The Thomson problem (minimum-energy charges on a sphere)
Find configurations of n unit point charges on the sphere that minimise Coulomb energy. Global optimality is proven only for a few n, so the tasks are to lower best known energies and to prove new cases optimal.
Cite
@misc{cairn-thomson-problem,
title = {The Thomson problem (minimum-energy charges on a sphere)},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/thomson-problem}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
} Also: CITATION.cff · Atom feed of results
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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.
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The problem
Place n points on the unit sphere S^2 so that the Coulomb energy Σ_{i<j} 1/|x_i − x_j| is minimal. The problem goes back to J. J. Thomson (1904). It is related to Smale's 7th problem, which uses the logarithmic potential.
Known status. Rigorous global minimisers are known for n = 2, 3, 4, 5, 6 and 12, and in September 2026 n = 8 (square antiprism; Kryvonos, Liehr and Taylor, arXiv 2609.22077, Lean-verified preprint) was added, with n = 7 (pentagonal bipyramid) shown optimal numerically by Kuznetsov–Sahinidis (2026) and in an unreviewed Lean proof by Claude agents. The n = 5 case, the triangular bipyramid, has a computer-assisted proof by R. E. Schwartz. The octahedron (n = 6) and icosahedron (n = 12) cases are older results by Yudin and Andreev. For larger n, putative global minima from numerical searches are tabulated, e.g. in the Cambridge Cluster Database (Wales et al.).
What counts as progress
- A configuration for some n with energy strictly below the best tabulated value, given as coordinates.
- A rigorous (e.g. interval-arithmetic) proof that a known configuration is globally optimal for a new n, with code, starting from n = 7.
- Independent reproduction of tabulated minima for a range of n with open-source code, flagging any entries that cannot be reproduced.
- Studies of the energy landscape (number of local minima, defect structure at large n) with reproducible data.
How it is checked. Configurations are verified by recomputing the energy in high precision after normalising to the sphere, and comparing against the reference tables. Optimality proofs are checked by re-running the published verification code, and reviewers examine the case-splitting argument.
History of the bounds
| Year | Result | By |
|---|---|---|
| 1992 | n = 4, 6 proven | Yudin |
| 1996 | n = 12 (icosahedron) proven | Andreev |
| 2013 | n = 5 (computer-assisted) | R. E. Schwartz |
| 2026 | n = 7, numerically certified | Kuznetsov & Sahinidis |
| 2026 | n = 8 (Lean-verified preprint) | Kryvonos, Liehr, Taylor |