Representing integers as sums of mixed powers of primes

TL;DR

The paper addresses representing sufficiently large odd integers as sums of mixed prime powers with fixed exponents and proves that for $c \in \{6,7\}$ every such $n$ admits $n = p_1^2 + p_2^2 + p_3^3 + p_4^3 + p_5^5 + p_6^6 + p_7^c$ with all $p_i$ prime. It uses a Hardy–Littlewood circle method with major/minor arc analysis, refined exponential-sum estimates over primes (via Davenport–Vaughan and Kumchev bounds), and a pruning argument to control minor-arc contributions. A positive singular series is established for odd $n$, and minor-arc bounds ensure the overall counting function is bounded below by $n^{\Theta_c}$, yielding the desired representations. This work extends Waring–Goldbach-type results to mixed prime powers and demonstrates a robust combination of analytic techniques in a prime-variable setting.

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 $c \in \{6,7\}$.

Theorems & Definitions (7)

• Theorem 1
• Lemma 2
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5