Resolution of Erdős Problem #728: a writeup of Aristotle's Lean proof

We provide a writeup of a resolution of Erdős Problem #728; this is the first Erdős problem (a problem proposed by Paul Erdős which has been collected in the Erdős Problems website) regarded as fully resolved autonomously by an AI system. The system in question is a combination of GPT-5.2 Pro by OpenAI and Aristotle by Harmonic, operated by Kevin Barreto. The final result of the system is a formal proof written in Lean, which we translate to informal mathematics in the present writeup for wider accessibility. The proved result is as follows. We show a logarithmic-gap phenomenon regarding factorial divisibility: For any constants $0