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

Zeév Rudnick; Sa'ar Zehavi

doi:10.4171/rmi/1234

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

Zeév Rudnick, Sa'ar Zehavi

Tel Aviv University

Research output: Contribution to journal › Article › peer-review

Abstract

A conjecture due to Cilleruelo states that for an irreducible polynomial f with integer coefficients of degree d ≥ 2, the least common multiple L_f(N) of the sequence f(1), f(2),..., f(N) has asymptotic growth log L_f(N) ∼ (d − 1)N log N as N → ∞. 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.

Original language	English
Pages (from-to)	1441-1458
Number of pages	18
Journal	Revista Matematica Iberoamericana
Volume	37
Issue number	4
DOIs	https://doi.org/10.4171/rmi/1234
State	Published - 1 Jan 2021

Keywords

Irreducible polynomial
Least common multiple
Primes

All Science Journal Classification (ASJC) codes

General Mathematics

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10.4171/rmi/1234

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abstract = "A conjecture due to Cilleruelo states that for an irreducible polynomial f with integer coefficients of degree d ≥ 2, the least common multiple Lf(N) of the sequence f(1), f(2),..., f(N) has asymptotic growth log Lf(N) ∼ (d − 1)N log N as N → ∞. 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.",

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N2 - A conjecture due to Cilleruelo states that for an irreducible polynomial f with integer coefficients of degree d ≥ 2, the least common multiple Lf(N) of the sequence f(1), f(2),..., f(N) has asymptotic growth log Lf(N) ∼ (d − 1)N log N as N → ∞. 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.

AB - A conjecture due to Cilleruelo states that for an irreducible polynomial f with integer coefficients of degree d ≥ 2, the least common multiple Lf(N) of the sequence f(1), f(2),..., f(N) has asymptotic growth log Lf(N) ∼ (d − 1)N log N as N → ∞. 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.

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

Abstract

Keywords

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