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Source. Theorem 2, printed p. 437 (PDF p. 3) of R. L. Graham, A theorem on partitions, J. Austral. Math. Soc. 3 (1963), no. 4, 435--441, DOI 10.1017/S1446788700039045; proof pp. 438--439. The copy read is an image-only scan; the statement was read on the rendered page images.

Statement

Theorem 2 (p. 437). Given any integer $m$, some threshold $r=r(m)$ has the property that every integer $n>r$ admits positive integers $k,a_1,\ldots,a_k$ satisfying the three conditions

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

For $m\le1$ it follows from Theorem 1 with $r=77$, as the paper notes (p. 438). For $m\ge1$ it is the case $\alpha=1$, $\beta=m$ of Theorem 3, which the paper proves from it.

Proof pointer and sketch

The proof (pp. 438--439) takes $m\ge2$ and rests on the Lemma of p. 438 (recorded on the card). By Dirichlet's theorem on primes in arithmetic progressions there is an $h$ with $mh-1$ a prime above $13$. The Lemma, applied with $t=m$ to the remainder after $\frac1m+\frac1{m(mh-1)}$ and a block of terms $\frac1{mq_i-1}$ with rapidly growing primes $q_1,\ldots,q_m$, completes a representation of $1$ whose remaining denominators also have the form $mc-1$. Splitting $\frac1{mq_i-1}$ as $\frac1{mq_i}+\frac1{mq_i(mq_i-1)}$ for the first $j$ of the $q_i$ ($1\le j\le m$) raises the denominator sum by $j$ modulo $m$, so the $m$ resulting representations have denominator sums $U_j$ covering every residue class modulo $m$. Replacing $\frac1m$ by $\sum\frac1{md_i}$, where $1=\sum\frac1{d_i}$ is a representation from Theorem 1 with denominator sum $U>77$ (the paper notes that its denominators involve only the primes $2,3,5,7,11,13$), gives a representation with denominator sum $mU+U_j-m$; hence every $n>78m+U_m-m$ is covered, all denominators exceed $m$, and they stay distinct because $mh-1$ and the $q_i$ are primes above $13$. Read for structure only; not verified here.

Dependencies and read depth

Same paper: Theorem 1 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); Dirichlet's theorem, cited from LeVeque, Topics in Number Theory (1956), p. 76. Read depth: claims checked (statement read clause by clause on the page image of p. 437); proof not verified.

Bears on

• Problem 351: the case $p(x)=x$. A representation $n=\sum a_i$ with distinct $a_i>m$ and $\sum1/a_i=1$ gives $n+1=\sum(a_i+1/a_i)$, a sum of distinct terms of $\{n+1/n\}$ avoiding every term of index at most $m$; taking $m$ above the indices of a finite set $B$ shows that every integer above $r(m)+1$ is a finite sum of distinct terms outside $B$. The polynomial case is not treated here.
• Problem 283: the case $p(x)=x$ with all denominators above a prescribed bound; it decides nothing for any other polynomial.