lean_workbook_plus_66729

⚠️ 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 root $\mod p$ is an integer $a$ such that the smallest integer $k,$ for which $a^k \equiv 1 (\mod p),$ is $k=p-1$ .

Why this node was retired

The posted statement is

theorem lean_workbook_plus_66729 (p : ℕ) (hp : p.Prime) (a : ℤ) : (∀ k : ℕ, a ^ k ≡ 1 [ZMOD p] → k ≥ p - 1) ∧ (a ^ (p - 1) ≡ 1 [ZMOD p])   :=  by sorry

The formal statement changes or incorrectly supplies part of the source problem; the accepted note identifies a concrete witness or definitional mechanism.

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

Proposed corrected statement

Define a to be a primitive root modulo prime p iff its nonzero residue has multiplicative order p−1; equivalently a^(p−1)≡1 and every positive exponent k with a^k≡1 satisfies k≥p−1. Do not assert this for arbitrary integers a.

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.