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Abstract.Various sequences related to one introduced by Michael Somos are shown to be periodic mod m for every m , although the problem remains open for the original sequence. Some observations are made concerning the prime divisors and the rate of growth of certain sequences.

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Introduction 1.1. Consider the problem of arranging n points on the u n i t sphere so as to maximize the m i n i m u m distance between a n y two of the points. I t is immaterial whether we consider spherical distances or Euclidean distances (measured in the embedding three-dimensional space). The problem has been solved previously 1) only for n ~ 9 a n d n :: 12. For n = 3, 4, 6, 12, the extremal configuration consists of the vertices of a regular polyhedron with triangular faces. Several proofs of this fact have been given by F~J~S T6THis perhaps the simplest, a n d is the most closely related to the present paper. I t is m e n t i o n e d in t h a t paper t h a t the result was discovered i n d e p e n d e n t l y b y H. HADWIGER. FEJES T6TH also noticed t h a t the critical distance is the same for n = 5 as for n = 6. The cases n = 7, 8, 9 were solved b y SCHiiTTE a n d VA~ DER WAERDEN [8]. A simplified proof for n = 7 appears in VA~ DER WAERDEN [11], where it is also noted t h a t this case was solved i n d e p e n d e n t l y b y G. FRAY. I n addition, the case n = 13 is discussed (but n o t completely solved) in SCHtiTTE a n d VA~ D~R WAERDEN [9] and in LE~C~ [5]. Asymptotic results are discussed in HABICHT a n d VA~ DER WAERDEN [4] a n d in VA~ DER WAERDEN [11]. No knowledge of a n y of these papers is assumed. I n this paper, a proof of the conjecture of VA~ DE~ WA]~RD~ for the case n = 24 is given. (This result was a n n o u n c e d in [7]. W i t h regard to the conjecture, see [8], pages 97, 108, 123.) I n the optimal arrangement, the points

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