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