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Claim. Every integer $n>77$ is a sum $a_1+\cdots+a_k$ of integers $1<a_1<\cdots<a_k$ with $\frac1{a_1}+\cdots+\frac1{a_k}=1$; the integer $77$ has no such partition. This is the case $p(x)=x$ of Problem 283, with the threshold made exact. The theorem is Theorem 1 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_1 records the statement of p. 435 and the proof's structure. The exclusion of $77$ is Lehmer's unpublished check, reported in the paper's Remarks (p. 441). The same paper's Theorem 3 gives the rational-$\alpha$ form of the same case: for positive rationals $\alpha$ and $\beta$, every large integer is a sum of distinct integers exceeding $\beta$ with reciprocal sum $\alpha$. The problem itself is the case $\alpha=\beta=1$ of the paper's conjecture $2'$ (Remarks, p. 441), which asks for the same with a polynomial in place of $x$.

Covers. The polynomial $p(x)=x$ only: for it the answer is yes, with $m>77$ as the exact range. The theorem decides nothing for any other polynomial; the general case is the full claim Price 2026.

Depends on. Nothing in this wiki; the theorem is the paper's own.

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 case $p(x)=x$. The problem's label, PROVED (LEAN), settles the whole problem through the Price argument rather than this case, so the curator's credit is not listed as reviewed evidence. The statement is checked against p. 435 and the proof is recorded for structure only; the proof is not independently reviewed. Van Doorn's 2025 preprint quantifies the threshold for general $\alpha$ and is recorded on the problem page.

Formalization. Van Doorn's Aristotle-generated ExplicitGraham.lean in the repository Woett/Lean-files, linked above at its commit of 27 March 2026, proves in its Part 2 the lemma ogGraham (every $n\ge78$ is a sum of distinct positive integers with reciprocal sum $1$) by the steps $n\to2n+2$ and $n\to2n+179$, with no sorry and before either of the file's two axioms is used. The exclusion of $77$ is not formalized, and the corpus has not built the file, so no formalized evidence follows.