Problem 287

Let $k\geq 2$ . Is it true that, for any distinct integers $1<n_1<\cdots <n_k$ such that

1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}

we must have $\max(n_{i+1}-n_i)\geq 3$ ?

Source: erdosproblems.com/287

Number theory
Unit fractions

Wiki page

Status

Open

No claim settles this problem.

Falsifiable on the site: the label is FALSIFIABLE (page last edited 23 January 2026), which the site explains as open but refutable by one finite counterexample. The standing in the frontmatter derives from the claim pages, both pending partial claims: the Lean developments Pr_Huang 2026 (largest denominator up to $4\times10^9$ , then about $3.74\times10^{60}$ ) and Ramji 2026 (up to about $8.59\times10^{23}$ , then a 959-digit limit) claim the statement for every representation whose largest denominator is below their limits, which would settle every $k$ up to about a third of the limit, and nothing claims the statement for all $k$ or a counterexample. The site's proof-claim tab is empty.

Er32: Erdős, P., Egy Kürschák-féle elemi számelméleti tétel általánosítása [Generalization of an elementary number-theoretic theorem of Kürschák]. Matematikai és Fizikai Lapok 39 (1932), eight-page offprint. The site's key misspells the title (Kürschak, számelméti, áltadánositása) and gives the journal as MAt. es Phys. Lapok (1932).
ErGr80: Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique 28, Université de Genève (1980), pp. 33--34.
Va99: Various, Some of Paul's favorite problems. Booklet produced for the conference "Paul Erdős and his mathematics", Budapest, July 1999 (1999). Listed by the site (booklet card); item 1.15 (printed p. 3) prints the statement as an assertion, "If $a_1<a_2<\dots<a_n$ are positive integers and $\sum\frac1{a_i}=1$ , then $\max_i(a_{i+1}-a_i)>2$ ", the site's $\ge3$ .
Er50: Erdős, P., Az $1/x_1+\cdots+1/x_n=a/b$ egyenlet egész számú megoldásairól. Mat. Lapok 1 (1950), 192--210, p. 194. The earliest known statement of the conjecture.

Problem 287

Status

Proof claims

Discussion