Computational Approaches to the Brocard-Ramanujan Equation n!+1=m²: Algorithms, Complexity, and Experimental Evidence

We survey and extend computational approaches to the Brocard--Ramanujan Diophantine equation n!+1=m^2, one of the oldest open problems linking factorials and perfect squares. After fixing notation and recalling the three known solutions---the Brown numbers (n, m) = (4, 5), (5, 11), (7, 71) ---we give a self-contained proof of Overholt's theorem that the abc conjecture implies finiteness of the solution set, and we quantify the probabilistic heuristic predicting that no further solutions exist. Our principal focus is algorithmic. We analyze the naive ``factorial-and-square-root'' search, establish its essentially quadratic running time, and contrast it with the quadratic-residue (QR) sieve of Berndt and Galway, in which a candidate exponent n is eliminated by exhibiting a prime p with n!+1p=-1. We prove that, under standard equidistribution heuristics, the sieve eliminates a non-solution after an expected two Legendre-symbol evaluations and attains near-linear total complexity---the decisive gain that has enabled searches to n<10^15. We then report original experiments: exact verification of the known solutions; validation of the sieve against brute force for n 10^4; confirmation that the number of prime tests a value passes is distributed as Bin (k, 12) to within 2. 3 10^-3; a screened search to n=10^6; and measured speedups exceeding 440 over the naive method. We close by situating the equation within the wider landscape of polynomial--factorial Diophantine problems and distributed computational number theory.