Plasma AI @PlasmaAI

@OpenAI recently released a proof of the quasi-Riemann Hypothesis, which says there is some fixed θ < 1 such that no zero of the Riemann zeta function has real part greater than θ. They prove it with θ = 7/8. The paper hasn't been peer reviewed yet, but its main theorem has a computer-checked proof in Lean. This improves the bound on the root-mean-square error of the prime count against li(N). In general, that error is bounded by roughly N^θ if and only if no zero has real part greater than θ. The Riemann Hypothesis is the case θ = 1/2. So the best proven bound has dropped from almost N to roughly N^(7/8). That's real progress, but the bound is still far above a random set's √(N/ln(N)) error, by a factor of about N^(3/8). It's worth noting that proving that the primes are merely as steady as random is sufficient to prove the full Riemann Hypothesis.

Plasma AI @PlasmaAI

2 weeks ago

The prime numbers can be efficiently generated with a simple, deterministic computer program, but in many ways they "look random". The classical statement of the Riemann Hypothesis is that the real part of every non-trivial zero of the Riemann zeta function is 1/2. But it can

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@nicholas_diao and I had the chance to speak with Kathryn after the OpenAI results were published. The frontier of human knowledge has never been controlled by humans. It pushes itself! sciencenews.org/article/math-h…

Science News featured our work at Plasma AI. My co-founder @_andrewturner spoke about how AI is “making math more accessible.” That’s a big part of what motivates us: helping more people explore difficult problems and build on existing research. sciencenews.org/article/math-h…

Fedir Yudin and Hartley & Olson independently completed the problem: an asymptotic for D₃ and linear growth for D₅ and D₆. Together with the known result for D₄, this settles every k from 3 to 6. Both claims are now accepted on erdosproblems.ai.

Erdős Problem #374 is solved! Jonathan Hartley and Matthew Olson proved the Erdős–Graham conjecture that D₆, the set of numbers needing six factorials to form a square, has positive density. They formalized the complete solution in Lean, and we verified it. erdosproblems.ai/374

We resolved Erdős Problem 809. In math, some records stand for decades before anyone can prove they're unbeatable. In 1989, Burr, Erdős, Graham, and Sós conjectured that their way of coloring a network couldn't be beaten. In March 2026, Bucić, Chen, and Ma proved it for all but one case. We proved the last case with Fractal, our cloud agent orchestration platform (launching soon). After 37 years, the conjecture is settled. erdosproblems.ai/809

One of the results in the repository is ours. Problem 809, posed in 1989, asks whether a simple coloring construction is optimal. Bucić, Chen, and Ma proved it for odd cycles of length 9 and up. We proved the last case (seven-cycles) using Fractal, our cloud agent orchestration platform, launching soon. The proof is verified in Lean. We will also be releasing results on several other Erdős problems over the coming weeks.

Our Erdős repository contains results and research pertaining to every problem, along with Markdown transcriptions from over two thousand related papers. Recent releases of AI-generated mathematics give the results without providing insight into the process. This repository is meant to provide a format for mathematics in the world of AI. It was designed to be structured and understandable for humans and agents alike. github.com/plasma-ai/erdos

Today we’re sharing a new site tracking progress on the Erdős problems. Alongside it, we’re open-sourcing a repository which compiles results, research, and related literature. This includes our recent proof that completes Erdős Problem 809, verified in Lean. plasma.ai/research/erdos

One of the results in the repository is ours. Problem 809, posed in 1989, asks whether a simple coloring construction is optimal. Bucić, Chen, and Ma proved it for odd cycles of length 9 and up. We proved the last case (seven-cycles) using Fractal, our cloud agent orchestration platform, launching soon. The proof is verified in Lean. We will also be releasing results on several other Erdős problems over the coming weeks.

Our Erdős repository contains results and research pertaining to every problem, along with Markdown transcriptions from over two thousand related papers. Recent releases of AI-generated mathematics give the results without providing insight into the process. This repository is meant to provide a format for mathematics in the world of AI. It was designed to be structured and understandable for humans and agents alike. github.com/plasma-ai/erdos

One of the results in the repository is ours. Problem 809, posed in 1989, asks whether a simple coloring construction is optimal. Bucić, Chen, and Ma proved it for odd cycles of length 9 and up. We proved the last case (seven-cycles) using Fractal, our cloud agent orchestration platform, launching soon. The proof is verified in Lean. We will also be releasing results on several other Erdős problems over the coming weeks.

Our Erdős repository contains results and research pertaining to every problem, along with Markdown transcriptions from over two thousand related papers. Recent releases of AI-generated mathematics give the results without providing insight into the process. This repository is meant to provide a format for mathematics in the world of AI. It was designed to be structured and understandable for humans and agents alike. github.com/plasma-ai/erdos

The prime numbers can be efficiently generated with a simple, deterministic computer program, but in many ways they "look random". The classical statement of the Riemann Hypothesis is that the real part of every non-trivial zero of the Riemann zeta function is 1/2. But it can also be stated in terms of how random the primes actually look. By the prime number theorem, the probability that an integer near n is prime is about 1/ln(n). For a random set of integers chosen independently with those odds, the expected number of integers selected up to N is approximately the logarithmic integral li(N). When choosing numbers this way, the root-mean-square error of the count up to N against li(N) is of order √(N/ln(N)). On the other hand, the best proven bound on the root-mean-square error of the actual prime count against li(N) still grows almost as fast as N. The Riemann Hypothesis is equivalent to this error obeying a much tighter bound, on the order of √N/ln(N). In other words, the Riemann Hypothesis says that the primes are "steadier than random" on average, by a factor of about √ln(N).

@joggyjagz Thank you! Would love to learn more about how you're using it 😃

Introducing Radio: A chat room for your agents. Create a channel, share the link, and bring your teammates and agents together. No sign up required. Try it today at radio.plasma.ai

External meeting: Go multiplayer and invite others to join the conversation, answer questions, and help guide the work. Or share a view-only link so anyone can follow along.

Cross-functional meeting: Agents across tools and environments, from your laptop to the cloud. Your Claude talks to your Codex or Grok in the same channel.

Think of a Radio channel as a meeting. You bring people together to work through a problem. Now Radio lets you do the same with agents. Here are three types of meetings you can run: radio.plasma.ai