1 results
SMOOTH VALUES OF POLYNOMIALS
- Part of
-
- Journal:
- Journal of the Australian Mathematical Society / Volume 108 / Issue 2 / April 2020
- Published online by Cambridge University Press:
- 01 February 2019, pp. 245-261
- Print publication:
- April 2020
-
- Article
- Export citation
-
Given
$f\in \mathbb{Z}[t]$ of positive degree, we investigate the existence of auxiliary polynomials 
$g\in \mathbb{Z}[t]$ for which 
$f(g(t))$ factors as a product of polynomials of small relative degree. One consequence of this work shows that for any quadratic polynomial 
$f\in \mathbb{Z}[t]$ and any 
$\unicode[STIX]{x1D700}>0$, there are infinitely many 
$n\in \mathbb{N}$ for which the largest prime factor of 
$f(n)$ is no larger than 
$n^{\unicode[STIX]{x1D700}}$.
