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Three obvious necessary conditions for the existence of a k-cycle system of order n are that if n > 1 then n 1 k, n is odd, and 2 k divides n(n -1). We show that if these necessary conditions are sufficient for all n satisfying k I n < 3k then they are sufficient for all n. In particular, there exists a 15-cycle system of order n if and only if n = 1, 15, 21, or 25 (mod 301, and there exists a 21-cycle system of order n if and only if n = 1, 7, 15, or 21 (mod 421, n.# 7, 15.

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