On Cilleruelo's conjecture for the least common multiple of polynomial sequences

Zeév Rudnick; Sa'ar Zehavi

On Cilleruelo's conjecture for the least common multiple of polynomial sequences

Number Theory 2019-04-16 v2

Authors: Zeév Rudnick , Sa'ar Zehavi

Abstract

A conjecture due to Cilleruelo states that for an irreducible polynomial $f$ with integer coefficients of degree $d\geq 2$ , the least common multiple $L_f(N)$ of the sequence $f(1), f(2), \dots, f(N)$ has asymptotic growth $\log L_f(N)\sim (d-1)N\log N$ as $N\to \infty$ . We establish a version of this conjecture for almost all shifts of a fixed polynomial, the range of $N$ depending on the range of shifts.

Keywords

cilleruelo conjecture least common multiple polynomial sequences irreducible polynomial asymptotic growth

Cite

@article{arxiv.1902.01102,
  title  = {On Cilleruelo's conjecture for the least common multiple of polynomial sequences},
  author = {Zeév Rudnick and Sa'ar Zehavi},
  journal= {arXiv preprint arXiv:1902.01102},
  year   = {2019}
}

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On Cilleruelo's conjecture for the least common multiple of polynomial sequences

Abstract

Keywords

Cite

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