lean_workbook_plus_47332

⚠️ Retired — specification defect

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

We can actually solve for integer pairs $(p,q)$ using $\left(2p^2+p\right)^2<(2q+1)^2<\left(2p^2+p+1\right)^2$ for every $p<-1$ and $p>2$ . If the calculation is done correctly, the solutions are $(-1,-1)$ , $(-1,0)$ , $(0,-1)$ , $(0,0)$ , $(2,-6)$ , and $(2,5)$ .

Why this node was retired

The posted statement is

theorem lean_workbook_plus_47332 (p q: ℤ) (h₁ : (2 * p ^ 2 + p) ^ 2 < (2 * q + 1) ^ 2) (h₂ : (2 * q + 1) ^ 2 < (2 * p ^ 2 + p + 1) ^ 2) : p = -1 ∧ q = -1 ∨ p = -1 ∧ q = 0 ∨ p = 0 ∧ q = -1 ∨ p = 0 ∧ q = 0 ∨ p = 2 ∧ q = -6 ∨ p = 2 ∧ q = 5   :=  by sorry

The original equation is absent from the excerpt. The formalization assumes an impossible sandwich for all parameters and does not certify any listed solution.

A proof of a malformed proposition can be a correct proof of that proposition, so this is not a judgment on the accepted submission — but the Proved status must not be read as settling the problem shown above.

Proposed corrected statement

Restore the original Diophantine equation and use the strict square sandwich only as an exclusion argument for p<−1 or p>2; verify the finite remaining solutions against that equation.

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.