2013/12/11 by Qingzhong Ji, Ji, Qingzhong, Hourong Qin +1
Computer Science · Mathematics · Social Sciences · #11T55 #12Y05 #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #Political and Social Issues #math.NT #msc:11T55 #msc:12Y05
paper · pdf · doi:10.48550/arxiv.1312.3107
12 pages
openalex publication_date 2013/12/11 · arxiv created 2016/12/15 · arxiv updated 2016/12/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we consider the function field analogue of the Lehmer's totient problem. Let p(x)∈\mathbbFq[x] and φ(q,p(x)) be the Euler's totient function of p(x) over \mathbbFq[x], where \mathbbFq is a finite field with q elements. We prove that φ(q,p(x))|(q^\rm deg(p(x))-1) if and only if (i) p(x) is irreducible; or (ii) q=3, p(x) is the product of any 2 non-associate irreducibes of degree 1; or (iii) q=2, p(x) is the product of all irreducibles of degree 1, all irreducibles of degree 1 and 2, and the product of any 3 irreducibles one each of degree 1, 2 and 3.