Nonlinear Sierpiński and Riesel numbers

In 1960, Sierpiński proved that there exist infinitely many odd positive integers k such that k⋅2n+1 is composite for all positive integers n. Such values of k are known as Sierpiński numbers. Extending the ideas of Sierpiński to a nonlinear situation, Chen showed that there exist infinitely many positive integers k such that kr⋅2n+d is composite for all positive integers n, where d∈{−1,1}, provided that r is a positive integer with . Filaseta, Finch and Kozek improved Chenʼs result by completely lifting the restrictions on r when d=1, and they asked if a similar result exists if kr is replaced by f(k), where f(x) is an arbitrary nonconstant polynomial in Z[x]. In this article, we address this question when f(x)=axr+bx+c∈Z[x]. In particular, we show, for various values of a, b, c, d and r, that there exist infinitely many positive integers k such that f(k)⋅2n+d is composite for all integers n⩾1. When d=1 or −1, we refer to such values of k as nonlinear Sierpiński or nonlinear Riesel numbers, respectively.