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Claim. M. Sekanina, Notes on the factorisation of infinite cyclic groups (in Russian), Czechoslovak Math. J. 9 (84) (1959), no. 4, 485-495. Remark 1.7 (p. 490) proves that the set of squares $\{n^2:n\ge0\}$, the full image of $f(X)=X^2$, is not a direct factor of $\mathbb Z$: there is no $A\subseteq\mathbb Z$ such that every integer is uniquely $a+n^2$ with $a\in A$ and $n\ge0$. The proof uses that $x^2-y^2=k$ is solvable exactly when $k\not\equiv2\pmod 4$. The same page leaves the higher powers $\{n^k:n\ge0\}$, $k>2$, unresolved, the question that the formal-conjectures monomial variant of Problem 477 records. Erdős and Graham cite the paper in the problem itself. The journal and Crossref records give no issue month, so the page carries 1 January 1959.

Covers. $f(X)=X^2$: the squares admit no tiling complement.

Depends on. Nothing in this wiki; the claim is the cited paper's remark.

Acceptance. Refereed: Czechoslovak Mathematical Journal 9 (1959).