bateman_horn_conjecture

⚠️ 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.

Bateman–Horn conjecture (1962): Quantitative version of Schinzel's Hypothesis H giving the asymptotic count of n ≤ N where k irreducible polynomials simultaneously take prime values. The predicted constant involves an Euler product over primes. Generalizes Hardy–Littlewood twin prime constant.

Why this node was retired

The posted statement is

import Mathlib

theorem bateman_horn_conjecture (k : ℕ) (hk : 1 ≤ k)
    (polys : Fin k → Polynomial ℤ)
    (hdeg : ∀ i, 1 ≤ (polys i).natDegree)
    (hlc : ∀ i, 0 < (polys i).leadingCoeff)
    (hirr : ∀ i, Irreducible (polys i))
    (omega_poly : ℕ → ℝ)
    (homega : ∀ p : ℕ, Nat.Prime p →
      omega_poly p = {r : Fin p | ∀ i, ¬(p : ℤ) ∣ (polys i).eval r}.ncard) :
    Filter.Tendsto (fun N : ℕ =>
      (∑ i ∈ Finset.range N,
        if ∀ j : Fin k, Nat.Prime ((polys j).eval (i : ℤ)).natAbs then (1 : ℝ) else 0) /
      ((N : ℝ) / (Real.log N) ^ k))
    Filter.atTop
    (nhds (∏' p : {p : ℕ // Nat.Prime p},
      (1 - omega_poly p / (p : ℝ)) / (1 - 1 / (p : ℝ)) ^ k)) := by
  sorry

The polynomial list permits duplicates, its omega counts nonroots rather than roots, and the normalization omits degree factors.

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

Proposed corrected statement

Require distinct irreducible polynomials and no fixed prime divisor; define ω(p) as the number of residues where their product vanishes modulo p. Use the standard Bateman–Horn singular product and the integral/main-term normalization including the product of degrees.

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.