article published yes Vivian Zieve Kuperberg author c3bac823-112d-11f0-a3f5-c264f852e697 We explore two questions about pseudo-polynomials, which are functions f : N → Z such that k divides f(n + k) − f(n) for all n, k. First, for certain arbitrarily sparse sets R, we construct pseudo-polynomials f with p|f(n) for some n only if p ∈ R. This implies that not all pseudo-polynomials satisfy an assumption of a recent paper of Kowalski and Soundararajan. We also consider α-primary pseudo-polynomials, where the pseudo-polynomial condition is only required for k lying in a set of primes of density α. We show that if an α-primary pseudo-polynomial is O(e(β−)n), where β = √7 3 − 1 6 ≈ 0.715, then it is a polynomial. Elsevier2022 eng Pseudo-polynomialsChinese remainder theoremRuzsa’s conjecture 0022-314X 2006.0252710.1016/j.jnt.2022.04.006 241531-541 yes 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> 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> 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. 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. V.Z. Kuperberg, Journal of Number Theory 241 (2022) 531–541. 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>. 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>. 221962026-06-29T12:58:07Z2026-07-14T11:08:14Z