---

Statement

Conjecture 4, as posed on p. 167:

Let $n_1<n_2<\cdots$ be an infinite sequence of positive integers such that $n_{i+1}/n_i>c>1$. Can the set of rationals $a/b$ for which

$$> \frac ab=\frac1{n_{i_1}}+\frac1{n_{i_2}}+\cdots+\frac1{n_{i_t}} >$$

is solvable for some $t$ contain all the rationals in some interval $(\alpha,\beta)$. [sic] We conjecture not.

The paper adds (p. 167): "If this conjecture is true then according to Graham [5] this is best possible." Its reference [5] (p. 168) is R. L. Graham, On finite sums of unit fractions, Proc. London Math. Soc. (3) 14 (1964), 193--207, which was not read for this card; it has its own card, graham_1964_finite_sums_unit_fractions.

Source. M. N. Bleicher and P. Erdős, Denominators of Egyptian fractions, J. Number Theory 8 (1976), 157--168; Conjecture 4 on printed p. 167 (PDF p. 11), the last of the four conjectures of Section IV, "Some Conjectures"; the bibliography on p. 168 (PDF p. 12). The copy read is a scan whose text layer garbles formulas; read on the page images.

Read depth. Claims checked: the statement was read clause by clause on the page image. It is a conjecture; there is no proof to check in this paper.

Dependencies

None.

Bears on

• Problem 355: the problem asks whether a lacunary sequence exists whose finite distinct-reciprocal sums represent every rational in some interval, which is the negation of this conjecture; the problem page records the status and the source that settles it (the 2025 paper filed as doorn_2025_lacunary_sequences_whose_reciprocal_sums_represent).