For five equal masses equally spaced in angle around the center of mass, the regular pentagon is claimed to be the only central configuration, but 15 of 16 proof regions are deferred to a missing supplement.
A simpler proof of regular polygon solutions of the N-body problem
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Uniqueness of star central configurations in the $5$-body problem
For five equal masses equally spaced in angle around the center of mass, the regular pentagon is claimed to be the only central configuration, but 15 of 16 proof regions are deferred to a missing supplement.