On the generalization of Golomb's conjecture

2020/05/24 by Chaohua Jia, Jia, Chaohua
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2005.11752

openalex publication_date 2020/05/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let p be a sufficiently large prime number, r be any given positive integer. Suppose that a₁, …, a_r are pairwise distinct and not zero modulo p. Let N(a₁, …, a_r; p) denote the number of α₁, …, α_r, β, which are primitive roots modulo p, such that α₁+β≡ a₁, …, α_r+β≡ a_r (\rm mod p). In the first version of this paper, we proved an asymptotic formula for N(a₁, …, a_r; p) so that we could answer an open problem of Wenpeng Zhang and Tingting Wang. But we found that our result had been included in a paper of L. Carlitz in 1956, which is explained in the additional remark below.

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