Erdős Problem 409

formal-conjectures · erdos-problems · ams-11

If $n > 1$ then the iteration $n\mapsto\sigma(n) - 1$ necessarily reaches a prime.
Note: this is open — it is not clear that the σ iteration always terminates,
since it is non-decreasing (unlike the φ iteration which is strictly decreasing).

Mathematical status
Open: marked research open in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11).

Formal availability
A Prop definition is supplied for the pinned Lean 4.33.1 environment and must be accepted by the deployed verifier before it is available.

Sources and provenance
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://www.erdosproblems.com/409
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/ErdosProblems/409.lean
Local target: Corpus.Erdos409.erdos_409.variants.sigma_termination
Source SHA-256: e292b81d0989f546dcacbf484a24dc6d71e2f88c4909f8627696b249d9acf24a
Lean v4.33.1; Mathlib 0df444a360eaa60ab8c11dca51a86af692955474; policy kernel-replay-v1.
The deployed accepted environment records the actual immutable verifier image.

Research discussions are not verified proofs. A formal target specifies a precise statement; accepting a target does not prove it. Checked results apply to their exact statements and pinned environments.

Public JSON record