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Statement

Section 4 (p. 206) says that Theorem 5 applies to many sequences $S$ satisfying its conditions (1) and (2), that the proofs of these applications are left to a later paper, and states the following five results. Throughout, $(x,y)$ is the greatest common divisor.

(1) (p. 206). Let $a$ and $b$ be arbitrary positive integers and $p/q$ a positive rational with $(p,q)=1$. Then

$$\frac pq=\sum_{i=1}^n\frac1{ak_i+b}$$

for some positive integers $n$ and $k_1<k_2<\cdots<k_n$ if and only if

$$\left(\frac{q}{(q,(a,b))},\frac{a}{(a,b)}\right)=1.$$

The paper adds: "(This result is obtained by considering the sequence $(a+1,2a+1,3a+1,\ldots)$.)" (p. 206).

(2) (p. 206; also announced in §1, p. 193). A rational $p/q$ is a finite sum of reciprocals of distinct squares of integers if and only if $p/q\in\bigl[0,\tfrac{\pi^2}6-1\bigr)\cup\bigl[1,\tfrac{\pi^2}6\bigr)$.

(3) (p. 207). For every positive integer $n$, every sufficiently small positive rational is a finite sum of reciprocals of distinct $n$th powers of integers.

(4) (p. 207). A positive rational $p/q$ with $(p,q)=1$ is a finite sum of reciprocals of distinct square-free integers if and only if $q$ is square-free.

(5) (p. 207). If $T$ is a set of integers containing all sufficiently large primes and all sufficiently large squares, then every positive rational is a finite sum of reciprocals of distinct integers from $T$.

The paper remarks (p. 207) that (1) and (5) settle two questions raised by H. S. Wilf (Reciprocal bases for the integers, Research problem 6, Bull. Amer. Math. Soc. 67 (1961), 456).

Source. R. L. Graham, On finite sums of unit fractions, Proc. London Math. Soc. (3) 14 (1964), no. 2, 193--207, doi:10.1112/plms/s3-14.2.193; §4, pp. 206--207. The edition read is named on the source card.

Read depth. Claims checked: the five statements were read clause by clause on the page images of the print. The paper proves none of them, so no proof was checked. Nothing here is independently reviewed.

Proof pointer

None in the paper: the proofs are left to a later paper (p. 206). Each is presented as an application of Theorem 5.

Dependencies

Theorem 5 of the same paper, as the paper indicates.

Bears on

• Problem 282: (1) is a criterion for which rationals are finite sums of distinct reciprocals of terms $ak+b$, $k\ge1$, of an arithmetic progression; for $a=2$ and $b=1$ its condition reads $(q,2)=1$, so it states that a reduced positive $p/q$ is a finite sum of reciprocals of distinct odd integers greater than $1$ exactly when $q$ is odd. (2) is the corresponding criterion for distinct squares. Both concern which rationals have a representation and are stated without proof; the paper says nothing about the greedy algorithm or its termination.