lean_workbook_plus_48868

⚠️ 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.

$p(p-1)\le 2n-2<p(p+1)$ $\iff$ $4p^2-4p+1\le 8n-7<4p^2+4p+1$

Why this node was retired

The posted statement is

theorem lean_workbook_plus_48868 : ∀ n : ℕ, 1 < n ∧ ∃ p : ℕ, p * (p - 1) < 2 * n ∧ 2 * n < p * (p + 1) ↔ 1 < n ∧ ∃ p : ℕ, 4 * p ^ 2 - 4 * p + 1 ≤ 8 * n - 7 ∧ 8 * n - 7 < 4 * p ^ 2 + 4 * p + 1   :=  by sorry

The original two inequalities p(p−1)≤2n−2<p(p+1) were changed in both offsets and strictness; scope and natural subtraction also require care.

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 natural n≥1 and p≥1 prove the exact equivalence p(p−1)≤2n−2<p(p+1) iff4p²−4p+1≤8n−7<4p²+4p+1, using integer arithmetic or justified natural subtractions and parentheses.

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.