[QuesnayJr]

Why would I try it to win an argument on HN? That's a bizarre suggestion. Just look at the degree. If it were degree 47 in 17 variables then it wouldn't be surprising, but here it's surprising.

Of course people tried hard to solve them all, which is why it's so surprising that they were open. If anything, the solutions have gotten easier. The unit distance graph solution relied on a famous theorem remote from graph theory. The cycle double cover solution relied on a standard theory in graph theory. The solution of the Jacobian conjecture required nothing beyond knowing the definition of the Jacobian.

We're just surprisingly bad at judging the difficulty of problems. It's probably something psychological. It's even a known phenomenon, where someone will be stuck on a proof, someone else will announce the result, and the first person will suddenly get unstuck on their proof and produce an independent proof of the same theorem.

[Davidzheng]

Sorry last comment was a bit emotional from me, but I do not think it's findable like you say--during my PhD I tried to find some ideals I knew existed in char 2 in 5 variables and low degree and I didn't think I ever got close. 3 variables, 7 degree, coefficients up to 6 is like 6^100 possibilities. You've got to narrow it down somewhat no? Even sparse is intractable I would guess.

I think the solutions which rely on the least amount of theory are the most telling of the AIs being higher in intelligence than humans today already. There's almost no theory to teach someone to understand the cycle double cover conjecture as you say, yet no one finds it. I don't think the conclusion is that it was "easy", but that it was in fact irreducibly difficult in a way that proofs developed with theory are not. Theory gives the human brain abstractions to simplify complex proofs to be understandable at our capacity--I think there are many proofs which probably are not of this form.

But I think our differences hinge on how hard we perceive these solutions to be--I think they are very hard to find!

[Davidzheng]

But actually my feeling is that the final solutions of these last two problems probably is hiding how the AI came up with them! For all we know it used a LOT of theory! As Dolly Parton says "it takes a lot of money to look this cheap" and it takes a lot of intelligence for the proofs to look this dumb. [In high school, I knew of this competition math kid joke where after you derive an inequality with various methods, you use standard results to write the original equations as just a sum-of-squares---like in a "are you stupid, it's >=0 bc it's a sum of squares" sort of way]

[meowface]

You can use that retroactive logic about any hard problem though. Unsolved murder cases, math, theoretical physics.

If tons of smart humans try for years and fail and then an LLM tries for a few weeks or hours and succeeds, the implications are clear. And these are by far the dumbest LLMs will ever be.

[QuesnayJr]

The retrospective view is important, though. In retrospect, these problems weren't that hard. (The unit distance graph problem was the hardest.) There are some problems that still seem hard, even when we know the answer. Nobody thinks that Fermat's last theorem is easy, even though now know it to be true.

Before AI, it was pretty rare that a problem that turned to be unexpectedly easy, so mathematicians thought they were pretty good judges of it. (The last pre-AI example I can remember is the Gaussian correlation conjecture.) So thanks to AI we have learned that we were overconfident in our ability to judge difficulty.

If a truly major problem falls, like the Riemann hypothesis, and the proof turns out to be 10 pages, then the lesson will be a different one -- mathematicians are bad at math, and they should turn to more natural domains for them, like folding and putting away towels.

[betafj]

If some LLM is able to come up with a simple proof of Fermat's Last Theorem (a solution that Fermat himself could come up with) in the future, would you still say that Fermat's Last Theorem is hard?

[QuesnayJr]

No. I would say that it was easy, and that something had gone terribly wrong with math research that we missed it.

[meowface]

This would seem to lead to absurd implications. What if in, say, 20 years, an AI is able to independently prove nearly everything important in under an hour, including the Riemann hypothesis, with no contamination from other proofs? (Let's say it's also free to write and run arbitrary code, as well.)

Whether it's 5, 10, 20, 50 years, obviously the takeaway cannot be "something had gone terribly wrong". The takeaway would be the smartest humans were never close to the theoretical intelligence and wisdom ceiling and never could've been. This will one day seem obvious in retrospect. There's no reason evolution by natural selection would've landed any species near such a ceiling.

[QuesnayJr]

That's not at all the scenario you proposed. You proposed a simple proof that Fermat himself would understand. We have developed a tremendous amount of math since Fermat, and if none of it was relevant that would be damning. If there's a simple proof of the Riemann hypothesis that Riemann would understand, then I would say the same thing.

That doesn't rule out an AI that makes a genuine breakthrough. If there's some new branch of math that no human has even imagined that answers the Riemann hypothesis, then that is exactly how I would expect it to go.