On the Least Common Multiple of Polynomial Sequences at Prime Arguments

TL;DR

This paper explores the behavior of the least common multiple of polynomial values at prime arguments, providing lower bounds and insights into the greatest prime divisors, extending conjectures on polynomial sequences.

Contribution

It introduces new lower bounds for the LCM of polynomial values at primes and investigates the prime divisors, advancing understanding of polynomial sequences at prime inputs.

Findings

Established non-trivial lower bounds for the LCM at prime arguments.

Provided results on the greatest prime divisor of polynomial values at primes.

Extended conjectures to the context of prime inputs for polynomial sequences.

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 $f(p).$