---

Source. Theorem 3, printed pp. 439--440 (PDF pp. 5--6) of R. L. Graham, A theorem on partitions, J. Austral. Math. Soc. 3 (1963), no. 4, 435--441, DOI 10.1017/S1446788700039045; proof pp. 440--441. The copy read is an image-only scan; the statement was read on the rendered page images on 2026-09-18.

Statement

Theorem 3 (pp. 439--440). Given positive rationals $\alpha$ and $\beta$, some threshold $r=r(\alpha,\beta)$ has the property that every integer $n>r$ admits positive integers $k,a_1,\ldots,a_k$ satisfying the three conditions

1. $\beta<a_1<a_2<\cdots<a_k$;
2. $n=a_1+a_2+\cdots+a_k$;
3. $\alpha=a_1^{-1}+a_2^{-1}+\cdots+a_k^{-1}$.

With $\alpha=1$ and $\beta=1$ this is Theorem 1 without its explicit threshold; Theorem 1 gives $r(1,1)\le77$ and the Remarks (p. 441) record Lehmer's unpublished check that $77$ itself has no such partition. The Remarks' conjecture $2'$, recorded on the card, asks for the same conclusion with condition 2 replaced by $n=f(a_1)+\cdots+f(a_k)$ for a polynomial $f$ under the hypotheses recorded there. Theorem 3 is the conjecture's case $f(x)=x$, and the conjecture's case $\alpha=\beta=1$ is the question of Problem 283 up to wording (the card records the difference).

Proof pointer and sketch

By the Lemma of p. 438 (used for Theorem 2) there are integers $\beta<c_1<\cdots<c_k$ with $\alpha=\sum1/c_i$. With $c=2c_k$, the last term is split as $1/c+1/c$ and one copy of $1/c$ is expanded through a representation $1=\sum_{i\le w}1/d_i$ into $\sum1/(cd_i)$ (display (1), p. 440); as $U=\sum d_i$ runs through all sufficiently large integers (by Theorem 1), the denominator sums of (1) cover all large integers in one residue class modulo $c$. Variants (2), (3), ..., ($c$) that replace $1/c$ by $1/(c+1)+1/(c(c+1))$, then by $1/(c+1)+1/(c(c+1)+1)+1/(c(c+1)(c(c+1)+1))$, and so on, with the $d_i$ restricted to be large (by Theorem 2, $U$ still runs through all sufficiently large integers), cover the remaining residue classes modulo $c$ (p. 441). Read for structure only; not verified here.

Dependencies and read depth

Same paper: Theorem 1, Theorem 2 and the Lemma of p. 438, which the paper cites as a special case of a theorem of the author's paper On finite sums of unit fractions (Proc. London Math. Soc., then to appear). Read depth: claims checked (statement read clause by clause on the page images of pp. 439--440); proof not verified.

Bears on

• Problem 283: the rational-$\alpha$ form of the case $p(x)=x$, and the existence result whose threshold van Doorn's quantitative bounds estimate (van Doorn, Theorem 1).