Polynomial analogue of the Smarandache function

JOURNAL OF NUMBER THEORY

卷:217

Article

Li, Xiumei¹  Sha, Min² 

[1] Qufu Normal Univ, Sch Math Sci, Qufu 273165, Shandong, Peoples R China

[2] Univ New South Wales, Sch Math & Stat, Sydney, NSW 2052, Australia

关键词: Erdos problem;    Factorial;    Smarandache function;    Polynomials over finite fields;   

DOI  :  10.1016/j.jnt.2020.05.015

来源: Elsevier

【 摘 要 】

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 F-q[t] over a finite field F-q and then define the Smarandache function of a non-zero polynomial f is an element of F-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 Erdos, which implies that for almost all polynomials f, S(f) = t(d), where d is the maximal degree of the irreducible factors of f . (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free