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SOME n−2 TERRACES FROM n POWER-SEQUENCES, n BEING AN ODD PRIME POWER

Published online by Cambridge University Press:  30 July 2009

IAN ANDERSON  and
D. A. PREECE
IAN ANDERSON
Affiliation:
Department of Mathematics, University of Glasgow, University Gardens, Glasgow G12 8QW, UK e-mail: [email protected]
D. A. PREECE
Affiliation:
Queen Mary University of London, School of Mathematical Sciences, Mile End Road, London E1 4NS, UK and Institute of Mathematics, Statistics and Actuarial Science, Cornwallis Building, University of Kent, Canterbury, Kent CT2 7NF, UK e-mail: [email protected]
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Abstract

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A terrace for m is an arrangement (a1, a2, . . . , am) of the m elements of m such that the sets of differences ai+1 − ai and ai − ai+1 (i = 1, 2, . . . , m − 1) between them contain each element of m \ {0} exactly twice. For m odd, many procedures are available for constructing power-sequence terraces for m; each such terrace may be partitioned into segments, one of which contains merely the zero element of m, whereas each other segment is either (a) a sequence of successive powers of an element of m or (b) such a sequence multiplied throughout by a constant. We now adapt this idea by using power-sequences in n, where n is an odd prime power, to obtain terraces for m, where m = n − 2. We write each element from n so that they lie in the interval [0, n − 1] and then delete 0 and n − 1 so that they leave n − 2 elements that may be interpreted as the elements of n−2. A segment of one of the new terraces may be of type (a) or (b), incorporating successive powers of 2, with each entry evaluated modulo n. Our constructions provide n−2 terraces for all odd primes n satisfying 0 < n < 1,000 except for n = 127, 241, 257, 337, 431, 601, 631, 673, 683, 911, 937 and 953. We also provide n−2 terraces for n = 3r (r > 1) and for some values n = p2, where p is prime.

Keywords

Primary 11A07secondary 05B30
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 52 , Issue 1 , January 2010 , pp. 65 - 85
Copyright
Copyright © Glasgow Mathematical Journal Trust 2009

References

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