The classical Brocard-Ramanujan problem, n!+1=k^2, is a long-standing open problem in number theory. This work explores the generalized Diophantine equation ₈=₀^a (n+i) !+1=k^2. The paper reports the discovery of a remarkable new solution, (n, k, a) = (4, 215, 4), found through systematic computational search. This script can be used to reproduce the theoretical magnitude curve plot (Figure 1) presented in the paper, which visualizes all seven known solutions. Key Contributions Discovery of an Exceptional Solution: The primary contribution is the discovery of a new, remarkable integer solution to the generalized equation: (n, k, a) = (4, 215, 4). This is the only known solution where a>1. Generalization of the Problem: The work extends the classic Brocard-Ramanujan problem (n!+1=k^2) to a more general family of Diophantine equations, ₈=₀^a (n+i) !+1=k^2. Extensive Computational Search: The paper reports on a systematic computational search conducted for n 30, 000 with a 100. This search validated the extreme rarity of solutions and found a total of seven solutions within these bounds. Development of a Theoretical Framework: A theoretical framework was developed to explain the scarcity of solutions, using prime factorization constraints, modular arguments, and probabilistic heuristics. Formulation of a New Conjecture: Based on extensive computational evidence and theoretical analysis, the paper proposes the conjecture that the seven identified solutions are the only positive integer solutions to the generalized equation. Included Files `generalized-brocard-ramanujan-problem-venkat-2025ᵥ2. pdf`: The full academic paper (Version 2). It presents the seven known solutions, the exceptional solution (4, 215, 4), and the theoretical framework. `search. py`: The main Python script used for the computational search. It validates solutions up to n=30, 000 and automatically generates the magnitude curve visualization. `solutions. txt`: The structured dataset output containing all discovered solutions and their metadata. `consoleₒutputₛearch. txt`: The raw terminal logs from the n=30, 000 run, serving as verification of the search performance and results. `magnitudecurve. pdf`: The visualization figure from the paper comparing known solutions against the theoretical growth curve. `requirements. txt`: List of Python dependencies required to run the code (`gmpy2`, `numpy`, `matplotlib`). Licensing This project uses a dual-license model: Source Code (. py file): MIT License All Other Files (Paper, documentation, results): Creative Commons Attribution 4. 0 International (CC BY 4. 0)
Arvind Naladiga Venkat (Tue,) studied this question.