  Elliott and Halberstam proved that $\sum_{p<n} 2^{\omega(n-p)}$ is asymptotic
to $\phi(n)$. In analogy to the Erd\H{o}s--Kac Theorem, Elliott conjectured
that if one restricts the summation to primes $p$ such that $\omega(n-p)\le 2
\log \log n+\lambda(2\log \log n)^{1/2}$ then the sum will be asymptotic to
$\phi(n)\int_{-\infty}^{\lambda} e^{-t^2/2}dt/\sqrt{2\pi}$. We show that this
conjecture follows from the Bombieri--Vinogradov Theorem. We further prove a
related result involving Poisson--Dirichlet distribution, employing deeper
lying level of distribution results of the primes.
