Squares in Polynomial Product Sequences
Number Theory
2011-07-12 v1
Abstract
Let F(n) be a polynomial of degree at least 2 with integer coefficients. We consider the products N_x=\prod_{1 \le n \le x} F(n) and show that N_x should only rarely be a perfect power. In particular, the number of x \le X for which N_x is a perfect power is O(X^c) for some explicit c<1. For certain F(n) we also prove that for only finitely many x will N_x be squarefull and, in the case of monic irreducible quadratic F(n), provide an explicit bound on the largest x for which N_x is squarefull.
Cite
@article{arxiv.1107.1730,
title = {Squares in Polynomial Product Sequences},
author = {Paul Spiegelhalter and Joseph Vandehey},
journal= {arXiv preprint arXiv:1107.1730},
year = {2011}
}