An Optimization-Theoretic Framework for Brocard's Problem: Factorial–Square Energy Minimization and Deviation Growth Analysis

Brocard's Problem is a classical Diophantine equation in number theory that asks whether there exist positive integers n and m satisfying the equation n! + 1 = m^2. Currently only three solutions are known: (4,5), (5,11), and (7,71), and it remains an open question whether additional solutions exist. In this paper, we propose a new research perspective based on optimization theory to study this problem. The original Diophantine equation is reformulated as a discrete optimization problem by introducing a factorial–square deviation function defined as F(n,m) = | m^2 − ( n! + 1 ) |. Based on this formulation, we further construct a factorial square energy function E(n) and a normalized energy measure to analyze the approximation behavior between factorial values and perfect squares. By combining factorial growth analysis, discrete gradient dynamics, and potential function modeling, we investigate the structural properties of the deviation function as n increases. The analysis shows that factorial growth dominates the spacing structure of square numbers, leading to rapidly increasing deviation values for large n. Furthermore, we derive a deviation growth inequality indicating that the distance between n! + 1 and the nearest square number grows at least on the order of sqrt(n!) for sufficiently large n. This optimization-theoretic framework provides a new analytical perspective for studying factorial-based Diophantine equations and helps explain the extreme rarity of solutions to Brocard's equation. The proposed approach may also be extended to other nonlinear integer equations involving rapidly growing arithmetic functions.