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, $\log \operatorname{lcm} {f(n)\mid n<x} \sim (d-1)x\log x.$ In this article, we investigate the analogue of prime arguments, namely, $\operatorname{lcm} {f(p)\mid p<x}$ 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 $f(p).$