lean_workbook_plus_15262

⚠️ 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 $p$ be a prime of the form $4k + 1$ . Prove, that there are no positive integers $a,b,c$ such that $a^2 + b^2 = c^2$ and $a^2 + (p + 1)b^2 = d^2$ .

Why this node was retired

The posted statement is

theorem lean_workbook_plus_15262 (p : ℕ) (hp : p.Prime) (hp1 : p % 4 = 1) : ¬ (∃ a b c d : ℕ, a^2 + b^2 = c^2 ∧ a^2 + (p + 1) * b^2 = d^2)   :=  by sorry

The source positive integer restriction was dropped, admitting the all-zero solution.

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≡1 mod4, state nonexistence with a,b,c,d>0 explicitly, retaining the two square equations; verify the corrected number-theoretic assertion 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.