Erdős Problem 418

formal-conjectures · erdos-problems · ams-11

The **Odd Noncototient Conjecture**: every non-cototient is even. Equivalently, every odd natural
number is of the form $n - \phi(n)$ for some $n$.

This is the unconditional form of erdos_418.variants.conditional, which derives the odd case from a
strengthening of the Goldbach conjecture. See [Wikipedia: Noncototient].

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/418
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://en.wikipedia.org/wiki/Noncototient
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/ErdosProblems/418.lean
Local target: Corpus.Erdos418.erdos_418.variants.odd_noncototient
Source SHA-256: 046766767810dbcb600b52f2e124562ad31024632643f174078fd7e6ae2c36fe
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