Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 299

claims/: The 1 claim page of Problem 299, one per claimant's result; the problem's standing derives from them.

Statement. Is there an infinite sequence $a_1<a_2<\cdots $ such that $a_{i+1}-a_i=O(1)$ and no finite sum of $\frac{1}{a_i}$ is equal to $1$ ?

Formulation. The terms are positive integers, and a finite sum is a sum $\sum_{i\in S}1/a_i$ over a finite set $S$ of indices, each index used at most once. This is the setting of the poser's text: Erdős and Graham [ErGr80], printed p. 36, state the question in their section on the sets $\mathscr X$ of distinct positive integers whose reciprocals sum to $1$ (defined on p. 32), with gaps $a_{i+1}-a_i\le k$ and sums $\sum_i\varepsilon_i/a_i$ , $\varepsilon_i\in\{0,1\}$ . They state it for finite sequences $a_1<\cdots<a_t$ : probably $a_t$ is bounded in terms of $a_1$ and $k$ , but they had not excluded an infinite sequence with this property, which is the site's question. The finite form has the same answer. With $a_1$ and $k$ fixed, every prefix of such a finite sequence is again one, and each term has at most $k$ possible successors; so if $a_t$ were unbounded, König's lemma would give an infinite such sequence.

Status. Disproved, in the site's label. Every such bounded-gap sequence has a finite reciprocal sum equal to one, by Bloom's positive-upper-density theorem. The site's label is DISPROVED (LEAN); the linked formalization evidence is specified below.

Source. T. F. Bloom, Erdős Problem #299, https://www.erdosproblems.com/299, accessed 2026-09-05. The original reference is [ErGr80, p. 36]. As of that date the discussion and proof-claim pages had no comments or proof claims.

References.

[ErGr80] P. Erdős and R. L. Graham, Old and new problems and results in combinatorial number theory, Monographies de L'Enseignement Mathématique (1980), p. 36, as cited by the site.
[Bl21] T. F. Bloom, On a density conjecture about unit fractions, arXiv:2112.03726 (2021), v2 (2023), Theorem 2; with Appendix B co-written by T. F. Bloom and B. Mehta; JEMS 27 (2025), 4563–4589.
[LiSa24] Y. P. Liu and M. Sawhney, On further questions regarding unit fractions, arXiv:2404.07113v1 (10 April 2024), Theorem 1.1.

Formalization. The Bloom–Mehta Lean 3 density proof, the Lean 4 reduction from sequences to density in Boris Alexeev's lean-proofs collection, and the Google DeepMind file's sequence and density declarations with sorry bodies are described under Existing formalization; the corpus has built none of them.

Current assessment

Claims. One accepted full claim settles the problem: the bounded-gap consequence of Bloom's theorem, which rests on the accepted claim page Bloom's positive-upper-density theorem and whose acceptance evidence is the refereed publication of the density theorem in J. Eur. Math. Soc. 27 (2025). The site's curator is the theorem's author, so the site's label is recorded on the claim page as the catalog's label and not as independent review; the author's own Lean formalizations, the Lean 3 density theorem and the Lean 4 reduction, are unbuilt here and give no formalized evidence. The frontmatter standing is derived from the claim page.

Bloom's solution to Problem 298 implies a negative answer. A bounded-gap increasing sequence has a set of values of positive lower density, so the stronger upper-density theorem applies. The library's bounded-gap consequence gives the explicit counting inequality and full deduction, linking the main proof once at Theorem 2. No additional number-theoretic lemma is needed for this reduction.

The theorem page records the explicitly sourced formalization variant used to handle a parameter discrepancy in the printed technical proof. This compilation detail is separate from the problem's established negative answer.

No independent review of the route is recorded.

Quantitative context

For related quantitative progress, Liu and Sawhney's Theorem 1.1 shows that, for every fixed $\varepsilon>0$ and sufficiently large integer $N$ , any $A\subseteq\{1,\ldots,N\}$ with reciprocal mass at least $(\log N)^{4/5+\varepsilon}$ has a unit subsum. The precise quantifiers are recorded on Problem 298. Its linked proof uses corrected sufficient forms of lemmas of arXiv:2404.07113v1, whose unrestricted printed forms are false. This refinement is not needed for the bounded-gap disproof above.

Existing formalization

The Google DeepMind file expresses the sequence as strictly increasing and positive with an eventual big-O gap bound. Its sequence and density declarations have sorry bodies; the external proof tag links the Bloom–Mehta Lean 3 density proof. Appendix B reports complete formal verification of Bloom's theorem. The reduction from sequences to density is formalized in Lean 4 on top of the Lean 4 port of that development: the file src/latest/ErdosProblems/Erdos299.lean of Boris Alexeev's lean-proofs collection, posted 13 May 2026 and linked at its pinned commit on the claim page, declares itself a formalization of Bloom's solution and names Bloom as informal author and Mehta and Bloom as formal authors; its not_erdos_299 shows that a strictly increasing sequence with gaps at most $C$ has upper density at least $1/(a_0+C+1)$ and applies the density theorem, recording the axioms propext, Classical.choice and Quot.sound. A vendored copy in Jayyhk/erdos-lean restates the result in the formal-conjectures form. The corpus has built none of these.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (34)

Problem 297 Problem 298 Positive upper density forces a unit reciprocal subsum problems/unit_fractions/E0299/claims Unit Fractions unit_fractions/bloom_2021_density_conjecture_about_unit_fractions Bounded-gap consequence of Theorem 2 Corollary 1: unit reciprocal sum in a regular smooth set Lemma 1: avoiding a prime interval Lemma 2: two separated small prime divisors Lemma 3: prime powers dividing two nearby integers Lemma 4: reciprocal mass forces many prime-power components Lemma 5: extracting a large coprime divisor Lemma 6: deleting sparse prime-power fibres Lemma 7: tuning reciprocal mass while preserving fibres Proposition 1: the main density criterion Proposition 2: Fourier detection of a reciprocal subsum Proposition 3: pruning or common divisibility in short intervals Theorem 2: positive upper density supplies a unit sum Theorem 3: a quantitative reciprocal-mass criterion Theorem 4: Pomerance’s reciprocal-mass lower bound unit_fractions/liu_2024_further_questions_regarding_unit_fractions Local expansion and decay of a Bernoulli Fourier factor Source claim on integers with many prime factors Removing large prime-power divisors Sifting an interval by a set of small primes Azuma--Hoeffding inequality Positive contribution from the major Fourier arc Lemma 5.1 — Extracting a large divisor with reciprocal mass Lemma 6.1: localization of reciprocal mass Lemma 6.2: pruning thin prime-power fibers Proposition 5.2 — The main reciprocal-sum representation criterion Theorem 1.1: the four-fifths reciprocal-mass threshold Prime reciprocal and prime-power product estimates

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