Experimental Validation of the Entropy--Sieve Method for Erdos79 problem

we performed numerical experiments
implementing Algorithm~1 (the entropy--sieve algorithm) for integers $n \leq N$
with $N$ up to $10^6$.

The results confirm two key predictions of our framework:
\begin{enumerate}
  \item The KL divergence $D(P\|Q)$ between the empirical joint law of
  $(X_p)_{p\le \sqrt{n}}$ and the product distribution of their marginals
  decays towards $0$ as $N$ grows (see Figure~\ref{fig:kl-divergence}).
  \item The entropy--sieve bound for $\Pr(S(n)=0)$ dominates the empirical
  frequency of exceptions, but remains of the same logarithmic scale (see
  Figure~\ref{fig:bound-vs-empirical}), illustrating the predictive power of
  the method.
\end{enumerate}