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arXiv · 1107.1730

Squares in Polynomial Product Sequences

Paul Spiegelhalter·Joseph Vandehey
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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.

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Paul Spiegelhalter, Joseph Vandehey. 2011-07-08. Squares in Polynomial Product Sequences. https://arxiv.org/abs/1107.1730

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