The Thomson problem

Where do $N$ electrons settle on a sphere when they repel each other by Coulomb's law? J. J. Thomson asked in 1904, and despite a century of computation the minimum-energy arrangement is proved only for a handful of $N$ . Any configuration's energy can be computed in milliseconds, which makes record tables easy to check and optimality proofs the real challenge.

Problem

For points $z_1, \dots, z_N$ on the unit sphere $S^2 \subset \mathbb{R}^3$ , the Coulomb energy is

E(z_1, \dots, z_N) = \sum_{1 \le i < j \le N} \frac{1}{\|z_i - z_j\|}.

Let $C(N)$ be its minimum over all configurations of $N$ points. Determine $C(N)$ and the minimizing configurations: prove optimality for new values of $N$ , find configurations with lower energy than the best known, and pin down the asymptotic expansion of $C(N)$ .

Background

Thomson posed the problem in his 1904 paper on the structure of the atom. It is the classical case of energy minimization on spheres, related to spherical designs and to Smale's seventh problem, and numerical searches for good configurations have been run for decades. Proofs of optimality are hard because the space of configurations is continuous: Schwartz's proof for five points needed a long computer-assisted argument, and Cohn and Kumar settled twelve points through the universal optimality of the icosahedron.

What is known

Proved cases: $N = 2, 3$ (elementary); $N = 4$ and $6$ (Yudin, 1992); $N = 5$ (Schwartz, 2013, computer-assisted); $N = 12$ (Cohn and Kumar, 2007); $N = 8$ , the square antiprism (Kryvonos, Liehr and Taylor, September 2026, computer-assisted and verified in Lean).
$N = 7$ , claimed: a Lean 4 proof that the pentagonal bipyramid is the unique minimizer was announced in September 2026, with code and a write-up on GitHub. It is not peer reviewed, and its author notes that the formal statement has not been checked by a human.
Record tables: Ballinger, Blekherman, Cohn, Giansiracusa, Kelly and Schürmann (2009) tabulated putative optima up to 64 points, and later benchmarks go beyond 200 points. Georgiev, Gómez-Serrano, Tao and Wagner (2025) matched these records to about $10^{-8}$ for $N$ up to 300 without improving them.
Asymptotics: $C(N) = \tfrac12 N^2 + O(N^{3/2})$ (Wagner). Kuijlaars and Saff conjectured the coefficient of $N^{3/2}$ to be $3\left(\tfrac{\sqrt{3}}{8\pi}\right)^{1/2} \zeta(1/2)\, L_{-3}(1/2) \approx -0.5530$ .

What would count as progress

A proof of optimality for $N = 9$ , $10$ or $11$ , ideally machine-checked as for $N = 8$ ; the Lean kernel then checks it in minutes.
An independent review of the $N = 7$ proof, in particular of its formal statement.
A configuration with lower energy than the best published value for some $N$ . Its energy is a sum of $N(N-1)/2$ terms, checked in milliseconds; small improvements need care with floating-point error, so report coordinates and the energy with enough digits.
Any rigorous progress on the $N^{3/2}$ coefficient.

References

V. A. Yudin, "The minimum of potential energy of a system of point charges", Discrete Mathematics and Applications 3 (1993), 75–81 (Russian original 1992). doi:10.1515/dma.1993.3.1.75
H. Cohn and A. Kumar, "Universally optimal distribution of points on spheres", Journal of the AMS 20 (2007), 99–148. doi:10.1090/s0894-0347-06-00546-7
B. Ballinger, G. Blekherman, H. Cohn, N. Giansiracusa, E. Kelly and A. Schürmann, "Experimental study of energy-minimizing point configurations on spheres", Experimental Mathematics 18 (2009), 257–283. doi:10.1080/10586458.2009.10129052
R. E. Schwartz, "The five-electron case of Thomson's problem", Experimental Mathematics 22 (2013), 157–186. doi:10.1080/10586458.2013.766570
L. Kryvonos, L. Liehr and M. A. Taylor, "Energy minimization for eight points on the sphere" (2026). arXiv:2609.22077

Source

Imported from AlphaEvolve Repository of Problems, problem 33, curated by Google DeepMind. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.