---
OA_place: repository
OA_type: green
_id: '22196'
abstract:
- lang: eng
  text: "We explore two questions about pseudo-polynomials, which\r\nare functions
    f : N → Z such that k divides f(n + k) −\r\nf(n) for all n, k. First, for certain
    arbitrarily sparse sets R, we\r\nconstruct pseudo-polynomials f with p|f(n) for
    some n only if\r\np ∈ R. This implies that not all pseudo-polynomials satisfy
    an\r\nassumption of a recent paper of Kowalski and Soundararajan.\r\nWe also consider
    α-primary pseudo-polynomials, where the\r\npseudo-polynomial condition is only
    required for k lying in\r\na set of primes of density α. We show that if an α-primary\r\npseudo-polynomial
    is O(e(β−)n), where β = √7\r\n3 − 1\r\n6 ≈ 0.715,\r\nthen it is a polynomial."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Vivian Zieve
  full_name: Kuperberg, Vivian Zieve
  id: c3bac823-112d-11f0-a3f5-c264f852e697
  last_name: Kuperberg
citation:
  ama: Kuperberg VZ. On pseudo-polynomials divisible only by a sparse set of primes
    and α-primary pseudo-polynomials. <i>Journal of Number Theory</i>. 2022;241:531-541.
    doi:<a href="https://doi.org/10.1016/j.jnt.2022.04.006">10.1016/j.jnt.2022.04.006</a>
  apa: Kuperberg, V. Z. (2022). On pseudo-polynomials divisible only by a sparse set
    of primes and α-primary pseudo-polynomials. <i>Journal of Number Theory</i>. Elsevier.
    <a href="https://doi.org/10.1016/j.jnt.2022.04.006">https://doi.org/10.1016/j.jnt.2022.04.006</a>
  chicago: Kuperberg, Vivian Zieve. “On Pseudo-Polynomials Divisible Only by a Sparse
    Set of Primes and α-Primary Pseudo-Polynomials.” <i>Journal of Number Theory</i>.
    Elsevier, 2022. <a href="https://doi.org/10.1016/j.jnt.2022.04.006">https://doi.org/10.1016/j.jnt.2022.04.006</a>.
  ieee: V. Z. Kuperberg, “On pseudo-polynomials divisible only by a sparse set of
    primes and α-primary pseudo-polynomials,” <i>Journal of Number Theory</i>, vol.
    241. Elsevier, pp. 531–541, 2022.
  ista: Kuperberg VZ. 2022. On pseudo-polynomials divisible only by a sparse set of
    primes and α-primary pseudo-polynomials. Journal of Number Theory. 241, 531–541.
  mla: Kuperberg, Vivian Zieve. “On Pseudo-Polynomials Divisible Only by a Sparse
    Set of Primes and α-Primary Pseudo-Polynomials.” <i>Journal of Number Theory</i>,
    vol. 241, Elsevier, 2022, pp. 531–41, doi:<a href="https://doi.org/10.1016/j.jnt.2022.04.006">10.1016/j.jnt.2022.04.006</a>.
  short: V.Z. Kuperberg, Journal of Number Theory 241 (2022) 531–541.
date_created: 2026-06-29T12:58:07Z
date_published: 2022-05-18T00:00:00Z
date_updated: 2026-07-14T11:08:14Z
day: '18'
doi: 10.1016/j.jnt.2022.04.006
extern: '1'
external_id:
  arxiv:
  - '2006.02527'
intvolume: '       241'
keyword:
- Pseudo-polynomials
- Chinese remainder theorem
- Ruzsa’s conjecture
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2006.02527
month: '05'
oa: 1
oa_version: Preprint
page: 531-541
publication: Journal of Number Theory
publication_identifier:
  issn:
  - 0022-314X
publication_status: published
publisher: Elsevier
quality_controlled: '1'
scopus_import: '1'
status: public
title: On pseudo-polynomials divisible only by a sparse set of primes and α-primary
  pseudo-polynomials
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 241
year: '2022'
...