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South East Regional Meeting On Numbers 2005
April 15-17, 2005
University of South Carolina; Department of Mathematics
Columbia, SC 29208, USA

Organizers
Michael Filaseta, Robert Murphy, Ognian Trifonov, and Gang Yu

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Sequences of Reducible { 0, 1 }-Polynomials Modulo a Prime
by
Lenny Jones
Shippensburg University
Coauthors: Judith Canner (Shippensburg University), Joe Purdom (Shippensburg University)

Abstract

Let p be a prime, let k be a positive integer and let f:=f(x) be a {0, 1}-polynomial with f(0) not divisible by p. Define a sequence, denoted (f, k, p), of {0, 1}-polynomials by: f1:=f and fi:=fi-1+xkn, for i larger than 1, where kn is the smallest multiple of k larger than the degree of fi-1, such that fi-1+xkn is reducible modulo p. Let M denote the set of positive integer multiples of k larger than the degree of f that are not degrees of terms in (f, k, p). We say (f, k, p) has the root pattern [r1, r2, ... , rt] modulo p if there exists a positive integer N such that fN+st+i(ri) is divisible by p for each i=1, 2, ... , t, and all nonnegative integers s. We investigate conditions on f, k and p which determine whether M is empty or finite, and which guarantee that (f, k, p) has a root pattern.

Date received: February 15, 2005

Copyright © 2005 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caqa-06.