Lets play over/under on an AI model proving (or counter exampling) the Riemann hypothesis?
I'm not sure what a good mark would be, but considering this result lets put it at 2027-08-10 (One year from today).
I'm not sure what a good mark would be, but considering this result lets put it at 2027-08-10 (One year from today).
Solving RH likely requires AI that is substantially more creative. But we haven't even solved the creativity problem for writing let alone mathematics. I believe that transformers are a trillion dollar local optimum that we will find it very hard to escape.
Let's wait for the models to produce a good novel first.
With writing it's just more obvious. LLMs don't write with personality. They don't create new and exciting worlds on their own. Everything they output feels derivative.
In mathematics you see the same effect. They are very good at finding results that humans missed, taking advantage of their broad knowledge and tireless work ethic.
But just as they have been unable to create new literary worlds, they also have so far been unable to create new mathematics.
I believe this lack of creativity is intrinsic to how these models are architected and trained. We want models that produce these in-distribution outputs because those types of models are more economically valuable. Nobody wants a coding agent with spontaneity, we want models that predictably and obediently solve problems - and that's what we got.
It is really the same with any art form.
There's no way you can conclude that. Yes, "Fable 2" or whatever this was probably won't. But we can't know what Fable 3/4/5/etc will be able to do.
If anything, if we have 1 or 2 more years of progress like the last 12 months, which have been insane, I'd say LLMs are likely to solve it.
Why? Because such explicit numerical improvements are not that interesting, which is best summed up in a review paper [0] of our efforts on RH spanning a century and a half,
> The pathetic attempts to enlarge the ridiculous zero free region in the critical strip is a perfect example of what brute force can do without fully exploiting fundamental arithmetic aspects of the problem. (italic added)
For outsiders, zero free region is another angle abundant with numerical improvements but no groundbreaking insights. For percentage people stop at ~40% because there is no need to proceed further, just like we are not interested in computing the googol-th digit of pi although in principle we could.
The groundbreaking results are like Selberg's that goes from zero to 0.01% (actually it is left unspecified, all we know is the percentage is positive), or Zhang's twin prime bound from infinity to 70,000,000. After this leap the pure numerical difference between 0.01%, 40%, 67% or even 100% is not substantial, and that's partly why Selberg did not even bother to compute it. Also RH will not follow from 100%, because in mathematics 100% does not mean all.
On the other hand, it is also wrong to dismiss such results all together. Riemann already know the real part of all zeros are bewteen zero and one, and RH says they equal 1/2. If someone or some LLM proved they are all less than 0.99, well this would be huge, and I'd bet they would easily get a Fields and be remembered forever. Innocent looking results could have drastically different technical depth behind them.
Alas math ppl tend to agree that RH will not be proved one bit at a time. The fundamental arithmetic aspects, once found out, will likely knock out not only RH but all the other L-functions in one go.
For example, even if Claude could prove the statement "100% of the zeroes lie on the critical line", that's strictly weaker than the Riemann Hypothesis, so even the best possible version of this result would fall short. (It's an asymptotic result, so it just means the percentage of counterexamples to the Riemann hypothesis goes to zero as their magnitude gets large.)
The singularity is approaching.
I'd accept AI likely became somewhat helpful to frontier AI research & development in early 2026.
I think for me though the real game changer moment will be when AI working autonomously is able to hypothesis and test algorithmic improvements at a faster rate than humans. This will be done to some extent by scale – lots of parallel agents coming up with lots of hypotheses and running the best candidates as tests. But also (and perhaps more importantly) by making more consequential algorithmic discoveries in the field of machine learning than humans – a bar we appear to have crossed or are crossing with math.
I suspect AIs today are super-human at finding performance improvements and minor iterations on current approaches. Whether they can solve some of the larger algorithmic challenges in the field however I'm not yet sure, although it seems likely that unreleased models are starting to make progress here.
An algorithm breakthrough on par in significance with the attention mechanism, primarily driven by automated AI research in say a field like continual learning would in my opinion be extremely significant and should leave no doubters that the singularity is here and will rapidly alter the world as we have known it.
Because algorithms have lower bounds, and the computational characteristics of LLMs are well-characterized by papers like https://arxiv.org/abs/2310.07923 . No amount of intelligence can make something faster than a mathematically-proven lower bound, any more than it could make 1+1=3 (that's why every single successful production transformer architecture has some form of O(N^2) attention layers, because it's mathematically impossible to achieve the same expressive power without any). There is room for speedup where current implementations are slower than the proven lower bound, but not when they're already close to it.
Sure, but I'm obviously not limiting research to improvements on current approaches only.
We know the brain is far more energy efficient and sample efficient than current AI. There is clearly better algorithms out there.
The question is who will find those next big algorithmic improvements like the transformer architecture? Will it be AI or humans?
My bet would be AI.
There have been apocalyptic preachers foretelling the end times for my entire life. Interesting to see how the language has changed, even as the predictions fail again and again.