On a problem of Romanoff type

TL;DR

This paper proves that a positive lower density of natural numbers can be expressed as the sum of a prime and two powers of two with square exponents, solving a problem posed by Chen and Yang in 2014.

Contribution

It establishes the positive lower density of numbers representable in the form involving primes and powers of two with squared exponents, advancing understanding of additive number theory.

Findings

Positive lower density of such representable numbers

Resolution of Chen and Yang's 2014 problem

Extension of Romanoff-type results

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.