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Sequences of Reducible {0,1}-Polynomials Modulo a Prime
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Judith Canner

Department of Statistics

North Carolina State University

Raleigh, North Carolina 27695-8203

USA

Lenny Jones and Joseph Purdom

Department of Mathematics

Shippensburg University

Shippensburg, Pennsylvania 17257

USA

**Abstract:**
Construct a recursive sequence of polynomials, staring with 1, in the
following way. Each new term in the sequence is determined by adding
the smallest power of *x* larger than the degree of the previous term,
such that the new polynomial is reducible over the rationals.
Filaseta, Finch and Nicol have shown that this sequence is finite. In
this paper we investigate variations of this problem over a finite
field. In particular, we allow the starting polynomial to be any
{0,1}-polynomial with nonzero constant term, and we
allow the exponent on the power of *x* added at each step to be chosen
from the set of multiples of a fixed positive integer *k*. Among our
results, we show that these sequences are always infinite. We develop
necessary and sufficient conditions on *k* and the characteristic *p*
of the field, so that the sequence starting with 1 uses every multiple
of *k* as an exponent in its construction. In addition, we prove for
*k*=1 and *p* >=5 that there exists a {0,1}-polynomial
*f* such that the sequence starting with *f* uses every positive
integer larger than the degree of *f* as an exponent in its
construction.

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Received December 19 2005;
revised version received July 19 2006.
Published in *Journal of Integer Sequences* July 19 2006.

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