Least m such that φ(m) = n!

Conjecture: unless n! + 1 is prime (i.e., n ∈ A002981), a(n) = p q where p is the least prime > √(n!) such that (p - 1) | n! and q = n!/p - 1 + 1 is prime. - M. F.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-oeis-55487,
  title        = {Least m such that φ(m) = n!},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-55487}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Conjecture: unless $n! + 1$ is prime (i.e., $n \in A002981$ ), $a (n) = pq$ where $p$ is the least prime $> n!$ such that $(p - 1) ∣ n!$ and $q = \frac{n !}{p - 1} + 1$ is prime.

M. F. Hasler, Oct 04 2009

The conclusion $(p (n)) . Prime$ asserts that such a prime $p$ exists: if no prime satisfies the search conditions then p n is sInf ∅ = 0, which is not prime.

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

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«55487».

theorem conjecture (n : ℕ) (hn : 1 ≤ n) (h_not_prime : ¬ isFactorialPrime n) :
    (p n).Prime ∧ a n = p n * q n

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

A055487

Source and licence

Imported from Formal Conjectures (OEIS), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.