waring_prime_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.

Waring–Goldbach problem: Is every sufficiently large integer a sum of s prime k-th powers? Hua (1938) proved s = 2^k + 1 suffices. The optimal s for each k is unknown.

Why this node was retired

The posted statement is

import Mathlib

theorem waring_prime_conjecture (k : ℕ) (hk : 2 ≤ k) :
    ∃ s : ℕ, ∀ n : ℕ, n ≥ 2 →
    ∃ (primes : Fin s → ℕ),
      (∀ i, Nat.Prime (primes i)) ∧
      n = ∑ i, (primes i) ^ k := by
  sorry

All n≥2 are required rather than sufficiently large admissible n.

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

Proposed corrected statement

Quantify a threshold N and an appropriate number of summands, requiring n≥N and the necessary local congruence conditions in a precise Waring–Goldbach theorem.

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.