For all large $k$, can $1$ be written as a sum of reciprocals over $k$ disjoint integer intervals? (Erdős #289)
Statement
Is it true that for all sufficiently large $k$ there exist finite intervals $I_1,\ldots,I_k\subset\mathbb{N}$ — pairwise distinct, mutually non-overlapping and non-adjacent (so consecutive intervals are separated by at least one omitted integer), each with $\lvert I_i\rvert\geq 2$ — such that $$1=\sum_{i=1}^k\sum_{n\in I_i}\frac1n?$$ Here each interval $I_i=[a_i,b_i]$ contributes $\sum_{n=a_i}^{b_i}1/n$.
Acceptance. FULLY RESOLVES (proof-shaped): a complete proof that for all sufficiently large $k$ there is a family of $k$ distinct, pairwise non-overlapping and non-adjacent intervals, each of length $\geq 2$, with reciprocal sum exactly $1$ — most naturally an explicit construction valid for every $k\geq k_0$ with a stated $k_0$ — or a proof that infinitely many $k$ admit no such family. ADVANCES (each independently checkable): (a) exhibit explicit valid interval families for a run of consecutive values of $k$ (a certified table of solutions, each checkable by summing the listed reciprocals), establishing the property for all $k$ in a stated range and raising the largest $k$ known to be realisable; (b) a construction covering an infinite arithmetic progression of $k$; (c) a proof for a natural relaxation of the constraints. Deliver the proof/construction, or the search code plus certified interval families.
Background
Posed by Erdős and Graham [ErGr80]. As originally stated the distinctness, non-overlap and non-adjacency conditions were omitted, but Kovač noted (in the site comments) that without them the representation is easy, so these restrictions are surely intended. For the analogous target value $2$, Hickerson and Montgomery — in their solution to Amer. Math. Monthly problem E2689, proposed by Hahn — found $2=\sum_{i=1}^5\sum_{n\in I_i}\frac1n$ with $I_1=[2,7]$, $I_2=[9,10]$, $I_3=[17,18]$, $I_4=[34,35]$, and $I_5=[84,85]$. There is no Erdős prize attached. Listed as open on erdosproblems.com/289 (fetched 2026-07-21, status 'open'); neighbouring venue problems on interval and unit-fraction reciprocal sums include Erdős #288, #148, and #287. Attacker's tool: for each target $k$, search for admissible disjoint-interval families summing to $1$ (backtracking or ILP over interval endpoints), and look for a general construction — e.g. an interval-splitting or merging template — that yields valid families for all large $k$, which would settle the problem.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #289 (T. F. Bloom) | website |
Investigations · 0
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