Graham 1963 theorem partitions

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theorem_1: States that every integer n > 77 is a sum of distinct positive integers greater than 1 whose reciprocals sum to 1, with 77 itself excluded by Lehmer's unpublished check.

theorem_2: States that for every integer m all sufficiently large integers are sums of distinct integers greater than m whose reciprocals sum to 1.

theorem_3: States that for positive rationals α and β every sufficiently large integer is a sum of distinct integers exceeding β whose reciprocals sum to α.

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R. L. Graham, A theorem on partitions, J. Austral. Math. Soc. 3 (1963), 435--441. Received 17 March 1963.

The copy read for this card is an image-only scan of the seven printed pages (physical PDF p. $n$ is printed p. $434+n$). It has no text layer; the title, author, received date and page numbers were confirmed on the page images, and the statements below were read there, with an OCR pass used only to locate them. Provenance: the download URL of that scan was not recorded; 303,466 bytes. No notice is printed (pp. 435 and 441 read on the page images); the journal's article page on Cambridge Core shows "Copyright © Australian Mathematical Society 1963" (DOI 10.1017/S1446788700039045, read 2026-10-02), every other right reserved.

Read status: claims checked. Theorems 1, 2 and 3, the Lemma of p. 438 and the Remarks of p. 441 were read clause by clause on the page images; no proof was checked.

Contents

• Theorem 1 (p. 435; result page): every integer $n>77$ is a sum $a_1+\cdots+a_k$ of integers $1<a_1<\cdots<a_k$ whose reciprocals sum to $1$. The proof (pp. 435--437) is a table of representations for every $n$ from $78$ to $167$ and the odd $n$ from $169$ to $333$ (the first transformation below supplies the even $n$ from $168$ to $334$), followed by the two transformations $1=\frac12+\sum\frac1{2d_i}$ and $1=\frac13+\frac17+\frac1{78}+\frac1{91}+\sum\frac1{2d_i}$, which carry a representation with denominator sum $U$ to ones with sums $2U+2$ and $2U+179$, with all denominators still distinct provided no $d_i$ equals $1$ or $39$.
• Theorem 2 (p. 437; result page): for any integer $m$ there exists $r=r(m)$ such that every integer $n>r$ is a sum $a_1+\cdots+a_k$ of positive integers with $m<a_1<\cdots<a_k$ and $1=a_1^{-1}+\cdots+a_k^{-1}$. The proof (pp. 438--439) rests on the Lemma (p. 438): for a positive rational $p/q$ and an integer $t$ coprime to $q$, and for every $s$, there are positive integers $k$ and $s<c_1<\cdots<c_k$ with $p/q=\sum_{i=1}^k1/(tc_i-1)$, a special case of a theorem of the author's paper on finite sums of unit fractions (the paper's [1], then to appear).
• Theorem 3 (pp. 439--440; result page): for any positive rationals $\alpha$ and $\beta$ there exists $r=r(\alpha,\beta)$ such that every integer $n>r$ is a sum $a_1+\cdots+a_k$ of positive integers with $\beta<a_1<\cdots<a_k$ and $\alpha=a_1^{-1}+\cdots+a_k^{-1}$. Proved on pp. 440--441 from the Lemma and Theorems 1 and 2.
• Remarks (p. 441): the least admissible $r(\alpha,\beta)$ seems hard to determine; Theorem 1 gives $r(1,1)\le77$, and unpublished work of D. H. Lehmer shows that $77$ is not a sum of distinct positive integers with reciprocal sum $1$, so $r(1,1)=77$. Conjecture $2'$: Theorem 3 should stay true with its condition 2 changed to $n=f(a_1)+\cdots+f(a_k)$, for every integer-valued polynomial $f$ with positive leading coefficient whose values have no common prime factor; "At present, however, very little is known about this problem."

Compiled scope

The statements above were checked on the page images of pp. 435, 437--441; the table and the proofs were not checked. Nothing here is independently reviewed.

Bears on. #283: the problem's statement is the case $\alpha=\beta=1$ of the conjecture $2'$ of the Remarks (p. 441), with "no $d\ge2$ divides every $p(n)$" in place of the prime-by-prime condition; Theorem 3 with $\alpha=1$ is the case $f(x)=x$ (and Theorem 1 its explicit form with threshold $77$). #351: Theorem 2 (or Theorem 3 with $\alpha=1$) and $n+1=\sum(a_i+1/a_i)$ give, for every bound, that all sufficiently large integers are sums $\sum(a_i+1/a_i)$ over distinct $a_i$ above that bound, so the case $p(x)=x$ of the problem's sequence $\{p(n)+1/n\}$ stays complete after removing any finite set of terms; the polynomial case is not treated here, and the completeness criterion for the values $p(n)$ alone is Graham 1964.

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