Prime avoiding numbers is a basis of order $2$

TL;DR

This paper proves that large integers can be expressed as sums of two numbers each far from the nearest prime, with the distance growing faster than the trivial logarithmic bound, advancing understanding of prime gaps.

Contribution

It establishes a new lower bound on the distance from integers to primes in representations of large numbers, improving previous results by a significant logarithmic factor.

Findings

Every large integer N can be written as N=n_1+n_2 with F(n_i) ≥ (log N)(log log N)^{1/325565}

Improves upon the trivial bound F(n_i) ≫ log N for prime gaps in such representations

Demonstrates that numbers can be expressed as sums of numbers avoiding primes by larger-than-expected gaps.

Abstract

For a positive integer $n$, we denote by $F(n)$ the distance from $n$ to the nearest prime number. We prove that every sufficiently large positive integer $N$ can be represented as the sum $N=n_1+n_2$, where $$ F(n_i) \geqslant (\log N)(\log\log N)^{1/325565}, $$ for $i=1,2$. This improves the corresponding "trivial" statement where only $F(n_i)\gg \log N$ is required.