Problem 287

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claims/: The 2 claim pages of Problem 287, one per claimant's result; the problem's standing derives from them.

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Statement. 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 $\max(n_{i+1}-n_i)\geq 3$?

Formulation. The denominators are integers greater than $1$; the site made this explicit after a comment of 26 December 2025 exhibited a twenty-term representation with negative denominators and every gap at most $2$, and the site's owner confirmed that all unit-fraction problems assume denominators above $1$ (the page was last edited 23 January 2026). The condition $k\ge2$ excludes $1=1/1$. A counterexample would be a single representation of $1$ by distinct integers above $1$ in which every consecutive gap is $1$ or $2$; the statement asserts that none exists. The representation $1=\tfrac12+\tfrac13+\tfrac16$ has gaps $1$ and $3$, so the bound $3$ cannot be raised.

Status. 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.

Source. erdosproblems.com/287, accessed 2026-09-17: the problem page (FALSIFIABLE; last edited 23 January 2026; source keys [Er32], [ErGr80], [Va99]; additional thanks to Gusarich and Terence Tao), its discussion thread (22 comments, 26 December 2025 to 10 September 2026) and its empty proof-claim tab. Cite as: T. F. Bloom, Erdős Problem #287, https://www.erdosproblems.com/287, accessed 2026-09-17.

References.

• [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.

Formalization. Statement only for the main question. The file ErdosProblems/287.lean of formal-conjectures,(the link is pinned to that commit), declares erdos_287 as answer(sorry) ↔ the statement for all $k\ge2$ and all strictly increasing s : Fin k → ℕ with 1 < s 0 and real reciprocal sum $1$, under category research open, proof sorry. The same file carries the variant gap_at_least_two (the bound $2$, research solved, with a formal_proof pointer added 2026-09-16 to the theorem gap_at_least_two of P287/gap_full.lean in the repository Zed-Rez/erdos-287-lean), the test best_possible for $(2,3,6)$ (proved by decide), the conjecture prime_conjecture (a prime $p\in[N,2N]$ with $(p+1)/2$ prime for all large $N$; research open) and prime_conjecture_implies (that conjecture implies the statement for all $k\ge k_0$; category textbook, proof sorry, with a formal_proof pointer added 2026-09-21 to PCI.prime_conjecture_implies of P287/Part3.lean in the same repository). The community database records a formalized statement since 24 June 2026 and no formal-proof URL. The finite-range Lean proofs are recorded on the claim pages; the corpus has not built any of these files.

Current assessment

The question. On 2026-09-17 the site asks the statement above, shows FALSIFIABLE, which it explains as open but refutable by a finite counterexample, cites [Er32], [ErGr80] and [Va99], and lists no proof exposition and no proof claim. Its commentary makes three points: the representation $1=\frac12+\frac13+\frac16$ shows that the bound $3$ cannot be raised; the weaker bound $2$ amounts to the fact, proved by Erdős [Er32], that $1$ is not a sum of reciprocals of consecutive integers; and the statement would hold with at most finitely many exceptions if every interval $[N,2N]$ with $N$ large contained a prime $p$ with $\frac{p+1}2$ prime. The label records the question's logical form, not its state of knowledge: a counterexample is one finite set of denominators whose gaps and reciprocal sum are checked by finite arithmetic, while a proof must cover all $k$.

Origin. Erdős stated the conjecture in 1950 (p. 194): by Kürschák's theorem the reciprocals of consecutive integers never sum to an integer, so every solution has a gap $x_{i+1}-x_i\ge2$; in his experience there is always a gap $\ge3$, which he writes he has not been able to prove; $2,3,6$ shows one cannot go further; and perhaps for every $c$ all solutions with enough terms have a gap $>c$. The 1980 monograph repeats it on printed p. 33, with $\mathcal X$ the monograph's notation for the finite sets $\{x_1<\cdots<x_n\}$ of positive integers whose reciprocals sum to $1$: the authors recall as well known that every such set has a gap $\max(x_{k+1}-x_k)>1$, since the reciprocals of consecutive integers never sum to $1$, or to any integer (citing [Th (15)], [Kü (18)] and [Er (32)]), and ask: "Is it true that $\max(x_{k+1}-x_k)\ge3$?" They add that $1=\frac12+\frac13+\frac16$ attains the bound and that they do not know whether equality occurs infinitely often, or ever again (monograph card).

