SOMEn−2TERRACES FROMnPOWER-SEQUENCES,nBEING AN ODD PRIME POWER

Abstract

AbstractA terrace formis an arrangement (a1,a2, . . . ,am) of themelements ofmsuch that the sets of differencesai+1−aiandai−ai+1(i= 1, 2, . . . ,m− 1) between them contain each element ofm\ {0} exactly twice. Formodd, many procedures are available for constructing power-sequence terraces form; each such terrace may be partitioned into segments, one of which contains merely the zero element ofm, whereas each other segment is either (a) a sequence of successive powers of an element ofmor (b) such a sequence multiplied throughout by a constant. We now adapt this idea by using power-sequences inn, wherenis an odd prime power, to obtain terraces form, wherem=n− 2. We write each element fromnso that they lie in the interval [0,n− 1] and then delete 0 andn− 1 so that they leaven− 2 elements that may be interpreted as the elements ofn−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 modulon. Our constructions providen−2terraces for all odd primesnsatisfying 0 <n< 1,000 except forn= 127, 241, 257, 337, 431, 601, 631, 673, 683, 911, 937 and 953. We also providen−2terraces forn= 3r(r> 1) and for some valuesn=p2, wherepis prime.