Polynomial analogue of the Smarandache function

2019/06/03 by Xiumei Li, Min Sha, Li, Xiumei +1
Mathematics · #Advanced Mathematical Identities #Advanced Mathematical Theories #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.1906.00510

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openalex publication_date 2019/06/03 · arxiv created 2020/07/11 · arxiv updated 2020/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In the integer case, the Smarandache function of a positive integer n is defined to be the smallest positive integer k such that n divides the factorial k!. In this paper, we first define a natural order for polynomials in \mathbbF_q[t] over a finite field \mathbbF_q and then define the Smarandache function of a non-zero polynomial f ∈ \mathbbF_q[t], denoted by S(f), to be the smallest polynomial g such that f divides the Carlitz factorial of g. In particular, we establish an analogue of a problem of Erd\H os, which implies that for almost all polynomials f, S(f)=t^d, where d is the maximal degree of the irreducible factors of f.

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