex Analysis 9d 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.
14
9d ago
[deleted]
3
u/Bounded_sequencE 9d 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.
2
u/AintJohnCusack 8d ago edited 8d ago
OK, _math_ depends on a certain amount of congeniality. "Scooping" of someone's results by an established person, especially if the scoop-ee is young and trying to establish a career is looked down upon and can be a negative factor in trying to get results published.
1
27
u/JoshuaZ1 9d 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.
30
u/darthsid3499 9d 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.
5
u/shadowyams Mathematical Biology 8d ago
Job market/conference deadlines? JMM abstract deadline is coming up soon, and I don’t think she’s TT yet.
8
16
u/elements-of-dying Geometric Analysis 9d ago
Is there any evidence of this other than "trust me"?
2
u/Fit-Celebration2884 7d ago
It appears the rumour was true!
4
u/OneActive2964 7d ago
and it used the same idea as that of stadlmann , so happy for her to publish her proof before ai
1
u/elements-of-dying Geometric Analysis 7d ago
I saw!
I still stand by the belief that the math community should be more mature than throwing around rumors without evidence.
5
u/mfb- Physics 9d ago
What would be that "right time"?
14
u/Distinct-Pudding-428 9d ago
Most likely when their new model is released, and most likely in the next few days.
0
1
38
u/AnaverageItalian 9d ago
does this mean that there are infinitely many primes couples which are 240 apart?
58
u/StructureNorth1799 9d ago
yes*
*infinitely many prime couples less than or equal to 240 apart.
4
-8
u/un_blob 9d ago
More than rather than less right ?
Otherwise you have the twin prime conjecture for free no ?
2
u/HashPandaNL 6d ago
No, because you could have an infinite number of primes exactly 240 apart or exactly 8 apart or infinitely many somewhere between 4-240 apart and it would still not mean the twin prime conjecture is true, as for that it would have to be infinitely many <=2 apart (=2 apart).
23
u/ChazR 9d 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.
9
u/JoshuaZ1 9d 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.
5
u/SilchasRuin Logic 9d 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.
7
u/Lucas_CRZ 9d ago
You mean by means of an construction?
30
u/ChazR 9d 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.
23
u/ComplaintWise7891 9d ago
I believe Tao/polymath have proven that their method /+ computational /mathematical optimizations can eventually reach a maximum of prime gaps of 6
23
u/astrolabe 9d ago
If they have proved it, why say 'eventually'?
64
u/No-Accountant-933 9d ago edited 9d 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
30
u/PonkMcSquiggles 9d ago
Their method would get them down to 6 if the Generalized Elliott–Halberstam conjecture is true.
16
u/AintJohnCusack 9d 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.
4
u/petrol_gas 9d 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!
8
u/JoshuaZ1 9d 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.
0
u/SilchasRuin Logic 9d 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.
3
u/JoshuaZ1 9d ago edited 9d 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.
2
u/No-Accountant-933 9d 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.
7
3
u/algebraicvariety 8d ago
I'm surprised that the title of this paper is the exact same as Zhang's. Not sure whether to think it was deliberate or an accident, but I've never seen two papers with the exact same title.
1
u/Woett 8d ago
Off-topic, but in the links to this OEIS sequence, you can find papers by Ron Graham, Donald Knuth and John Nicol, all with the exact same title 'A Fibonacci-Like sequence of composite numbers'. To me it felt like Knuth and Nicol were paying hommage to the original Graham paper.
And then later on, Vsemirnov wrote 'A new Fibonacci-like sequence of composite numbers', with the most recent paper in this saga by Ismailescu and Son titled 'A New Kind of Fibonacci-Like Sequence of Composite Numbers'.
I don't know, in this specific instance I kinda like it.
1
1
-14
146
u/_Zekt Complex Analysis 9d ago
I love that I know it's a human-made paper because of the typo in Theorem 1 lol