OpenAI releases solutions to many open problems in mathematics. They provide both lean proofs and preprints in a git repo. We knew they were working on this, now it has been released.
Some results that I thought were interesting include (numbers from this document):
#4: Hilbert’s 10th problem was resolved negatively 1970. It asks if there is an algorithm to determine whether a multi-variable polynomial with integer coefficients has an integer root. This is a generalization of that result to rational roots: It shows that no algorithm that can decide whether a polynomial with integer coefficients has a rational root.
#25: Every rational can be written as an Egyptian fraction of length .
#28: The Gaussian moat conjecture is true. Here is what it says: Pretend the Gaussian primes are islands and the rest of is ocean. Suppose we can kayak a distance before we need to stop and rest at an island. Then there is no value of large enough to allow us to make a journey from the origin to infinity.
#103: L = RL = BPL, i.e. probabilistic algorithms with logarithmic space complexity can be de-randomized to deterministic algorithms with logarithmic space complexity.
#107: Matrix multiplication in time.
#109: Integer multiplication (slightly) faster than .
#130: Fourier transforms (slightly) faster than . As measured by the number of arithmetic operations.
#137: One-tape Turing machines that run in time can be simulated in space.
#155: There is a tile that forces aperiodic tiling of and under the rule that we’re not allowed to rotate the tiles.
#158: The chromatic number of (the unit distance graph of) the plane is greater than 5.
#241: “Rigidity of Turing Degrees”: No non-trivial relabelling of Turing degrees preserves their order.
#261: Characterize spectrum of lattice Anderson model in dimension 2 vs 3 or more dimensions.
#363: Demonstrate two solutions of the Boltzmann equation for hard spheres, with the same initial conditions.
The most surprising thing to me was the claim of an average of 3 hours of thinking time per solved problem. To me it seems that the model has passed human level abilities which allow it to scoop up these results with relatively little effort. If a model can be trained on open problems and presumably create it’s own open problems from the result of solving open problems I’m not sure where it will end.
Solving problems they’re good at, but creating its own open problems remains to be seen. Some, like Noam Brown, think AI not being able to come up with good problems to train on is a plausible (not likely, but plausible) way for things to plateau.
Woah. Some additional context for #107, matrix multiplication time complexity:
From 1990 to August 2026, the best known exponent improved only from about 2.3755 to 2.3711. Then this new result just jumps straight to 2.25. This is a famous problem that a huge number of researchers have worked on.
And #109, integer multiplication, is just as bizarre. Most of the experts in the field had believed for a long time, maybe 50 years, that O(n log n) was the asymptotic barrier, and it would be a very natural one at that, and what’s left is just figuring out how to get there. And then it turns out it wasn’t even optimal?
Yup. Human mathematicians reached in 2019. I remember reading this post reporting on the release of an algorithm at the time. I assume people probably mostly stopped working on it at that point, if they assumed the natural limit had been reached. The network coding conjecture mentioned in that post, which implied , must be false.
Here are Scott Aaronson’s thoughts on the resolution of Unique Games, and some of the other theoretical CS problems: https://scottaaronson.blog/?p=10169
Update on #109: this post complains that the AI’s algorithm “cheats” by using unbounded coefficients, making counting exact arithmetic operations unfaithful to other models of computation (because multiplying by really big numbers makes each arithmetic op pretty expensive, as you might guess). The main interesting thing from the post is that it links this paper, which is short but very nice: Morgenstern, 1973, which actually establishes an lower bound when coefficients are bounded, which is pretty cool. Morgenstern’s proof doesn’t seem to carry over the the case of computing a DFT with unitary normalization, though, which is kind-of suspicious.
OpenAI releases solutions to many open problems in mathematics. They provide both lean proofs and preprints in a git repo. We knew they were working on this, now it has been released.
Some results that I thought were interesting include (numbers from this document):
#4: Hilbert’s 10th problem was resolved negatively 1970. It asks if there is an algorithm to determine whether a multi-variable polynomial with integer coefficients has an integer root. This is a generalization of that result to rational roots: It shows that no algorithm that can decide whether a polynomial with integer coefficients has a rational root.
#5: Catalan’s constant
#17: The irrationality exponent of
#25: Every rational
#28: The Gaussian moat conjecture is true. Here is what it says: Pretend the Gaussian primes are islands and the rest of
#102: The Unique Games Conjecture is true.
#103: L = RL = BPL, i.e. probabilistic algorithms with logarithmic space complexity can be de-randomized to deterministic algorithms with logarithmic space complexity.
#107: Matrix multiplication in
#109: Integer multiplication (slightly) faster than
#130: Fourier transforms (slightly) faster than
#137: One-tape Turing machines that run in time
#155: There is a tile that forces aperiodic tiling of
#158: The chromatic number of (the unit distance graph of) the plane is greater than 5.
#241: “Rigidity of Turing Degrees”: No non-trivial relabelling of Turing degrees preserves their order.
#261: Characterize spectrum of lattice Anderson model in dimension 2 vs 3 or more dimensions.
#363: Demonstrate two solutions of the Boltzmann equation for hard spheres, with the same initial conditions.
And the Quasi-Riemann Hypothesis!
The most surprising thing to me was the claim of an average of 3 hours of thinking time per solved problem. To me it seems that the model has passed human level abilities which allow it to scoop up these results with relatively little effort. If a model can be trained on open problems and presumably create it’s own open problems from the result of solving open problems I’m not sure where it will end.
Solving problems they’re good at, but creating its own open problems remains to be seen. Some, like Noam Brown, think AI not being able to come up with good problems to train on is a plausible (not likely, but plausible) way for things to plateau.
Woah. Some additional context for #107, matrix multiplication time complexity:
From 1990 to August 2026, the best known exponent improved only from about 2.3755 to 2.3711. Then this new result just jumps straight to 2.25. This is a famous problem that a huge number of researchers have worked on.
And #109, integer multiplication, is just as bizarre. Most of the experts in the field had believed for a long time, maybe 50 years, that O(n log n) was the asymptotic barrier, and it would be a very natural one at that, and what’s left is just figuring out how to get there. And then it turns out it wasn’t even optimal?
Yup. Human mathematicians reached
Here are Scott Aaronson’s thoughts on the resolution of Unique Games, and some of the other theoretical CS problems: https://scottaaronson.blog/?p=10169
Update on #109: this post complains that the AI’s algorithm “cheats” by using unbounded coefficients, making counting exact arithmetic operations unfaithful to other models of computation (because multiplying by really big numbers makes each arithmetic op pretty expensive, as you might guess). The main interesting thing from the post is that it links this paper, which is short but very nice: Morgenstern, 1973, which actually establishes an