lean_workbook_plus_30941

⚠️ 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.

Let $a_0=k$ and $a_{n+1}=a_n^2+k$ for each $\geq{0}$ . For how many $k\in{\mathbb{Z^+}}$ is $\gcd(a_{444},a_{2013})=1$ ?

Why this node was retired

The posted statement is

theorem lean_workbook_plus_30941 (f: ℕ → ℕ) (k : ℤ) (h₁ : 0 < k) (h₂ : ∀ n:ℕ, f n = f (n-1)^2 + k) : (Nat.gcd (f 444) (f 2013) = 1)   :=  by sorry

The initial value a₀=k is missing, the predecessor recurrence includes n=0 and is impossible for k>0, and a counting question became a universal gcd claim.

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

For each positive integer k define a₀=k and a_(n+1)=a_n²+k for n≥0. Determine the number of k for which gcd(a_444,a_2013)=1; do not insert a universal gcd answer without solving this classification.

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.