integer_sums_of_three_cubes

⚠️ 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.

Every integer (not ≡ ±4 mod 9) is a sum of three cubes: Can every such integer be represented as a³+b³+c³? Proved for all n up to 10^{15} for small n. Hardest open cases like 114 still unknown. Hardy-Littlewood conjecture says yes for all except ±4 mod 9.

Why this node was retired

The posted statement is

import Mathlib

theorem integer_sums_of_three_cubes :
    ∀ n : ℤ, ¬(n % 9 = 4 ∨ n % 9 = -4) →
    ∃ a b c : ℤ, a ^ 3 + b ^ 3 + c ^ 3 = n := by
  sorry

The residue exclusion uses integer−4 rather than the canonical remainder5 modulo9.

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

Proposed corrected statement

Exclude n%9=4 and n%9=5 (or use congruence to±4 in ZMod9).

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.