For some p we define "rings" as circular arrangements of the numbers from 0 through p-1. By "circular" we mean that rotation of an arrangement does not change the ring it represents (e.g. 02341 and 34102 represent the same ring). Obviously, there are (p-1)! different rings. From each ring we draw an arrow towards the ring one obtains when each number is increased by 1, mod p. Because a succession of p such transformations transforms a ring into itself, the arrows form cycles the lengths of which are divisors of p.
Hence, if p is prime, 1 and p are the only possible cycle lengths. Because a cycle of length 1 corresponds to a ring with a constant difference mod p between each number and its clockwise neighbour and that difference may range from 1 through p-1, exactly p-1 rings occur in a cycle of length 1. Hence, the remaining (p-1)!-(p-1) rings occur in cycles of length p, i.e. for any prime p (p-1)!-(p-1) is a multiple of p.
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30 May 1980 prof.dr. Edsger W. Dijkstra Burroughs Research Fellow |
Last revised on Mon, 30 Jun 2003.