[{"arxiv":1,"month":"05","oa":1,"page":"531-541","volume":241,"status":"public","title":"On pseudo-polynomials divisible only by a sparse set of primes and α-primary pseudo-polynomials","quality_controlled":"1","date_updated":"2026-07-14T11:08:14Z","_id":"22196","scopus_import":"1","date_published":"2022-05-18T00:00:00Z","fulldoi":"https://doi.org/10.1016/j.jnt.2022.04.006","publication_identifier":{"issn":["0022-314X"]},"article_type":"original","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."}],"date_created":"2026-06-29T12:58:07Z","OA_place":"repository","OA_type":"green","author":[{"first_name":"Vivian Zieve","id":"c3bac823-112d-11f0-a3f5-c264f852e697","last_name":"Kuperberg","full_name":"Kuperberg, Vivian Zieve"}],"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2006.02527"}],"publication":"Journal of Number Theory","article_processing_charge":"No","keyword":["Pseudo-polynomials","Chinese remainder theorem","Ruzsa’s conjecture"],"citation":{"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>.","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>.","short":"V.Z. Kuperberg, Journal of Number Theory 241 (2022) 531–541.","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>","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.","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>"},"publisher":"Elsevier","doi":"10.1016/j.jnt.2022.04.006","publication_status":"published","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","language":[{"iso":"eng"}],"year":"2022","type":"journal_article","day":"18","oa_version":"Preprint","intvolume":"       241","extern":"1","external_id":{"arxiv":["2006.02527"]}}]