Partitions of natural numbers with the intersection not empty (Q1952735)

Summary: Let \(\mathbb{N}\) be the set of nonnegative integers. For a given set \(A \subset \mathbb{N}\) the representation functions \(R_2(A, n)\), \(R_3(A, n)\) are defined as the number of solutions of the equation \(n = a+a'\), \(a,a' \in A\) with condition \(a < a'\), \(a \leq a'\), respectively. In this paper, we prove that if \(\mathbb{N} = A \cup B\) and \(A \cap B = \{8k : k \in \mathbb{N}\}\), then \(R_i(A, n) = R_i(B, n)\) cannot hold for all sufficiently large integers \(n\), where \(i = 2,3\).