What is proved. For all $k$, only the bound $2$. Two pending Lean claims, recorded under Finite ranges below, assert the statement for every $k$ up to about $10^{958}$; neither is accepted, and the corpus has not built either. Erdős's 1932 theorem (theorem (1), in the Hungarian offprint) states that $1/a+1/(a+d)+\cdots+1/(a+nd)$ is never an integer for positive integers $a,d,n$; its case $d=1$ is Kürschák's theorem (Mat. Fiz. Lapok 27 (1918), 299, cited there), and it is what excludes a representation with all gaps equal to $1$. The theorem covers complete arithmetic progressions only: denominators whose gaps mix $1$ and $2$ are not a progression, so apart from the case $d=2$ of [Er32], which excludes all gaps equal to $2$, nothing in [Er32] or [Er50] bears on the bound $3$ beyond stating it. The site attributes the bound $2$ to [Er32]; [Er32] itself credits the consecutive case to Theisinger (1915) and Kürschák (1918).

The conditional route. The monograph (pp. 33--34) says that a special case of Schinzel's hypothesis H, namely that between $x$ and $2x$ there are eventually always $k$ consecutive integers of the form $q_1,2q_2,\ldots,kq_k$ with the $q_i$ prime, implies that $\max(x_{j+1}-x_j)\le k$ "can hold for only finitely many $\{x_1,\ldots,x_n\}\in\mathcal X$". The site's commentary states the case $k=2$: a prime $p\in[N,2N]$ with $(p+1)/2$ prime for all large $N$ would give the statement with at most finitely many exceptions. The formal file records the same implication as prime_conjecture_implies, whose formal_proof pointer (2026-09-21) is the theorem PCI.prime_conjecture_implies of the development recorded as Ramji 2026. The prime conjecture is itself open, and a finite set of exceptions would still have to be excluded, so this route proves nothing unconditional; the implication is the monograph's and the site's observation, with a public Lean proof the corpus has not built, and it decides no instance of the question, so it is recorded here and on the Ramji page rather than as a conditional claim. A discussion comment of 5 May 2026 (posted as "Woett") explains the mechanism: if $n_1\ge12$ and all gaps are at most $2$ then $2n_1\le n_k\le8n_1$, and a "good" prime $p\in(M/2+1,M)$, $M=n_k$, with $(p-1)/2$ or $(p+1)/2$ prime forces $p$ or $2q$ into the denominators, where the $p$-adic or $q$-adic valuation of the sum cannot cancel; it also proposes the generalization that if for all large $N$ some $n\in[N,2N]$ has $n+j$ divisible by a prime $>3N/\log N$ for $1\le j\le m$, then all but finitely many solutions have a gap $>m$. This is a forum argument, not a published one; it decides no instance of the question and has no claim page.

Finite ranges. Two public Lean developments claim the statement for every representation whose largest denominator $M$ lies below a limit, and each is a pending partial claim: since a counterexample with $k$ terms has $n_1<k$ and $M\le3(k-1)$, a range $M\le X$ settles every $k\le X/3+1$. Pr_Huang 2026 (research note and Lean project of 26 August 2026 in the repository RexHannes/erdos-287-proof-search; $M\le4\times10^9$, extended on 10 September 2026 to about $3.74\times10^{60}$) and Ramji 2026 (the repository Zed-Rez/erdos-287-lean, an AI-generated development of 2 September 2026; $M\le8.59\times10^{23}$, extended on 21 September 2026 to a 959-digit limit). Neither is on the proof-claim tab, neither has been accepted by the site or built by the corpus, and both say the problem remains open.

