For every integer k≥ 2 and every strictly increasing sequence of natural numbers 1<s(0)<…<s(k-1) satisfying ∑_i=0^k-1frac1s(i)=1, the maximum consecutive gap max_0≤ i<k-1bigl(s(i+1)-s(i)bigr) is at least 3.

Erdős Problem #287 — Gaps in Egyptian Fraction Representations of One

About 1 year old · traced to 2025

For every integer $k\geq 2$ and every strictly increasing sequence of natural numbers $1<s(0)<\cdots<s(k-1)$ satisfying $\sum_{i=0}^{k-1}\frac{1}{s(i)}=1$ , the maximum consecutive gap

\max_{0\leq i<k-1}\bigl(s(i+1)-s(i)\bigr)

is at least $3$ .

References

Primary source

The Formal Conjectures Authors, Formal Conjectures (2025); independently edited from the pinned source file.

Additional references

Pinned Formal Conjectures source, Apache-2.0.

number-theory unit-fractions

InErdős Problems

Posted 1 month ago · edited 3 weeks ago

Progress summary

Refreshed 1 month ago

Open

No public proof, disproof, or substantive progress on this Erdős problem was found.

The problem asks whether every strictly increasing sequence of at least two positive integers whose reciprocals sum to one must contain a gap of at least three. Its machine-readable Lean 4 formulation was catalogued by the Formal Conjectures project in 2025; no proposer or original date was identified in the retrieved sources.

Current status (as of September 2026): the conjecture appears open, with no recorded public proof, disproof, or substantive progress.

Sources

No solutions have been posted yet.