Carmichael's totient function conjecture

formal-conjectures · wikipedia · ams-11

For every positive natural number $n$, there exists a natural number $m$ with $m ≠ n$, such that
$φ(n) = φ(m)$ where $φ$ is the Euler totient function.

*Carmichael's totient function conjecture*: For every positive natural number $n$,
there exists a natural number $m$ with $m ≠ n$, such that $φ(n) = φ(m)$.

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://en.wikipedia.org/wiki/Carmichael%27s_totient_function_conjecture
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://arxiv.org/abs/1104.3264
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/Wikipedia/CarmichaelTotient.lean
Local target: Corpus.WikipediaCarmichaelTotient.charmichaelTotient
Source SHA-256: 33ba3638a67ae7a2db414c5ae3e1385722b7b22e54d3ad87df37846c168f1a3e
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