On integers as the sum of a prime and a $k$-th power

TL;DR

This paper explores the representation of integers as the sum of a prime and a k-th power, aiming to prove a longstanding conjecture for all k ≥ 2 using Diophantine equations.

Contribution

It introduces a new approach based on Diophantine equations to address the Hardy-Littlewood conjecture for all k ≥ 2.

Findings

Provides an alternative method towards proving the conjecture

Suggests the conjecture holds for all k ≥ 2

Offers insights into the structure of sums involving primes and k-th powers

Abstract

Let $\mathcal{R}_k(n)$ be the number of representations of an integer $n$ as the sum of a prime and a $k$-th power. Define E_k(X) := |\{n \le X, n \in I_k, n\text{not a sum of a prime and a $k$-th power}\}|. Hardy and Littlewood conjectured that for $k = 2$ and $k=3$, E_k(X) \ll_{k} 1. In this note we present an alternative approach grounded in the theory of Diophantine equations towards a proof of the conjecture for all $k \ge 2$.