On coverings of the integers associated with an irreducibility theorem of A. Schinzel
by
Michael Filaseta
University of South Carolina

A well-known result of Sierpinski is that there is a positive integers k such that 2ⁿk + 1 is composite for every positive integer n. A polynomial analogue of this result is that there is a polynomial f(x) with positive integer coefficients such that f(x) xⁿ + 1 is reducible over the rationals for every positive integer n. However, whether or not this analogue is actually true remains an open problem. Schinzel established a connection between the existence of such an f(x) and coverings of the integers (a covering is a finite system of congruences x \equiv a_j mod m_j such that every integer satisfies at least one such congruence). In particular, Schinzel's result implies that if such an f(x) exists, then there is an odd covering of the integers (i.e., a covering consisting of distinct odd moduli > 1). Last year, at the AMS Regional Meeting in Illinois, the speaker discussed a new proof by K. Ford, S. Konyagin, and the speaker of an irreducibility result of Schinzel, a result which Schinzel used to establish the connection between the existence of such f(x) and coverings. This self-contained talk explains the rest of the story, how one uses irreducibility to connect the existence of such f(x) with coverings. Some related results will be discussed. In particular, we will consider the more general problem of determining a in Z for which there are f(x) with positive integer coefficients satisfying that f(x) xⁿ + a is reducible over the rationals for every positive integer n.

http://www.math.sc.edu/~filaseta/

Date received: March 13, 2000

Copyright © 2000 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 # cadx-89.