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Claim. Let $B=\{m^3:m\in\mathbb Z\}$. There is a set $A\subseteq\mathbb Z$ such that every integer has exactly one representation $a+m^3$ with $a\in A$ and $m\in\mathbb Z$; since cubing is injective, this answers Problem 477 affirmatively with $f(X)=X^3$. The claim also asserts that no integer-valued quadratic polynomial has such a complement, so that degree three is the least degree for which the question has a positive answer. The claim, submitted on 25 September 2026 under the username Dongdong, names Hongyu Shan, Dongdong Xu, Di Liang, Chang Dai and Han Chen as claimants and GPT 5.6 Sol as the system.

Submission note. Posted to erdosproblems.com as a proof claim by Hongyu Shan, Dongdong Xu, Di Liang, Chang Dai and Han Chen (account Dongdong) on 25 September 2026, giving "GPT 5.6 Sol" as the AI used:

We prove that the set of integer cubes is a direct additive factor of $\mathbb Z$: there exists $A\subseteq\mathbb Z$ such that every integer has a unique representation $a+m^3$. The proof reduces this to showing that, for every fixed noncube $k$, the set of $n$ for which $n^3-k$ is a difference of two cubes has density zero. Writing $n^3-k=y^3-x^3$, we split according to the gap $y-x$, treating small gaps with the square sieve and mixed cubic character sums, and large gaps with a fixed-level estimate for diagonal ternary cubic forms; this gives $O_{k,\delta}(N^{18/19+\delta})$ exceptional $n\in[N,2N)$. A two-sided greedy construction then produces $A$, while a separate obstruction for integer-valued quadratic polynomials shows that degree three is the least possible degree in the Erdős–Graham problem.

The argument as claimed. The construction reduces to showing that, for each fixed non-cube $k$, the integers $n$ with $n^3-k\in B-B$ have density zero; a two-sided greedy construction then produces $A$. Writing $n^3-k=y^3-x^3$, the claim splits by the gap $y-x$: small gaps are handled with the square sieve and mixed cubic character sums, large gaps with a fixed-level estimate for diagonal ternary cubic forms, giving $O_{k,\delta}(N^{18/19+\delta})$ exceptional $n\in[N,2N)$.

Standing. The claim is pending, and this page does not rest on the linked Overleaf manuscript's proof. The thread carried no comment and no curator response as of 7 October 2026, and no acceptance evidence exists. The site's earlier remark that a quadratic polynomial cannot work is an argument given in the problem's comments, not a dated manuscript, and is recorded on the problem page. The existence question is settled independently by the thirteenth-power construction on its claim page.