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

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