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On the Navier–Stokes Millennium Prize Problem
We’re sharing an AI-generated solution to the Navier–Stokes Millennium Prize Problem, including a writeup and a formal proof in Lean.
Most cited
Posts that commenters keep linking to, ranked by how many different people have cited them in Hacker News comments. 28 posts.
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Data Compression Explained (2012)
Copyright (C) 2010-2012, Dell, Inc . You are permitted to copy and distribute material from this book provided (1) any material you distribute includes this license, (2) the material is not modified, and (3) you do not charge a fee or require any other considerations for copies or for any works that incorporate…
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An OpenAI model has disproved a central conjecture in discrete geometry
An OpenAI model solved the 80-year-old unit distance problem, disproving a major conjecture in discrete geometry and marking a milestone in AI-driven mathematics.
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A Mathematical Framework for Transformer Circuits (2021)
Transformer language models are an emerging technology that is gaining increasingly broad real-world use, for example in systems like GPT-3 , LaMDA , Codex , Meena , Gopher , and similar models. However, as these models scale, their open-endedness and high capacity creates an increasing scope for unexpected and…
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Learning more about Claude's mathematical capabilities
An unreleased Claude model improved the lower bound for the fraction of zeroes of the Riemann zeta function that satisfy the hypothesis, raising it from 41.6% to 67.2%.
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Squares in Squares
The following pictures show $n$ unit squares packed inside the smallest known square (of side length $s$). If a pictured packing has multiple numbers in its label above, the picture represents the largest; each smaller is represented by removing any square. For the $n ≤ 324$ not pictured, the trivial packing (with no…
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A recent experience with ChatGPT 5.5 Pro
We are all having to keep revising upwards our assessments of the mathematical capabilities of large language models. I have just made a fairly large revision as a result of ChatGPT 5.5 Pro, to whi…
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Quantum computing bombshells that are not April Fools
For those of you who haven’t seen, there were actually two “bombshell” QC announcements this week. One, from Caltech, including friend-of-the-blog John Preskill, showed how to do quantum faul…
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The Mathocalypse
More Updates (Oct. 12): Jeffrey Ketchersid has used Claude Code agents to show that BusyBeaver(10) exceeds Graham’s number, beating the previous best result that BB(13) does so. In response t…
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Cloudflare targets 2029 for full post-quantum security
Recent advances in quantum hardware and software have accelerated the timeline on which quantum attack might happen. Cloudflare is responding by moving our target for full post-quantum security to 2029.
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A digestion of the Jacobian conjecture counterexample
The notorious Jacobian conjecture can be formulated concretely over the complex numbers as follows. Conjecture 1 (Jacobian Conjecture) Let $latex {F:{\bf C}^n \rightarrow {\bf C}^n}&fg=000000$ …
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The case against geometric algebra (2024)
Every once in a while the internet gets talking about Geometric Algebra (henceforth GA) and how it’s a new theory of math that fixes everything that’s wrong with linear algebra and multivariable calculus. When I come across this stance I am compelled to respond with something like: “wait wait, it’s not true! GA is…
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Rathbun's Operator
First, I’m a human typing this post. I’m not going to tell you who I am. Sure, many will say that is cowardly and fair, but I actually don’t think it would bring much value. What matters more is that I describe why, and what I did and didn’t do during the last two weeks or so with the OpenClaw agent I set up, known as…
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Ten advances in mathematics and theoretical computer science
OpenAI shares new results on long-standing open problems in mathematics and theoretical computer science, including advances in geometry, cryptography, and complexity.
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Decision trees – the unreasonable power of nested decision rules
An introduction to the Decision Trees, Entropy, and Information Gain.
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Human mathematicians are being outcounterexampled
It’s been an interesting few weeks for counterexamples. This post is basically my perspective of what has been going on in the world of formalization, AI tools and, in particular, counterexam…
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Nicholas Polson has authored 258 academic papers in 2026 so far
Nicholas Polson has, by my count using his SSRN page, already written 258 working papers in 2026 alone. He’s already written (or at least published to SSRN), six papers today, August 26, 2026.
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Formalizing Fermat's Last Theorem
Anthropic is an AI safety and research company that's working to build reliable, interpretable, and steerable AI systems.
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Improving Heuristics for A* Pathfinding
On my Introduction to A* page I cover the basics of the A* graph search algorithm for finding shortest paths. Here’s an example from the town of Denerim in Dragon Age Origins. Try moving the start and goal to see A* in action:
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Why I didn’t sign the Fields medallists’ letter
[This post has been cross-posted to Terence Tao’s blog.] When I was around 11 I heard for the first time about Fermat’s Last Theorem. I was immediately captivated by the problem stateme…
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I vibed a proof of Conway's conjecture
You can just prove things, apparently.
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Avoiding Trigonometry (2013)
Tutorials and articles of Inigo Quilez on computer graphics, fractals, math, art, demoscene, shaders and more.
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Monte-Carlo Simulations
Monte Carlo simulations (or methods ) is the technique of applying randomness and the Law of large numbers to the solution of various scientific and engineering problems. One of its first documented uses was by Stanislaw Ulam and John von Neumann for nuclear weapon simulations after WWII [1] .
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A Misalignment of AI in Mathematics
I am proud to be among the list of 25 initial signatories — all Fields Medallists — to the declaration below, which grew out of discussions between ourselves over the last week. We have…
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Factoring RSA 260
How Devin and a Cognition researcher built the world’s highest-performance GPU lattice siever, to make factoring numbers 10x cheaper than the previous…
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Responsible Release of AI-Generated Mathematics
Responsible Release of AI-Generated Mathematics September 29, 2026 Back to main page Download PDF At present, some frontier AI labs are testing advanced mathematical problems on proprietary models …
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Floating-Point Printing and Parsing Can Be Simple and Fast
A floating point number f has the form f = m · 2 e where m is called the mantissa and e is a signed integer exponent . We like to read numbers scaled by powers of ten, not two, so computers need algorithms to convert binary floating-point to and from decimal text. My 2011 post “ Floating Point to Decimal Conversion is…