lehmer_totient_problem

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

Lehmer's totient problem (1932): Does φ(n) | n-1 imply n is prime? The converse is Fermat's little theorem. No composite n with φ(n) | n-1 is known; such a number would be an absolute Euler pseudoprime. Conditional result: if φ(n) | n-1 and n is odd, then n is prime (proved).

Why this node was retired

The posted statement is

import Mathlib

theorem lehmer_totient_problem :
    ∀ n : ℕ, 1 ≤ n → (n.totient ∣ n - 1) → Nat.Prime n := by
  sorry

The domain includes1, which has totient1 but is not prime.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

Require n>1, keeping the divisibility implication.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.