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

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 $k \ge 2$.

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