On a problem of Romanoff type

Abstract

Let $\mathcal{P}$ and $\mathbb{N}$ be the sets of all primes and natural numbers, respectively. In this article, it is proved that there has a positive lower density of the natural numbers which can be represented by the form $$p+2^{m_1^2}+2^{m_2^2},p\in \mathcal{P},m_1,m_2\in \mathbb{N}.$$ This solves a problem of Chen and Yang in 2014.