lean_workbook_plus_36818

⚠️ 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 $n\in\mathbb N$ and define $P(x,y)=x^n+xy+y^n$ . Show that we cannot obtain two non-constant polynomials $G(x,y)$ and $H(x,y)$ with real coefficients such that $P(x,y)=G(x,y)\cdot H(x,y)$ .

Why this node was retired

The posted statement is

theorem lean_workbook_plus_36818 (n : ℕ) (P : ℝ → ℝ → ℝ) (hP : P = fun x y => x^n + x*y + y^n) : ¬ (∃ G H : ℝ → ℝ → ℝ, P = G * H)   :=  by sorry

Polynomial factors become arbitrary functions and nonconstancy is omitted; every function factors as itself times 1.

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

Proposed corrected statement

Use actual bivariate polynomials over ℝ and assert irreducibility/no factorization into two nonconstant polynomials. Audit the allowed n values and exclude any source boundary counterexamples before claiming irreducibility.

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.