Integers Represented as a Sum of Primes and Powers of Two

TL;DR

The paper proves that large even integers can be expressed as the sum of two primes and a fixed number of powers of two, introducing a new technique to bound exponential sums without relying on explicit zero-free regions.

Contribution

It presents a novel method to bound exponential sums, reducing the number of powers of two needed under the Generalized Riemann Hypothesis.

Findings

Every sufficiently large even integer is a sum of two primes and 13 powers of two.

Under GRH, the number of powers of two can be reduced to 7.

The proof avoids explicit zero-free region calculations of Dirichlet L-functions.

Abstract

It is shown that every sufficiently large even integer is a sum of two primes and exactly 13 powers of 2. Under the Generalized Rieman Hypothesis one can replace 13 by 7. Unlike previous work on this problem, the proof avoids numerical calculations with explicit zero-free regions of Dirichlet L-functions. The argument uses a new technique to bound the measure of the set on which the exponential sum formed from powers of 2 is large.