A ternary additive problem

摘要

The problem of representing a large integer (n) in the form (n=m^2+x^3+y^5) has been studied by a number of authors in the past decades. In this paper, we restrict (m) to square-free integers, and (x, y) to primes, and show that there is such a representation for all (nle N) with at most (O(N^{1-frac{1}{45}+varepsilon })) exceptions. We also improve the recent results of Liu (Acta Math Hungar 130(1–2):118–139, 2011) and Bauer (J Math 38(4):1073–1090, 2008) on related problems.