Some terraces from power-sequences, n being an odd prime

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• 作者：Ian Anderson ; D.A. Preece
• 关键词：2-Sequencings ; Number theory ; Power-sequence terraces ; Primitive roots
• 刊名：Discrete Mathematics
• 出版年：2008
• 出版时间：28 September 2008
• 年：2008
• 卷：308
• 期：18
• 页码：4086-4107
• 全文大小：237 K

文摘

A terrace for is an arrangement (a₁,a₂,…,a_m) of the m elements of such that the sets of differences b10f86f0a2f995eb886017647f"" title=""Click to view the MathML source"" alt=""Click to view the MathML source"">a_i+1-a_i and a_i-a_i+1 (i=1,2,…,m-1) between them contain each element of exactly twice. For m odd, many procedures are available for constructing power-sequence terraces for ; each terrace of this sort may be partitioned into segments one of which contains merely the zero element of , whereas each other segment is either (a) a sequence of successive powers of an element of or (b) such a sequence multiplied throughout by a constant. We now extend this idea by using power-sequences in , where n is an odd prime, to obtain terraces for where m=n+2. Our technique needs each of the n-1 elements from to be written so as to lie in the interval (0,n) and for three further elements 0, n and n+1 then to be introduced. A segment of one of the new terraces may contain just a single element from the set or it may be of type (a) or (b) with m=n and containing successive powers of 2, each evaluated modulo n. Also, a segment based on successive powers of 2 may be broken in one, two or three places by putting a different element from in each break. We provide terraces for all odd primes n satisfying 0<n<1000 except for 32b1475607"" title=""Click to view the MathML source"" alt=""Click to view the MathML source"">n=127,601,683.