Prime factors of values of polynomials
Volume 122 / 2011
Colloquium Mathematicum 122 (2011), 135-138
MSC: Primary 11N32; Secondary 11R11, 11R27.
DOI: 10.4064/cm122-1-12
Abstract
We prove that for every quadratic binomial $f(x)=rx^2+s\in{\mathbb Z}[x]$ there are pairs $\langle a,b\rangle\in{\mathbb N}^2$ such that $a\ne b,$ $f(a)$ and $f(b)$ have the same prime factors and $\min\{a,b\}$ is arbitrarily large. We prove the same result for every monic quadratic trinomial over ${\mathbb Z}.$
