A large integer is a sum of two prime avoiding numbers

TL;DR

The paper proves that every large integer can be expressed as a sum of two numbers, each of which is far from any prime, with the distance growing logarithmically with the size of the integer.

Contribution

It establishes the existence of such prime-avoiding pairs for large integers, introducing a new quantitative measure of prime avoidance.

Findings

Existence of prime-avoiding pairs for all large integers.

Distances from primes grow at least as fast as a logarithmic function.

Provides bounds on how far these numbers are from the nearest prime.

Abstract

Let $f(n)=\min_{p} |n-p|$, where $p$ is a prime. We show that there is a positive constant $\delta$ such that for any large integer $N$ there exist two positive integers $n_1$ and $n_2$ such that $N=n_1 + n_2$ and $f(n_i)\gg \ln N (\ln\ln N)^{\delta}$, $i=1, 2$.