lean_workbook_plus_63523

⚠️ 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.

Let $(x,y)$ be the pair satisfying $x^2-pqy^2=1$ with minimal $|xy|>0$ . Note that $pqy^2=(x-1)(x+1)$ . \n\n After a modulo $4$ analysis, it follows that $x-1=2u^2\delta$ and $x+1=2v^2(pq/\delta)$ for some divisor $\delta$ of $pq$ and integers $u,v$ satisfying $2uv=y$ . Therefore, $v^2\cdot\frac{pq}{\delta}-u^2\cdot\delta=1.$ If $\delta\in\{p,q\}$ then we get the desired identity. If $\delta=pq$ then as $|uv|=|y|/2<|xy|$ we reach a contradiction. Finally, if $\delta=1$ then $pq\mid u^2+1$ which is impossible, as $p\equiv q\equiv 3\bmod{4}$ , so the proof is finished.

Why this node was retired

The posted statement is

theorem lean_workbook_plus_63523 (p q : ℕ) (hp : p.Prime) (hq : q.Prime) (hpq : p ≠ q) : ∃ x y : ℤ, x^2 - p * q * y^2 = 1 ∧ |x * y| > 0 ∧ ∀ x' y' : ℤ, x'^2 - p * q * y'^2 = 1 → |x' * y'| ≥ |x * y|   :=  by sorry

Restrict the comparison solutions by |x′y′|>0 as well: minimize among nontrivial Pell solutions only.

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

Proposed corrected statement

Restrict the comparison solutions by |x′y′|>0 as well: minimize among nontrivial Pell solutions only.

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.