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

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 $k\ge 2$.

arXiv commentThis paper has been withdrawn by the author due to several errors in the manuscript, a prominent problem being that it has been known at least since Tarski that in real numbers there exists a deterministic Turing machine which determines if a variety is empty or nonempty