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Oct 3 at 18:10 comment added Per Alexandersson I still have flashbacks about this problem - Boris Shapiro gave me this problem to work on, first year as an undergraduate student. I did not manage to get far :)
Sep 15 at 13:10 comment added Alexandre Eremenko @Mark Lewko: more precisely, they did not prove finiteness, but assuming finiteness, they proved un upper bound. As far as I know, the problem is unsolved even for 3 charges. I've a computer assisted for 3 equal charges, but did not verify that it is correct, and it is not published.
Sep 14 at 16:58 comment added Mark Lewko It seems (from the paper of Gabrielov, Novikov and Shapiro: arxiv.org/pdf/math-ph/0409009) that Maxwell asserted without proof an upper bound (in fact $(n-1)^2$) on the number of equilibrium points in his treatise, rather than posing it as a question/conjecture. That said, it is a good example particularly given that it is precisely formulated in the original source.
Sep 14 at 10:11 history answered fedja CC BY-SA 4.0