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Statement

With $U_n$ the least number of distinct unit fractions with sum $1$ and largest term at most $1/n$ (p. 302), the solution of Erdős and Straus ends on p. 303 with the unnumbered remark: "It seems to us certain that $U_n-(e-1)n\to\infty$ as $n\to\infty$ but we have not proved this."

The editorial note after the solution (p. 303) adds a second, separate conjecture, credited to no one by name: "Several solvers conjecture that $2n$ is an upper bound for $U_n$." The note lists the other solvers who found bounds (Bohigian, Gardner, Kelly, Lossers, Reich, Schmeichel and the proposer) without saying which of them made this conjecture. Inequality (1) gives $U_n<2n$ for all sufficiently large $n$, since $e-1<2$; this is drawn here, not in the print, and leaves the small $n$ unchecked.

Source. H. D. Ruderman (proposer), P. Erdős and E. Straus (solvers), E2232, Representation of 1 by Egyptian fractions, Amer. Math. Monthly 78 (1971), no. 3, 302--303, doi:10.2307/2317539; the remark and the editorial note on p. 303. Edition and provenance are on the source card.

Read depth. Claims checked: both sentences were read on the page image of p. 303. They are conjectures; there is no proof.

Proof pointer

None. The solution proves only the two-sided bound (1), whose lower half shows that $U_n-(e-1)n$ is bounded below.

Dependencies

None (a conjecture).

Bears on

• Problem 295: the first remark states, as a belief, the divergence that the problem asks about; the second conjecture is not the problem's question.