On sums of powers of almost equal primes

TL;DR

This paper proves that large integers satisfying certain conditions can be expressed as sums of powers of primes that are almost equal, with improved bounds on the primes' proximity compared to previous results.

Contribution

It extends earlier results by establishing new bounds on the closeness of primes in sums of prime powers, under specific local and size conditions.

Findings

Expressibility of large integers as sums of prime powers with almost equal primes

Improved bounds on the proximity of primes compared to previous work

Conditions on the number of summands and local constraints

Abstract

Let $k \ge 2$ and $s$ be positive integers, and let $n$ be a large positive integer subject to certain local conditions. We prove that if $s \ge k^2+k+1$ and $\theta > 31/40$, then $n$ can be expressed as a sum $p_1^k + \dots + p_s^k$, where $p_1, \dots, p_s$ are primes with $|p_j - (n/s)^{1/k}| \le n^{\theta/k}$. This improves on earlier work by Wei and Wooley and by Huang who proved similar theorems when $\theta > 19/24$.