Entropy—Sieve Methods and Energy Functionals in the Erdős Problem [Er79] on Quadratic Prime Representations

In 1979, Erdős asked whether every sufficiently large integer $n$ admits a representation \[ n = a p^2 + b, \qquad p \in \mathbb{P},\ a \geq 1,\ 0 \leq b < p. \] Classical sieve theory (Brun--Selberg, Barban--Davenport--Halberstam) shows that almost all $n$ have such a representation, but the finiteness of the exceptional set $\mathcal{E}$ has remained open. We develop a new \emph{entropy--sieve method} that blends upper-bound sieve techniques with information-theoretic invariants. At its core is a reduction from Kullback--Leibler divergence to a quadratic energy functional of residue distributions. This framework yields two main advances: \begin{itemize} \item Unconditionally, we obtain power-saving upper bounds for $|\mathcal{E}(x)|$ under the Uniformity Hypothesis (UH), improving on classical sieve exponents. \item Conditionally, we show that the Strong Uniformity Hypothesis (sUH) implies finiteness of $\mathcal{E}$, and further that sUH follows from either the Elliott--Halberstam conjecture or the Generalized Riemann Hypothesis. \end{itemize}Thus Erdős’s problem reduces to uniformity estimates for second moments of arithmetic progression errors, connecting it with deep conjectures in prime distribution. Finally, we provide numerical validation of the entropy--sieve method, illustrating experimentally that the KL divergence decays as predicted. The accompanying codebase (Zenodo, 2025) allows exploration up to $N\approx 10^{16}$, confirming the sharpness of our analytic reductions. This establishes a rigorous analytic and computational framework for Erdős’s problem, unifying sieve theory, entropy methods, and conjectural inputs from analytic number theory.