On partitions of ℤ_m with the same representation function

2020/07/01 by Cui-Fang Sun, Sun, Cui-Fang, Meng-Chi Xiong +1
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.2007.00414

9 pages

arxiv created 2020/07/01 · arxiv updated 2020/07/02

Abstract

For any positive integer m, let ℤ_m be the set of residue classes modulo m. For A⊆ ℤ_m and n∈ ℤ_m, let R_A(n) denote the number of solutions of n=a+a' with unordered pairs (a, a')∈ A × A. In this paper, we prove that if m=2^α with α≠ 2, A∪ B=ℤ_m and |A∩ B|=2, then R_A(n)=R_A(n) for all n∈ ℤ_m if and only if B=A+(m)/(2).

Related

• On a problem of partitions of ℤ_m with the same representation functions
• Additive and subtractive bases of ℤ_m in average
• Partitions of nonnegative integers with identical representation functions
• Products in Residue Classes
• Partitions of finite nonnegative integer sets with identical representation functions