primitive_prime_divisor_conjecture

⚠️ 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.

Primitive prime divisor theorem (Zsygmondy 1892): For integers a > b > 0 with gcd(a,b)=1, aⁿ - bⁿ has a prime factor not dividing aᵏ - bᵏ for k < n, with finitely many exceptions. Used in group theory to classify groups; proved but with specific exception characterization open.

Why this node was retired

The posted statement is

import Mathlib

theorem primitive_prime_divisor_conjecture (a b : ℤ) (ha : 1 < a.natAbs) (hb : 0 < b.natAbs)
    (hcop : Nat.Coprime a.natAbs b.natAbs) :
    ∀ n : ℕ, 3 ≤ n →
    ∃ p : ℕ, Nat.Prime p ∧ p ∣ (a^n - b^n).natAbs ∧
      ∀ k : ℕ, 0 < k → k < n → ¬(p ∣ (a^k - b^k).natAbs) := by
  sorry

The classical exceptional exponent and ordered-positive-base hypotheses are omitted.

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 a>b>0 and gcd(a,b)=1. For the existing n≥3 range, exclude(a,b,n)=(2,1,6); if extending to n=2 also include the power-of-two a+b exception.

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.