Smooth Values of Polynomials

Fretwell, Dan and Wooley, Trevor and Bober, Jonathan and Martin, Greg (2020) Smooth Values of Polynomials. Journal of the Australian Mathematical Society, 108 (2). pp. 245-261. ISSN 1446-7887

Abstract

Given f Z[t] of positive degree, we investigate the existence of auxiliary polynomials g Z[t] for which factors as a product of polynomials of small relative degree. One consequence of this work shows that for any quadratic polynomial f Z[t] and any ϵ > 0, there are infinitely many for which the largest prime factor of f(n) is no larger than n.