On integers of the form $$2^{g(j_1)}+2^{g(j_2)}+p$$

Abstract

Let \(k\ge 3\) be a positive integer and let \(g(x)=a_{k}x^{k}+a_{k-1}x^{k-1}+\cdots +a_0\in \mathbb {Z}[x]\) with \(\gcd (a_{0}, \ldots , a_{k-1},a_{k})=1, a_{k}>0\). In this paper, we investigate the density of natural numbers which can be represented by the form \(2^{g(j_1)}+2^{g(j_2)}+p\), where \(j_1,j_2\) are positive integers and p is an odd prime.