On the Least Common Multiple of Polynomial Sequences at Prime Arguments

Abstract

Cilleruelo conjectured that if $f\in \mathbb{Z}[x]$ is an irreducible polynomial of degree $d\ge 2$ then, $\log \mathrm{lcm}\{f(n)\mid n<x\}\sim (d-1)x\log x.$ In this article, we investigate the analogue of prime arguments, namely, $\mathrm{lcm}\{f(p)\mid p<x\}$ where $p$ denotes a prime and obtain non-trivial lower bounds on it. Further, we also show some results regarding the greatest prime divisor of $f(p).$

10 pages, 1 table; to appear in Int. J. Number Theory