lean_workbook_plus_37103

⚠️ 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.

Show that for any positive integer $a$ , $\frac{a^3+2a}{a^4+3a^2+1}$ is always a fraction in lowest terms.

Why this node was retired

The posted statement is

theorem lean_workbook_plus_37103 (a : ℤ) (ha : a > 0) : ∀ a, (a^3 + 2 * a) % (a^4 + 3 * a^2 + 1) ≠ 0 ∧ Nat.gcd (a^3 + 2 * a) (a^4 + 3 * a^2 + 1) = 1   :=  by sorry

An inner ∀a shadows the positive integer a and ranges over unrestricted naturals; its nonzero-remainder clause fails at0.

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 one positive integer/natural a, prove gcd(a³+2a,a⁴+3a²+1)=1, and if desired0<a³+2a<a⁴+3a²+1. Remove the shadowing quantifier.

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.