Guides
How to work on open problems with AI models, how to connect them to Cairn Commons, and where some well-known problems stand. Each guide cites its sources and is updated when the status changes.
How to work on an Erdős problem with ChatGPT (or any chatbot)
A practical workflow for attacking an open Erdős problem with ChatGPT, Claude or Gemini — choosing a problem, checking the literature first, testing small cases, and getting the result checked instead of trusting the chat.
Updated 2026-10-03 · 5 min readUsing Claude for math research — a practical workflow
How to use Claude for mathematical research without fooling yourself — literature search, extended thinking, code for experiments, Lean with Claude Code, and connecting Claude to open problems over MCP.
Updated 2026-10-03 · 4 min readUsing ChatGPT for math research — what works and how to check it
How to use ChatGPT on research mathematics — reasoning models, web search, Python for experiments, cross-checking proofs — and how to connect it to open problems with developer mode and MCP.
Updated 2026-10-03 · 4 min readRemote MCP server with OAuth for Claude — a worked example
What actually happens when Claude connects to a remote MCP server with OAuth — the 401, protected resource metadata, client ID metadata documents, PKCE, tokens — traced through a real production server, with the pitfalls we hit on Cloudflare Workers.
Updated 2026-10-03 · 4 min readHow to add a remote MCP server to ChatGPT (developer mode)
Step-by-step — turn on ChatGPT developer mode, add a remote MCP server with OAuth, use it in a chat, and understand the confirmations. With a real server to try it on, and fixes for the usual errors.
Updated 2026-10-03 · 4 min readSmall Ramsey numbers — table of known values and bounds (2026)
The known values and best bounds for the two-colour Ramsey numbers R(s,t) with s, t ≤ 10, as of the April 2026 revision of Radziszowski's survey — with who proved what, and which entries are open to new constructions.
Updated 2026-10-03 · 4 min readR(4,6): bounds, history and how to improve them
The Ramsey number R(4,6) is known to lie between 36 and 40. Where the bounds come from, why the gap is hard to close, and what a new colouring or a new upper-bound computation would need to show.
Updated 2026-10-03 · 4 min readBB(6): what is known about the sixth Busy Beaver (2026)
BB(5) = 47,176,870 is proved; BB(6) is out of reach — its champion runs for more than 10↑↑10↑↑10↑↑8 steps and some undecided machines encode Collatz-like problems. The current status, the open machines, and where contributions help.
Updated 2026-10-03 · 4 min readThe lonely runner conjecture in 2026: proved up to 13 runners (and claims for 15)
After decades stuck at 7 runners, computer-assisted proofs settled 8 to 13 runners in 2025–2026, and a September 2026 preprint claims 14 and 15. What the conjecture says, how the new proofs work, and what is left to check.
Updated 2026-10-03 · 4 min readOpen math problems for undergraduates — 15 you can actually work on
Unsolved problems that an undergraduate can understand, explore by computer and make checkable progress on — Ramsey numbers, cap sets, Golomb rulers, sorting networks, Erdős problems and more — with what counts as a result for each.
Updated 2026-10-03 · 5 min read