IMO 2025 Problem 4: iteration of proper divisors
Follow the latest progress by email.
Starting from a positive integer a₁, repeatedly replace aₙ by the sum of its three largest proper divisors. A proper divisor is a positive divisor smaller than the number. Classify a₁ for which this process continues indefinitely and every term has at least three proper divisors.
Why it matters and what counts as progress
Why it matters: The problem asks which arithmetic starting values avoid eventual breakdown under a divisor-based iteration.
What counts as progress: A necessary-and-sufficient description of starting integers admitting indefinite iteration, with proof that no later term loses its three proper divisors.
How the problem developed
- 16 Jul 2025 · Contest problem [1]
IMO jury The problem was set on this day of the official IMO 2025 contest.
Google DeepMind and Gemini Deep Think DeepMind reported an officially graded five-of-six result (35/42); its published solution collection includes a proof for this problem.
AI contributions 2
Aristotle formalizes five IMO 2025 solutions
Harmonic reports that Aristotle generated machine-verified solutions for 2025 IMO Problems 1–5 from human-formalized problem statements, with the Lean proofs published.
Gemini Deep Think solves five IMO 2025 problems
An advanced Gemini Deep Think model produced natural-language solutions to five of the six 2025 International Mathematical Olympiad problems, earning 35 of 42 points under the official coordinators’ grading.
Sources
Website timeline
Publication dates and editorial changes.