We shaved a whole: 1/6129982163463555433433388108601236734474956488734408704 off the nlogn
I laughed out loud at the n lg n ^ (1 - 2^{-182}). It is so funny.
2^-182 is very funny but it's bigger than 0 and that's going to shatter a lot of people's conjectures.
Wowzers!<p>This also reaffirms my (wishful) thinking that <i>if</i> there’s a way to do FTL communication it’ll be something with an absurdly tiny factor like 2^-182 with a slight asymmetry in a probability somewhere.<p>Then you’re not violating FTL, just gaining a very slight chance that you <i>might</i> know something FTL – probably.
Given that c is the speed of causality itself, FTL communications would effectively be like predicting the future.<p>From that angle, beating light speed by some absurdly tiny factor would probably correspond to a means of predicting the future at some almost absurdly tiny factor better than random guessing.<p>Edit: Actually...it doesn't make sense to call this FTL communication, it's just predicting the future state of a system given some previous state. FTL comms would have to be predicting the future state of a system without information about the previous state.<p>Practically speaking predictive modeling would be a means of compensating for light speed comms, kind of like branch prediction in processors or speculative decoding in LLMs, but that wouldn't actually be FTL comms.
What's FTL?
If there was a way to do FTL communications you’d expect that Jane Street would have found it already
That would only mean we calculated c wrong
I mean, being pedantic a little, we don't actually know if c is constant, since measuring c is rather difficult. If c is not in fact constant in some medium or environment, then a huge number of things get very weird very fast.<p>So, yes, we could've measured c wrong. We just would have no idea if we did.<p>Source: Veritasium did a very fascinating video explaining this problem.
Dangit! I was betting on -183.
You didn't believe!
This is better
Why is that funny?
The relative difference is so absurdly small to be irrelevant at any realisable input size. At least that's my read; e.g. even at n=10^80 (~number atoms in universe), the relative difference is ~0. That's still probably underselling how similar this is to n log n.
The -182 feels highly arbitrary.
Is there an associated machine-checked proof of this?<p>We're in full vibe-code mode at work, so I understand both how powerful frontier models can be and how often they can over-confidently state subtly (or not so subtly) wrong things, even when you're taking great efforts to try to keep that from happening.<p>So without a Lean development or extensive human verification, I guess I'm a little bit skeptical, and even sort of hoping this is wrong - not just because of my not so positive feelings about AI, but by my disposition towards beauty in math. n log n is an awful lot nicer than what we have here.
Agree 100% on wanting machinr verification of AI generated math.<p>But in regards to beauty, i feel like multiplication already has a lot of non beautiful exponents. Best known matrix multiply is O(n^2.371). For integer factorization, the inverse of this problem, general number field sieve is a crazy subexponential.<p>If factorization is just barely subexponential, is it really that surprising that multiplication is just barely sub n lg n ?
We can just wait for whomever they stole THIS proof from to come forward with threatening emails sent by OpenAI.
For the uninitiated, why is this interesting given it doesn't seem to be so much below the threshold?
O(n lg n) is a bit of a threshold value. For a lot of algorithms, this is the best you can do, even in theory (similar to how O(n^2) is also a threshold for many algorithms). So for many algorithms, people stop trying when they get close to O(n lg n) on the belief that you'll never do better than that.<p>The fact that you can in principle go faster than n lg n, even if just by an almost imperceptible amount, is kind of surprising. It raises the question of, if n lg n isn't the limit, what is? How far down can we get the speed? If we can get it a little past n lg n, maybe we can go a lot further.<p>[or at least that is my understanding. not a theoretical computer scientist]
It's interesting because people wondered if it was possible to go below the threshold at all, that's all. Many suspected it was not possible.
this is perfect for when i have an array of at LEAST 2^118000 items<p>i will NEVER care about proposed multiplication speedups unless they are truly generalized
If you view them as "theories of computational limits" instead of "proposed practical speedups" they can be a lot more interesting.<p>It's most interesting when the lower bound can actually be proven. In lack of that, we have to guess what the best possible algorithm might yield (generalized or not). This tells us that need not be O(n log n) and we have the opportunity to still find better algorithms than we typically thought would be possible. This does the latter, which is interesting, but it just leaves us to hunger more for what the real limit must be :).
i understand this, but it always feels like we are being tricked when they say "integer multiplication below nlogn" because we intuit that that must mean "faster integer multiplication below nlogn EVERYWHERE!". but in reality it comes with 15 asterisks about the conditions that must be true for their statement to hold true.<p>Your issue is that I am viewing this proof as what it really is in terms of progressing the field and not from an imaginative perspective. I think that it is important to ground our selves somewhat in reality when discussing research like this because at the end of the day open ai is not doing for fun either.<p>openai wants to show the world what their product can do and i am simply not impressed
I can respect your opinion if it's consistent- i.e. it's not just directed at OpenAI's results. But this seems to be an example of a common phenomenon with AI discourse: while disparaging LLM achievements, you indirectly insult the careers/accomplishments of 99% of mathematicians for whom this would easily be the crown jewel of their CV.
This progresses the field a great deal, just perhaps not the field you're interested in? There's nothing wrong with a "and what can I apply that to in my life tomorrow" approach but it's certainly not the only approach worth having in the world.
Well, that’s true for a lot of TCS algorithms. The n log n algorithm prior was also not very practical for any numbers relevant to humans.
I love this, entirely separate from any applications or even understanding. It's incredible that we needed this trillion-dollar technology to learn about a faster way to multiply two numbers!<p>Math is incredibly rich, and even the simplest things have insanely complicated structure when you zoom in. However this all ends up, math is bigger than LLMs, and the people who claim it is getting "solved" and we are running out of open problems haven't stared into the abyss enough.
There are several of these "exponent used to be 1.0, we reduced it to 0.9999999" results in the "catalog".<p>There are also a bunch of other stinkers, like building a Turing machine out of Navier-Stokes fluids -- except that it only works if you can encode literally infinite amounts of data in the relative positions of two particles. I.e. assuming physics is based on set-theoretic real numbers, something we've known is wildly false for over a century: <a href="https://en.wikipedia.org/wiki/Banach-Tarski_paradox" rel="nofollow">https://en.wikipedia.org/wiki/Banach-Tarski_paradox</a><p>Just like vuln reporting, the AI industry has put zero effort into triage here, and the models are really good at making their findings sound more important than they really are.<p>The cynic in me suspects this is a smokescreen for the Navier-Stokes tokenstream plagarism fiasco.
Why is an openai release in .pdf? Isn't all ai in .md now?
I wonder if the AI spent extra time on this without being told to
I mean, cracking <i>anything</i> below the nlogn bound implies that there might be much more room for improvement. Often a very minor win over the theory opens up enough extra attention to later truly move the needle.
Related:<p><i>Sharing AI Progress in Mathematics</i><p><a href="https://news.ycombinator.com/item?id=49984923">https://news.ycombinator.com/item?id=49984923</a>
This is pretty remarkable, IF someone can understand it :)