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Claim. The set $\{n+1/n:n\in\mathbb{N}\}$ is strongly complete: for every finite set $B$, every sufficiently large integer is a sum of distinct elements of the set outside $B$. This is the case $p(x)=x$ of Problem 351. It follows from Theorem 3 of R. L. Graham, A theorem on partitions, J. Austral. Math. Soc. 3 (1963), no. 4, 435--441, received 17 March 1963, the date this page carries; the library's result page Theorem 3 records the statement. With $\alpha=1$ the theorem says that for every $\beta>0$ there is $r(\beta)$ such that every integer $n>r(\beta)$ is a sum $a_1+\cdots+a_k$ of distinct integers $a_i>\beta$ with $\sum1/a_i=1$. Then $n+1=\sum_i(a_i+1/a_i)$ is a sum of distinct elements of the set with indices above $\beta$. Taking $\beta$ above every index of the finite set $B$ gives every integer above $r(\beta)+1$ as a sum of distinct elements outside $B$, which is strong completeness.

Covers. The polynomial $p(x)=x$ only. The theorem decides nothing for any other polynomial; the general case is the full claim on the Price–Barreto page, and the case $p(x)=x^2$ is van Doorn's claim.

Depends on. Nothing in this wiki; the claim rests on the cited paper.

Acceptance. Refereed: the paper appeared in the Journal of the Australian Mathematical Society, volume 3 (1963), a refereed journal, and the site's commentary records that Graham proved the statement when $p(n)=n$. The site's label, PROVED (LEAN), settles the whole problem through the Price–Barreto argument rather than this case, so the curator's credit is not listed as reviewed evidence. The proof is not independently reviewed in this corpus.