A Sharper Explicit Result for the Sum of Two Almost Primes

Abstract

We prove that for every integer $N\geq2$, there exist positive integers $a$ and $b$ such that $N=a+b$ and $Ω(ab)\leq33$, where $Ω(n)$ denotes the number of prime factors of $n$, counted with multiplicity. This improves the previous bound of $40$ obtained by Dudek and Dunn. The proof applies the explicit Friedlander--Iwaniec $Λ^-Λ^2$ lower-bound sieve to a sequence derived from the products $n(N-n)$. The main new ingredient is pre-sieving at the prime $3$, which eliminates the extremal small-prime case in the dimension condition while keeping the resulting remainder terms under explicit control. We complete the proof using analytic estimates for large $N$, finite verification over an intermediate range, and explicit prime-gap data for small $N$.