A ternary additive problem

Abstract

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 $$n\le N$$ 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.