A Steiner triple system (briefly STS) is a pair where S is a finite set and is a collection of 3-subsets of S (called triples) such that every pair of distinct elements of S belongs to exactly one triple of . The number |S| is called the order of . It is well-known that there is an STS of order if and only if or 3 (mod 6). Therefore in saying that a certain property concerning STS is true for all it is understood that or 3 (mod 6). An STS of order v will sometimes be denoted by .
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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