#477: every K-th power set, K ≥ 3, tiles the integers — cubes included
The question. Erdős and Graham (1980) asked whether there is a polynomial f of degree at least 2 and a set A ⊂ ℤ such that every integer is uniquely a + f(k) — a "tiling complement" for the values of f. For f(k) = k² the answer is no (a congruence argument; Sekanina 1959 for the squares, AlphaProof and Adenwalla for a family of quadratics). Sekanina asked whether it fails for every power kK, K ≥ 2. The Formal Conjectures Lean file says of the cubes: "Probably there is no such A."
What happened, in order. On 28 June 2026 Peng et al. (pipeline-math) gave a tiling complement for K = 13 by an S-unit construction — that alone answers the headline question yes, and it is why the community database records #477 as solved. On 23 July we published a proof that every K ≥ 3 works: a finite-avoidance reduction plus two counting theorems of Heath-Brown and Browning–Heath-Brown, balanced so the number of "bad" n ≤ X is O(Xβ) with β = 18/19 for cubes and β < 1 for every K. On 26 August Liam Price's "Large Powers Tile the Integers" (even K ≥ 6) was formalized in Lean, and in September the Formal Conjectures file flipped to solved on that basis — with K = 3, 4, 5 and every odd K still marked open.
What is ours. Not the closure of #477 — Peng et al. had that first. The theorem for every K ≥ 3, cubes included, which is strictly stronger than what the Lean record cites and contradicts its stated expectation for X³.
How it was checked. Construction via GPT-5.6 Sol (ChatGPT Pro), attacked twice in fresh adversarial sessions in July (one real but cosmetic genus-formula error found and fixed). On 11 October 2026, with the community having moved to Lean-or-exposition, we re-refereed it from scratch: Claude (Fable 5.1) re-derived the gap-balancing bookkeeping and read both external theorems in the primary PDFs — they say exactly what the proof uses — and a separately-briefed hostile pass read the whole chain and the Annals-2002 fallback, closed a dangling Fermat's-Last-Theorem clause in Heath-Brown's own paper, and counted the "bad" n exactly up to 32,000 for several c (sublinear, matching the predicted families). Verdict: no gap in the load-bearing chain. It also found several non-load-bearing defects in our write-up — a misquoted theorem, an overclaimed framing, a stale note about the Lean scope — all corrected in the repository the same day.
The honest line. No human mathematician has reviewed this, and a machine referee is not community acceptance. erdosproblems.com no longer takes proof claims; its new route is an exposition sent to Bloom, and the Formal Conjectures maintainers own the "probably no" label. Both are the next step. Until then this page says CLOSED because the problem is closed — by others — and says plainly that the larger theorem is ours and unreviewed.