lean_workbook_plus_2490

⚠️ 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.

Find a matrix $A$ over $\mathbb Z/p\mathbb Z$ such that $A^{p^2}=I_n$ but $A^k\ne I_n,\ \forall k<p^2$ .

Why this node was retired

The posted statement is

theorem lean_workbook_plus_2490 (p : ℕ) (hp : p.Prime) (n : ℕ) : ∃ A : Matrix (Fin n) (Fin n) (ZMod p), A ^ (p ^ 2) = 1 ∧ ∀ k < p ^ 2, A ^ k ≠ 1   :=  by sorry

The least-order clause includes k=0, impossible for every matrix, and arbitrary dimensions may not support order p².

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

Proposed corrected statement

For prime p and n≥p+1 construct a unipotent matrix with one Jordan block of size p+1 and identity remainder, and prove exact multiplicative order p² by requiring A^k≠I only for1≤k<p². Investigate smaller dimensions separately.

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.