An Explicit Result for the Sum of Two Almost Primes

TL;DR

This paper proves that every integer greater than or equal to 2 can be expressed as the sum of two positive integers with a restricted number of prime factors, using explicit sieve methods and computational optimization.

Contribution

It provides an explicit bound on the prime factorization complexity of sums of two almost primes for all integers above 2.

Findings

Every N ≥ 2 can be written as a + b with Ω(ab) ≤ 40.

Uses explicit lower bound Selberg sieve for the proof.

Combines computational methods with theoretical sieve techniques.

Abstract

We show that every $N \geq 2$ can be written as the sum of positive integers $a$ and $b$ where $\Omega(ab) \leq 40$. The result is obtained through the direct application of an explicit lower bound Selberg sieve along with some computation and optimisation.