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 \[ n = a p^2 + b, p ∈ P,\ a ≥ 1,\ 0 ≤ b < p. \] Classical sieve theory (Brun--Selberg, Barban--Davenport--Halberstam) shows that almost all n have such a representation, but the finiteness of the exceptional set E has remained open. We develop a new 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: {itemize} Unconditionally, we obtain power-saving upper bounds for |E(x)| under the Uniformity Hypothesis (UH), improving on classical sieve exponents. Conditionally, we show that the Strong Uniformity Hypothesis (sUH) implies finiteness of E, and further that sUH follows from either the Elliott--Halberstam conjecture or the Generalized Riemann Hypothesis. {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≈ 10¹⁶, 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.