科研之友
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摘要
For any positive integer k and any set A of nonnegative integers, let r(1,k) (A, n) denote the number of solutions (a(1), a(2)) of the equation n = a(1) + k(a2) with a(1), a (2) is an element of A. Let k, l >= 2 be two distinct integers. We prove that there exists a set A subset of 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 >= n(0) 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 subset of N satisfying r(1,k) (A, n) = r(1,k) (N \ A, n) for all n >= n(0), we have r(1,k) (A, n) -> infinity as n -> infinity.
- 出版日期2016
- 单位华东师范大学
