On the nearest irreducible lacunary neighbour to an integer polynomial
Volume 162 / 2020
Colloquium Mathematicum 162 (2020), 121-134
MSC: Primary 11C08; Secondary 11R09, 12E05.
DOI: 10.4064/cm7978-8-2019
Published online: 15 April 2020
Abstract
There is an absolute constant $D_{0} \gt 0$ such that if $f(x)$ is an integer polynomial, then there is an integer $\lambda $ with $|\lambda | \le D_{0}$ such that $x^{n}+f(x)+\lambda $ is irreducible over the rationals for infinitely many integers $n\ge 1$. Furthermore, if $\deg f \le 25$, then there is a $\lambda $ with $\lambda \in \{-2,-1,0,1,2,3\}$ such that $x^{n}+f(x)+\lambda $ is irreducible over the rationals for infinitely many integers $n\ge 1$. These problems arise in connection with an irreducibility theorem of Andrzej Schinzel associated with coverings of integers and an irreducibility conjecture of Pál Turán.
