Daisy chains - a fruitful combinatorial concept
نویسنده
چکیده
“Daisy, Daisy, give me your answer, do!” [10] For any positive integer n, the units of Zn are those elements of Zn \ {0} that are coprime with n. The number of units in Zn is given by Euler’s totient function φ(n). If n is odd, a daisy chain for the units of Zn is obtained by arranging the units of Zn on a circle in some order [a1, a2, . . . , aφ(n)] such that the set of differences bi = ai+1 − ai (i = 1, 2, . . . , φ(n), with aφ(n)+1 = a1) is itself the set of units. Various constructions are given for daisy chains for odd values of n that have the prime-power decompositions p (i ≥ 1), pq (i ≥ 1, j ≥ 1) and pqr (where p, q and r are distinct odd primes). The paper’s emphasis is on values of n lying in the range 1 < n < 300, within which every primepower decomposition of an odd value n is of one of the types just given. The concept of fertile daisy chains is defined, and the link between daisy chains and terraces is briefly outlined.
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عنوان ژورنال:
- Australasian J. Combinatorics
دوره 41 شماره
صفحات -
تاریخ انتشار 2008