James Maynard [a] and Ze'ev Rudnick [b]
A lower bound on the least common multiple of polynomial sequences
Pages: 143-150
Received: 28 October 2019
Accepted in revised form: 19 March 2020
Mathematics Subject Classification (2010): 11N32.
Keywords: Prime factor, polynomial, Chebotarev density Theorem.
Authors address:
[a]: Mathematical Institute, Radcliffe observatory quarter,
Woodstock Road, Oxford OX2 6GG, England
[b]: Raymond and Beverly Sackler School of Mathematical Sciences,
Tel Aviv University, Tel Aviv 69978, Israel
This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement \({\rm n}^{o}\) 786758).
Abstract: For an irreducible polynomial \(f\in \mathbb{Z}[x]\) of degree \(d\geq 2\), Cilleruelo conjectured that the least common multiple of the values of the polynomial at the first \(N\) integers satisfies \(\log \operatorname{lcm} (f(1),\dots, f(N)) \sim (d-1) N\log N\) as \(N\to \infty\). This is only known for degree \(d=2\). We give a lower bound for all degrees \(d\geq 2\) which is consistent with the conjecture: \(\log \operatorname{lcm} (f(1),\dots, f(N)) \gg N\log N\).