2020/06/30 by Cui-Fang Sun, Sun, Cui-Fang, Meng-Chi Xiong +1
Mathematics · #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.2006.16513
8 pages
arxiv created 2020/06/30 · openalex publication_date 2020/06/30 · arxiv updated 2020/07/01 · openalex created_date 2020/07/10 · openalex updated_date 2026/07/28
For any positive integer m, let ℤm be the set of residue classes modulo m. For A⊆ ℤm and n∈ ℤm, let representation function RA(n) denote the number of solutions of the equation n=a+a' with ordered pairs (a, a')∈ A × A. In this paper, we determine all sets A, B⊆ ℤm with A∪ B=ℤm and |A∩ B|=2 or m-2 such that RA(n)=RB(n) for all n∈ ℤm. We also prove that if m is a positive integer with 4|m, then there exist two distinct sets A, B⊆ ℤm with A∪ B=ℤm and |A∩ B|=4 or m-4, B≠ A+(m)/(2) such that RA(n)=RB(n) for all n∈ ℤm. If m is a positive integer with 2‖m, A∪ B=ℤm and |A∩ B|=4 or m-4, then RA(n)=RB(n) for all n∈ ℤm if and only if B=A+(m)/(2).