Problem 477

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claims/: The 4 claim pages of Problem 477, one per claimant's result; the problem's standing derives from them.

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Statement. Is there a polynomial $f:\mathbb{Z}\to \mathbb{Z}$ of degree at least $2$ and a set $A\subset \mathbb{Z}$ such that for any $n\in \mathbb{Z}$ there is exactly one $a\in A$ and $b\in \{ f(k) : k\in\mathbb{Z}\}$ such that $n=a+b$?

Formulation. The question concerns the full image $f(\mathbb Z)$ and uniqueness of the pair $(a,b)$, not uniqueness of a polynomial input when different inputs have the same value.

Status. The site labels the problem SOLVED on its problem page as accessed 2026-09-09, the page last edited 5 September 2026 (OPEN in the site's export of 2026-09-04). Its commentary credits GPT, prompted independently by Price and by pipeline-math, with proving that such an $A$ exists for $f(n)=n^d$ and every even $d\ge6$; that is the range in which Price's construction tiles the full image, while the pipeline-math manuscript treats $f(X)=X^{13}$ only. The claim pages pipeline-math and Price record the results and their acceptance; Sekanina's records the refereed negative answer for $f(X)=X^2$, and a fourth claim is pending. The derived claim value is proved where the site's SOLVED reads as answered, because the question asks whether such an $f$ and $A$ exist and the accepted claims prove that they do.

Source. erdosproblems.com/477, accessed 2026-09-09; the statement above is unchanged from the 2026-09-04 import. Cite as: T. F. Bloom, Erdős Problem #477, https://www.erdosproblems.com/477.

Formalization. Statement in formal-conjectures. This mutable statement link supplies no local verification of the solution.

Current assessment

The pipeline-math manuscript, version of 29 June 2026, constructs a tiling complement for $\{m^{13}:m\in\mathbb Z\}$. Its canonical source digest records the pinned manuscript version and the source's identity. The extracted theorem and its five essential same-source result pages give a complete author-recorded reconstruction with two explicit external theorem interfaces. The repository's own compilation review of that reconstruction, relative to the two external premises, is recorded on the card and awards no standing here.

The site labels the problem solved; its page was last edited on 5 September 2026. Its commentary credits GPT, prompted independently by Price and by pipeline-math, with proving, against the expectation of Erdős and Graham, that such an $A$ exists for $f(n)=n^d$ and every even $d\ge6$, and Thomas Bloom's signed exposition of the same day expounds Price's construction and names pipeline-math's as independently found. That credit is the curator's public acceptance of the existence conclusion; it is not a line-by-line review of either manuscript, and the frontmatter's proved records the affirmative answer to the literal existence question.

The status search of 9 September 2026 covered the live problem and proof claim, the signed exposition, their linked manuscript and GitHub history, an author's account, arXiv and cited journal sources, and a targeted X search; that of 7 October 2026 covered the site's page and forum thread. The reconstruction on the card covers all six manuscript pages and checks the external theorem statements its argument uses at the versions the result page identifies, without reconstructing their proofs; it is at author-recorded standing. It repairs the manuscript's whole-box restatement of Heath-Brown's shell estimate by a fixed-shift shell sum and a bound on the smaller scales, and records the inferred nonconstant-family convention. These qualifications are on the card's result pages.

Four results are claimed for the problem, each with a claim page. The thirteenth-power construction (pipeline-math, 28 June 2026) settles the problem and is accepted on the curator's credit, which names pipeline-math among the provers of the existence conclusion for even $d\ge6$ and does not mention thirteenth powers; its forum claim of 12 September 2026 links a third-party Lean project, not built here, that describes itself as a formalization of the manuscript's theorem with Heath-Brown's bound and the Brownawell-Masser unit bound as axioms. Liam Price's complements for every power $d\ge5$ with nonnegative inputs (Price, 27 July 2026) are accepted on the same credit for the even exponents $d\ge6$, where the tiled set is the full image and $d=6$ settles the problem; the case $d=5$ and the odd exponents rest on the claimant alone, the manuscript's shared link showed only its application shell in the status search of 9 September 2026, and the exposition writes the construction with positive inputs. Beside the axiom-taking project, Boris Alexeev's repository holds a Lean file, with Codex as its formal author, that declares itself a formalization of Price's result for $f(X)=X^6$ and develops its estimates unconditionally; it is linked on Price's page and has not been built or audited here. Sekanina's 1959 negative answer for the squares is described below. Pending is the cubes claim of Shan, Xu, Liang, Dai and Chen (25 September 2026), which asserts that $\{m^3:m\in\mathbb Z\}$ has a tiling complement and that no integer-valued quadratic polynomial does, and had no response on the thread as of 7 October 2026. The site's own comments give an argument, credited to AlphaProof and Adenwalla, that no quadratic $f$ works: for infinite $A$ two members differ by a multiple of $2c_1$ (or of $4c_2$ when $c_1=0$), and that difference is a difference of two values of $f$, so some integer has two representations; the same argument excludes every $f$ whose value differences contain all multiples of a fixed $q\ge1$.

Historical formulation and other exponent ranges

Erdős and Graham's 1980 monograph, printed pp. 95–96, defines $S=\{f(1),f(2),\ldots\}$ and asks whether a direct decomposition $\mathbb Z=S+T$ can never occur. This positive-input formulation differs from the catalog's full image $f(\mathbb Z)$. The Sekanina (1959) paper cited there proves, in its Remark 1.7, that the squares have no tiling complement and leaves higher powers open; see its claim page. The historical expectation was negative, whereas the present catalog existence question is answered positively.

Liam Price's proof claim 154, submitted 27 July 2026 (claim page), states that for every integer $d\geq5$ a computable set $A_d$ complements $\{n^d:n\geq0\}$, and explicitly specializes to all-integer sixth powers. The linked Overleaf manuscript showed only its application shell at the shared link in the status search, so no page here rests on its proof and the link is unpinned. Bloom's exposition writes the tiled set as $\{n^d:n\geq1\}$, while his commentary states the result for the full image $f(\mathbb Z)$ with $f(n)=n^d$ and even $d\ge6$; the exposition is not a local proof for the nonnegative or all-integer even-power image until that endpoint is reconciled, and no page here checks its cited Salberger theorem. These limitations do not affect the all-integer thirteenth-power statement above.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• corvaja_zannier_2011_abcd_function_fields
• corvaja_zannier_2011_abcd_function_fields / recalled_abc_abcd_bounds
• heath_brown_2009_sums_differences_three_kth_powers
• heath_brown_2009_sums_differences_three_kth_powers / theorem_2
• pipeline_math_2026_tiling_complement
• pipeline_math_2026_tiling_complement / corollary_1_5
• pipeline_math_2026_tiling_complement / lemma_1_4
• pipeline_math_2026_tiling_complement / lemma_1_7
• pipeline_math_2026_tiling_complement / proposition_1_6
• pipeline_math_2026_tiling_complement / proposition_1_8
• pipeline_math_2026_tiling_complement / theorem_1_1