Least $m$ such that $\phi(m) = n!$

formal-conjectures · oeis · ams-11

The smallest positive integer $m$ whose Euler totient equals $n!$.

Conjecture: unless $n! + 1$ is prime (i.e., $n \in \text{A002981}$), $a(n) = p q$ where $p$ is the
least prime $> \sqrt{n!}$ such that $(p - 1) \mid n!$ and $q = \frac{n!}{p - 1} + 1$ is prime.
- M. F. Hasler, Oct 04 2009

We assume $a(n) \ne 0$ and $(p(n)).\text{Prime}$ to ensure the sInf searches are non-empty
and do not collapse to $0 = 0$.

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://oeis.org/A055487
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/OEIS/55487.lean
Local target: Corpus.OEIS55487.conjecture
Source SHA-256: 5b9a26205c7a71124d6994d53e6c7da0539c26ebabf4b7a4479eeb696c219b80
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