PreprintCC BY-NC-ND 4.0arXiv
On the probability that the integer parts of polynomials are coprime
Abstract
We prove that for any s≥ 2 and polynomials f1(x),…,fs−1(x)∈ ℝ[x] satisfying a certain equidistribution condition, the natural density of the numbers n∈ ℕ such that n,⌊ f1(n)⌋,…,⌊ fs−1(n)⌋ are mutually coprime is 1/ζ(s). This generalises a recent result of Champagne, in which this was proven under the assumption that f1 is linear.
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BibTeX
@misc{holdridge2026probability,
title = {{On the probability that the integer parts of polynomials are coprime}},
author = {Philippa Holdridge},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02223},
eprint = {2611.02223},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02223},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - On the probability that the integer parts of polynomials are coprime
AU - Philippa Holdridge
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.02223
UR - https://arxiv.org/abs/2611.02223
AB - We prove that for any s≥ 2 and polynomials f_1(x),…,f_s−1(x)∈ ℝ[x] satisfying a certain equidistribution condition, the natural density of the numbers n∈ ℕ such that n,⌊ f_1(n)⌋,…,⌊ f_s−1(n)⌋ are mutually coprime is 1/ζ(s). This generalises a recent result of Champagne, in which this was proven under the assumption that f_1 is linear.
ER -