AI 中文总结
该数论研究将两个殆素数之和对应的素因子计重数和的上界从40改进至33，通过应用显式筛法并引入素数3的预筛等技术完成证明。
AI 中文摘要
我们证明，对于每个整数$N\geq2$，都存在正整数$a$和$b$使得$N=a+b$，且$\Omega(ab)\leq33$，其中$\Omega(n)$表示$n$的素因子个数（计重数）。这改进了Dudek和Dunn得到的此前界值40。该证明将显式的Friedlander–Iwaniec $\Lambda^-\Lambda^2$下界筛法应用于由乘积$n(N-n)$导出的序列。主要新要素是对素数3进行预筛，这在保持所得余项受显式控制的同时，消除了维数条件中的极端小素数情形。我们通过大$N$的解析估计、中间范围的有限验证，以及小$N$的显式素数间隙数据完成了证明。
英文摘要
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 $N$.
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