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Claim. Let $B=\{m^{13}:m\in\mathbb Z\}$. There is a set $A\subseteq\mathbb Z$ such that every integer $n$ has exactly one representation $n=a+b$ with $a\in A$ and $b\in B$. Since $m\mapsto m^{13}$ is injective, the pair $(a,m)$ with $n=a+m^{13}$ is unique as well, so $f(X)=X^{13}$ answers the existence question of Problem 477 affirmatively, against the expectation Erdős and Graham recorded. The manuscript's result is Theorem 1.1 on the pipeline-math card; the manuscript was first added to the project's repository on 28 June 2026 and last changed on 29 June 2026, the version the link pins. The PDF prints no byline; the project attributes the proof to GPT 5.5 Pro with polishing and checking by its contributors, and the forum claim of 12 September 2026, claimed by Diyi Liu, Binghui Peng, Hantao Yu, Runzhou Tao and Steven Wang and submitted by Diyi Liu, names that system.

Submission note. Posted to erdosproblems.com as a proof claim by Diyi Liu, Binghui Peng, Hantao Yu, Runzhou Tao, Steven Wang (account diyiliu) on 12 September 2026, giving "GPT 5.5 Pro" as the AI used:

We prove that the set of thirteenth powers

$$> B=\{m^{13}:m\in\mathbb Z\} >$$

has a tiling complement in $\mathbb Z$: there exists a set $A\subseteq\mathbb Z$ such that every integer has a unique representation $a+b$ with $a\in A$ and $b\in B$. The proof combines an inductive tiling construction with an algebraic-geometric estimate. A sufficient condition for $B$ to tile $\mathbb Z$ is that, for every finite set $C\subseteq\mathbb Z\setminus B$, there exists $b\in B$ such that

$$> (C-b)\cap(B-B)=\varnothing. >$$

This condition allows any finite family of pairwise disjoint translates of $B$ to be extended to cover a prescribed uncovered integer without creating overlaps. Enumerating the integers and repeating this step therefore produces a tiling. For the thirteenth powers, we establish the required condition using an algebraic-geometric estimate for nonconstant polynomial parametrizations.

The argument. A finite-avoidance criterion (Lemma 1.7) shows that $B$ tiles $\mathbb Z$ as soon as every finite $C\subseteq\mathbb Z\setminus B$ admits $b\in B$ with $(C-b)\cap(B-B)=\varnothing$: translates of $B$ are then added one at a time to cover the next uncovered integer without overlap. The criterion is supplied by a count (Proposition 1.6): for each fixed $c\notin B$ only $O_c(T^{5/6})$ parameters $|t|\le T$ make $t^{13}-c$ a difference of two thirteenth powers. That count rests on Heath-Brown's 2009 bound for integral points on diagonal ternary surfaces and on the exclusion of rational curves on them through the three- and four-term unit-equation bounds recalled by Corvaja and Zannier (Lemma 1.4). The card's result pages hold the reconstruction, including the places where it departs from the manuscript's wording of the Heath-Brown input.

Formalization. The forum claim links a Lean project whose notes describe it as a formalization of the manuscript's main theorem, named there under a title and six-author list that the manuscript PDF does not print, with Heath-Brown's theorem and the Brownawell-Masser unit bound taken as axioms. It is a formalization of this result and so a link on this page, not a separate claim. This repository has not built or audited it, and it is not formalized evidence; the formal-conjectures file for the problem states the question and is not a proof.

Acceptance. The site's curator, Thomas Bloom, marked the problem solved on 5 September 2026; Bloom's 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 $d\ge6$, and Bloom's signed exposition names the pipeline-math construction as independently found beside Price's, expounding a proof for positive inputs and every exponent $d\ge5$ rather than reviewing this manuscript line by line. The credit is of the existence conclusion and of the independence of this construction; it is stated for the even exponents $d\ge6$, a range that does not contain $13$, and no curator text mentions thirteenth powers. It is the reviewed evidence here because the curator's label settles the problem and names pipeline-math among those who proved it; it is not a review of the thirteenth-power theorem itself, which rests on the manuscript. No journal publication or arXiv posting is known. The repository's own compilation review of the reconstructed proof, relative to the two external premises, is recorded on the card and awards no standing here.