Representing integers as sums of mixed powers of primes

TL;DR

This paper proves that all sufficiently large odd integers can be expressed as sums of mixed prime powers, extending classical Waring-Goldbach representations with new combinations involving primes raised to various powers.

Contribution

The paper introduces two new representations of large odd integers as sums of mixed prime powers, expanding the scope of Waring-Goldbach type results.

Findings

Every sufficiently large odd integer can be expressed as a sum involving primes raised to powers 2, 3, 5, 6, and 7.

New representations include sums with primes to the sixth and seventh powers.

The results extend classical additive prime number theory results.

Abstract

We establish two new Waring--Goldbach type representations: every sufficiently large odd integer $n$ can be expressed as \[ n = p_1^2 + p_2^2 + p_3^3 + p_4^3 + p_5^5 + p_6^6 + p_7^c, \] where each $p_i$ is prime and $c \in \{6,7\}$.