Representation of even integers as a sum of squares of primes and powers of two

TL;DR

This paper improves a result related to representing large even numbers as sums of squares of primes and powers of two, reducing the maximum number of powers of two needed from 46 to 31.

Contribution

The paper presents a significant reduction in the number of powers of two required for such representations, advancing the understanding of additive number theory related to primes and powers of two.

Findings

Reduced the maximum number of powers of two from 46 to 31.

Confirmed the representation for sufficiently large even numbers.

Contributed to the ongoing effort to approach Goldbach-like problems.

Abstract

In 1951, Linnik proved the existence of a constant $K$ such that every sufficiently large even number is the sum of two primes and at most $K$ powers of 2. Since then, this style of approximation has been considered for problems similar to the Goldbach conjecture. One such problem is the representation of a sufficiently large even number as a sum of four squares of primes and at most $k$ powers of two. In 2014, Zhao proved this to be true with $k = 46$. In this paper, we reduce this to $k = 31$.