Other thread items (none is progress). The discussion thread also contains finite checks and proposals, most with a disclosed AI-assistance statement, none accepted by the site or reviewed by the corpus; none is a dated manuscript, so none has a claim page. In order of strength claimed:

• An exhaustive exact-arithmetic search (5 May 2026) finds no representation with all gaps in $\{1,2\}$ and $n_1\le12$; a later comment (26 August 2026) reports none with $n_1\le15$ and none among the 199 representations with denominators $\le30$.
• "Good-prime chain" certificates (18 May and 27--28 May 2026) apply the mechanism above to chains $13<23<37<\cdots$ of primes $p$ with $p_{j+1}\le2p_j-3$ and $(p\pm1)/2$ prime, concluding that a counterexample needs $n_1>390{,}948{,}230{,}429$, then $n_1>409{,}938{,}931{,}585{,}744{,}826$, then $n_1>3.99\times10^{19}$, and (since $k\ge n_1$ and by a harmonic-sum estimate) $k\ge6.86\times10^{19}$; the primality checks are the commenters' own.
• A Lean development (27 May 2026) is said to verify the statement for each $2\le k\le18$ by finite enumeration with native_decide, leaving $k\ge19$ as sorry; the file was offered on request and is not public.

None of these changes the status: a bound on $n_1$ or on $k$ for a hypothetical counterexample is not a proof for all $k$, and the checks are the commenters' own computations; the two public Lean ranges above are recorded as claims because a verified range settles every $k$ below a third of it.

Formalization detail. The external pointer in the formal file for the bound-$2$ variant is the file P287/gap_full.lean of the repository Zed-Rez/erdos-287-lean at its first commit, which contains the theorems gap_at_least_two and gap_at_least_two_upstream_shape matching the variant's statement, with no sorry, admit or native_decide in its text; the pointer for prime_conjecture_implies is P287/Part3.lean of the same repository at its commit of 2026-09-19. These are facts about the files' text; the corpus has not built them, and no local kernel credit follows. The main statement erdos_287 has no formal proof for all $k$; its finite ranges are the two Lean claims above.

Search scope. The status rests on these routes; none found a proof, a counterexample or a proof claim.

• The site: problem page, discussion thread, proof-claim tab (empty); the community database record.
• To 2026-09-21: formal-conjectures 287.lean and the external Lean files it points to (see Formalization), and the two Lean repositories of the claim pages at the commits pinned there, the latest of 2026-09-21 (READMEs and theorem statements).
• The primary sources read as stated: [Er32] (pp. 1--8), [ErGr80] (pp. 32--34), [Er50] (p. 194).
• arXiv API metadata search (abs:"unit fractions" OR abs:"Egyptian fraction" OR abs:"Egyptian fractions") AND (abs:consecutive OR abs:gaps OR abs:gap): four records, none on this question. The API searches titles and abstracts only, so this zero is weak.
• A general web search engine: "Erdős problem 287" with the gap terms; nothing beyond the site and the arXiv items already listed.

Not searched: MathSciNet, zbMATH, Google Scholar full text, X. Not held: Theisinger (1915), Obláth (1918) and Kürschák (1918), cited by [Er32].

Proof coverage. There is nothing to compile for the statement itself. The bound $2$ rests on Erdős's 1932 theorem, recorded at statement level (claims checked; the proof has not been reviewed). The 1950 conjecture page and the monograph card record the question's history.

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.

• erdos_1980_old_new_problems_results_combinatorial_number_theory
• guy_1991_western_number_theory_problems
• guy_1991_western_number_theory_problems / problem_91_17
• various_1999_some_pauls_favorite_problems
• erdos_1932_egy_kurschak_fele_elemi
• erdos_1932_egy_kurschak_fele_elemi / theorem_1
• erdos_1950_az_egyenlet_egesz_szamu_megoldasairol_diophantine
• erdos_1950_az_egyenlet_egesz_szamu_megoldasairol_diophantine / conjectures_p194