Comment by vb-8448
10 hours ago
Maybe it's silly, but from someone who is ignorant on this topics, what are the consequences of this kind of "discoveries"?
Is it something "revolutionary" or just another small brick that will pile up until something really "revolutionary" will happen?
To me, this shows that extremely talented and qualified mathematicians (can) use frontier-level LLMs to automate their personal grind-y workloads that would otherwise (probably) take more time to accomplish with natural intelligence.
By itself, no consequence. But over time, provided we keep pumping out talented and qualified mathematicians and keep subsidizing costs, we could maybe hit a breakthrough... somewhere... that has real impact.
It's an indicator of AI progress. The solutions aren't especially revolutionary, but no person had been able to solve them after decades of collective attempts.
To be fair I don’t think there were too many people really trying to. Symbolically, one could make a parameterization of the Jacobian determinant and then brute force a solution, if one had known such a polynomial existed in only three dimensions.
This is not true at all. The parameter space is absolutely MASSIVE. The counterexample is a degree 7 polynomial in 3 variables, which means 360 coefficients. There's no particular way to bound these coefficients or even the degree or number of variables apriori, but assume you somehow did. Also assume you were confident that it would work with integer coefficients bounded from -12 to 12. Now you have to iterate over 360 degrees of freedom, verify that the Jacobian is a nonzero constant, and somehow show uninvertibility of the transformation, which is not a particularly simple task.
If you searched for coefficients from -12 to 12, this would be 25^360 = 2 * 10^503 different possibilities. A common reference point is that there are 10^80 atoms in the observable universe. Sure you could probably reduce this a bit with clever tricks, but the starting point makes the method completely unviable, even with the knowledge: A) a counterexample exists, B) it's in 3 variables, C) it's in degree 7 or less, D) it's in integer coefficients, E) those coefficients are 12 or lower.
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Oh yes there were. The Jacobian conjecture is "notorious for the large number of published and unpublished false proofs which turned out to contain subtle errors."
It's not quite the Reimann hypothesis, but many prominent mathematicians have spent years working on this problem. Yitang Zhang wrote his PhD thesis on it.
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Practically, from this specific one? Nothing, it's very much a math thing. It's like art or music at this level. Are there consequences to a van Gogh?
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"Hey Fable, please generate me the next 1000 undiscovered bitcoin hashes"
You're joking, but perhaps LLMs will find a way to mathematically break the complexity of factorization.
Maybe they'll find a solution where P=NP.
That could really throw a wrench into the whole internet thing.
It seems they need an expert human driver for now.
I'm sorry, I can't do that, but here is the design for a stable quantum computing platform that should allow you to generate those keys yourself...
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