Polynomials and Smooth Numbers
by
Greg Martin
Institute for Advanced Study

Although we expect to find many smooth numbers (i.e., numbers with no large prime factors) among the values taken by a polynomial with integer coefficients, it is unclear what the asymptotic number of such smooth values should be; this is in contrast to the related problem of counting the number of prime values of a polynomial, for which Bateman and Horn proposed a conjectured asymptotic formula that is widely believed to be true. We discuss how to employ the Bateman-Horn conjecture to derive an asymptotic formula for the number of smooth values of a polynomial with the smoothness parameter in a non-trivial range. This conditional result provides a believable heuristic for the number of smooth integers among all values {F(n)} as well as among the values {F(p)} on prime arguments only.

Date received: April 6, 1998

Copyright © 1998 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 # caar-03.