All 25,000 transitive groups of degree 24 realized as Galois groups
Klüners and Malle win without AI against mostly AI-assisted teams
Result confirmed
The takeaway
A crowdsourced competition set up by Terence Tao and the LMFDB solved the inverse Galois problem in degree 24 by July 2026. The winning pair computed their polynomials without AI, while ‘Almost all the other participants relied on the technology’.
Status
- Claim
Result confirmed
- Our reporting
- Medium confidence
- Verification
- Unverified claim
- Importance
- 3 of 5
- Last verified
- 14 October 2026
Your AI and this story
- GPT-6 Astra133 days after its cutoff
- Claude Opus 5.572 days after its cutoff
- Gemini 4 Argon163 days after its cutoff
- Grok 4.7102 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 72 days before it.
Key facts
- Announcement: Terence Tao’s blog, 16 June 2026, ‘Third SAIR competition: inverse Galois challenge’; organizers John Jones, Jen Paulhus, David Roe, Andrew Sutherland and Tao (per search summaries of the post); verification by Magma and PARI/GP rather than Lean
- Scope: 25,000 groups and 165,836 signatures; at the start the LMFDB held concrete polynomials for only 286 groups and 622 signatures (Paderborn)
- Outcome: ‘By July, all 25,000 groups had been solved’, so the degree-24 inverse Galois problem was fully resolved for these groups; participants submitted millions of integer polynomials (Paderborn)
- Winners: Jürgen Klüners (Paderborn) and Gunter Malle (RPTU Kaiserslautern), more than 143,000 relevant polynomials
- AI: ‘Almost all the other participants relied on the technology. Nevertheless, the team from Paderborn and Kaiserslautern managed to win the competition without using AI’; they used AI only to help write submission scripts (Paderborn)
- Context: Tao’s 9 Oct Math 2.0 lecture cites IGP24 as an example of new experimental mathematics
What happened
In June 2026 Terence Tao announced the third SAIR Foundation competition, run with the L-functions and Modular Forms Database (LMFDB): find explicit polynomials whose Galois groups are each of the transitive permutation groups of degree 24, almost the first open degree. Submissions were verified by computer algebra (Magma, PARI/GP).
Paderborn University’s press release of 10 Sep 2026 says the problem was finished: “By July, all 25,000 groups had been solved”, from a starting point of polynomials for only 286 groups in the LMFDB. Jürgen Klüners and Gunter Malle won with more than 143,000 relevant polynomials. The release frames it as humans against AI: “Almost all the other participants relied on the technology. Nevertheless, the team from Paderborn and Kaiserslautern managed to win the competition without using AI.” They used AI only for scripts that eased submission.
The organizers’ final report and the competition leaderboard were not read for this entry, so the ‘all solved’ claim rests on the university release.
Why it matters
IGP24 is one of the large crowdsourced experiments Tao has promoted as a new way to do mathematics with AI in the loop. Its result is a useful counterpoint to the many AI-credited results of autumn 2026: in a computational search problem, deep human expertise in Galois theory beat AI-assisted teams.
Sources
3 sources from 3 sites. Numbers match the chips in the text.
3 sources: 2 primary, 1 press
Primary
- Terence Tao: Third SAIR competition: inverse Galois challenge (16 June 2026)terrytao.wordpress.com, official
- SAIR competition page: IGP24competition.sair.foundation, official
Press
Changes
- Filed from lead of 2026-10-10 from Tao’s Math 2.0 slides; found through Paderborn’s press release