A note on representation functions with different weights

• Autorzy: Zhenhua Qu
• Języki publikacji: EN

Abstrakty

EN

For any positive integer k and any set A of nonnegative integers, let $r_{1,k}(A,n)$ denote the number of solutions (a₁,a₂) of the equation n = a₁ + ka₂ with a₁,a₂ ∈ A. Let k,l ≥ 2 be two distinct integers. We prove that there exists a set A ⊆ ℕ such that both $r_{1,k}(A,n) = r_{1,k}(ℕ ∖ A,n)$ and $r_{1,l}(A,n) = r_{1,l}(ℕ ∖ A,n)$ hold for all n ≥ n₀ if and only if log k/log l = a/b for some odd positive integers a,b, disproving a conjecture of Yang. We also show that for any set A ⊆ ℕ satisfying $r_{1,k}(A,n) = r_{1,k}(ℕ ∖ A,n)$ for all n ≥ n₀, we have $r_{1,k}(A,n) → ∞$ as n → ∞.

Kategorie tematyczne

• 11B34: Representation functions
• 05A17: Partitions of integers

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2016
• Tom: 143
• Numer: 1
• Strony: 105-112

Daty

wydano

2016

Twórcy

autor

Zhenhua Qu

• Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, 500 Dongchuan Rd., Shanghai 200241, China

Identyfikatory

DOI

10.4064/cm6512-12-2015