On the Least Common Multiple of Polynomial Sequences at Prime Arguments

Ayan Nath; Abhishek Jha

doi:10.1142/S1793042122500622

On the Least Common Multiple of Polynomial Sequences at Prime Arguments

Number Theory 2022-02-01 v4

Authors: Ayan Nath , Abhishek Jha

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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 $f(p).$

Keywords

least common multiple polynomial sequences prime arguments irreducible polynomial greatest prime divisor

Cite

@article{arxiv.2106.06782,
  title  = {On the Least Common Multiple of Polynomial Sequences at Prime Arguments},
  author = {Ayan Nath and Abhishek Jha},
  journal= {arXiv preprint arXiv:2106.06782},
  year   = {2022}
}

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

On the Least Common Multiple of Polynomial Sequences at Prime Arguments

Abstract

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