On the integers of the form $p^2+b^2+2^n$ and $b_1^2+b_2^2+2^{n^2}$

TL;DR

This paper investigates the density and distribution of integers formed by sums of squares and powers of two, proving positive density for certain sumsets and identifying residue classes devoid of such integers.

Contribution

It establishes positive lower density for the sumset involving primes and squares plus powers of two, and constructs residue classes with no such integers, extending understanding of these additive structures.

Findings

Sumset {p^2+b^2+2^n} has positive lower density.

Existence of residue classes with no integers of the form p^2+b^2+2^n.

Similar results for sumset {b_1^2+b_2^2+2^{n^2}}.

Abstract

We prove that the sumset {p^2+b^2+2^n: p is prime and b,n\in N} has positive lower density. We also construct a residue class with odd modulo, which contains no integer of the form p^2+b^2+2^n. And similar results are established for the sumset {b_1^2+b_2^2+2^{n^2}: b_1,b_2,n\in N}.