Short intervals asymptotic formulae for binary problems with prime powers

TL;DR

This paper establishes asymptotic formulas for the average number of representations of integers as sums of prime powers within short intervals, covering various exponents and fixed integer conditions.

Contribution

It provides new asymptotic results for binary problems involving prime powers with specific exponents, extending previous understanding of such representations.

Findings

Asymptotic formulas proven for sums of prime powers with exponents 2 and 3.

Results cover a range of exponents up to 11 for one of the primes.

The formulas hold in short intervals for the specified binary problems.

Abstract

We prove results about the asymptotic formulae in short intervals for the average number of representations of integers of the forms $n=p_{1}^{\ell_1}+p_{2}^{\ell_2}$, with $\ell_1, \ell_2\in\{2,3\}$, $\ell_1+\ell_2\le 5$ are fixed integers, and $n=p^{\ell_1} + m^{\ell_2}$, with $\ell_1=2$ and $2\le \ell_2\le 11$ or $\ell_1=3$ and $ \ell_2=2$ are fixed integers, $p,p_1,p_2$ are prime numbers and $m$ is an integer.