[NiloCK]

Recent LLM dingers like the Jacobian Conjecture counterexample have challenged the efficient mathematics hypothesis. The JC counterexample was so small in degree and coefficient. It should have been a "low fruit" in the scheme of things, but alas, unpicked for 50+ years with considerable attention from good mathematicians.

With respect to FLT, my hopes have modestly increased that a truly marvelous demonstration of this proposition does in fact exist, that Fermat actually had it, and that it may someday be recovered!

edit: some emphasis on modest. But let me be romantic here!

[danbruc]

There are 120 monomials of degree at most 7 in three variables. If we restrict the coefficients to the integers from -10 to +10, that makes 21^120 possible polynomials. And we need three of them, that makes 10^476. And we would still miss the specific counterexample because it includes a coefficient of 12 outside of our range. So I would say that you will never find this specific counterexample by chance and whether you could accidentally trip over any counterexample really depends on their density. And we have of course not addressed the question why you would search this specific region of the parameter space, why dimension 3, degree 7 and small integer coefficients? There might be good mathematical reason to look at this region, but it is probably non-trivial to even figure out where to look.

[clircle]

efficient mathematics hypothesis? What's that?

[fancyfredbot]

It's a pun on "efficient markets hypothesis".

The efficient markets hypothesis says the market prices incorporate all available information and there's no such thing as a cheap stock, so there's no easy money to be made by trading.

The math equivalent is presumably that all easy problems have been solved and all open problems should be very very hard. This has turned out not to be the case as LLMs have found simple counterexamples to long held conjectures.

[mathisfun123]

They're the same thing; knowledge markets are markets.

[gosub100]

Not really, the incentives for each are not binary values. There are bounties on the 10 millennium problems (that haven't been increased in 25 years), but there isn't much motivation beyond that. You get some street cred and your name in a prestigious journal. That's not-nothing but can't hold a candle to the potential fortunes of the market.

[rheniumclass]

> There are bounties on the 10 millennium problems (that haven't been increased in 25 years) ...

Strictly speaking, if you constructively prove P=NP and the solver has reasonable time complexity, e.g. quadratic, it could make you incredibly rich. If you don't attract the wrong kind of attention from the world's intelligence agencies first...

[mathisfun123]

i think you have no idea what you're saying...

> That's not-nothing but can't hold a candle to the potential fortunes of the market.

a market is where things are sold and bought. there are lots of markets which aren't as well capitalized as financial markets - your local farmer's market doesn't "hold a candle to the potential fortunes of the market" but it's still a market (and presumably the efficient market hypothesis still applies).