Problem 961

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claims/: The 3 claim pages of Problem 961, one per claimant's result; the problem's standing derives from them.

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Statement. Let $f(k)$ be the minimal $n$ such that every set of $n$ consecutive integers $>k$ contains an integer divisible by a prime $>k$. Estimate $f(k)$.

Formulation. The site's wording (page last edited 3 April 2026). A set of consecutive integers $>k$ is a block $u+1,\ldots,u+n$ with $u\ge k$; $f(k)$ is the least block length that forces a prime factor larger than $k$, so $f(k)-1$ is the longest run of consecutive $k$-smooth integers above $k$ (the site rephrases the question as the longest possible run of consecutive $k$-smooth integers). The definition is Erdős's own from 1955 ("the least integer so that the product of $f(k)$ consecutive integers, each greater than $k$ always contains a prime greater than $k$", printed p. 124) and 1976 (printed p. 271). The site's source key is [Er76e, p. 271]. The site calls the problem essentially equivalent to Problem 683, the site's problem on the largest prime factor of a binomial coefficient; the equivalence is the site's remark and is not made precise here.

Status. Open. The site labels the problem OPEN (page last edited 3 April 2026). No source determines the order of $f(k)$; what is in hand is a two-sided bound with a large gap. Upper bounds: $f(k)\le k$ (Sylvester--Schur, reproved by Erdős in 1934; claim page), $f(k)\le c_1k/\log k$ (Erdős 1955, Theorem 1; Erdős's 1976 survey restates it as $f(k)<3k/\log k$; claim page), and $f(k)<c_1k\log\log\log k/(\log k\log\log k)$, the bound the site credits to Jutila [Ju74] and Ramachandra and Shorey [RaSh73], whose papers are not held here: it is quoted from Erdős's 1976 survey, which attributes it jointly to Jutila, Ramachandra and Shorey, and from the site (claim page). Each bound is an accepted partial claim on its journal publication. Lower bound: from Rankin's prime-gap theorem, Erdős's 1955 display (3), $f(k)>c_3\log k\log\log k\log\log\log\log k/(\log\log\log k)^2$. Erdős expected $f(k)=o(k^\epsilon)$ and probably at most a power of $\log k$ (1976), and guessed $f(k)=(1+o(1))(\log k)^2$ on Cramér's conjecture (1955). No later source improving either bound was found in the search whose scope the Current assessment records; this is a bounded negative finding, not a certificate of openness.

Source. erdosproblems.com/961, accessed 2026-09-18: the problem page (OPEN, with the site's note that no finite computation can settle it; last edited 3 April 2026; source key [Er76e, p. 271]; commentary citing [Er34], [Er55d], [Ju74], [RaSh73] and Problem 683), its one-comment discussion thread (29 January 2026) and its empty proof-claim tab. Cite as: T. F. Bloom, Erdős Problem #961, https://www.erdosproblems.com/961, accessed 2026-09-18.

References.

• [Er76e] Erdős, P., Problems and results on consecutive integers. Publ. Math. Debrecen 23 (1976), no. 3--4, 271--282, DOI 10.5486/pmd.1976.23.3-4.15 (Crossref record); the definition of $f(k)$, the two classical bounds and display (1) on printed p. 271; the reference list, printed p. 282. Library home: erdos_1976_problems_results_consecutive_integers; result page display (1).
• [Er34] Erdős, Paul, A theorem of Sylvester and Schur. J. London Math. Soc. 9 (1934), no. 4, 282--288, DOI 10.1112/jlms/s1-9.4.282 (Crossref record); the theorem, printed p. 282, its binomial form and lemma, p. 283. Library home: erdos_1934_theorem_sylvester_schur; result page theorem.
• [Er55d] Erdős, P., On consecutive integers. Nieuw Arch. Wisk. (3) 3 (1955), 124--128; Theorem 1 and displays (1)--(3), printed p. 124, display (4) and the small values, p. 125. Library home: erdos_1955_consecutive_integers; result pages Theorem 1 and inequality (3).
• [Ju74] Jutila, Matti, On numbers with a large prime factor. II. J. Indian Math. Soc. (N.S.) 38 (1974), 125--130. Not held; no Crossref record (the journal carries no DOIs for that volume); the journal's current web host is listed under Search scope (below). Quoted second-hand from [Er76e] and the site.
• [RaSh73] Ramachandra, K. and Shorey, T. N., On gaps between numbers with a large prime factor. Acta Arith. 24 (1973), no. 1, 99--111, DOI 10.4064/aa-24-1-99-111 (Crossref record). Not held; the EuDML record of the paper names a PDF at the ICM digital library. Quoted second-hand from [Er76e] and the site. [Er76e]'s reference [15] pairs these two papers with Shorey, T. N., On gaps between numbers with a large prime factor. II. Acta Arith. 25 (1974), no. 4, 365--373 (DOI 10.4064/aa-25-4-365-373), which the site does not cite.
• [vDTa26] van Doorn, W. and Tang, Q., Consecutive integers free of certain prime factors. arXiv:2606.19863v1 (18 June 2026), 5 pp. Adjacent context only: its $n_k$ concerns prime factors in $(k,2k)$ (Problem 451), not $f(k)$. Library home: doorn_2026_consecutive_integers_free_certain_prime_factors.
• [OEIS] Sloane, N. J. A., Sequence A213253, The On-Line Encyclopedia of Integer Sequences (2012; entry last modified 28 June 2026, server time): $f(n)$ for $n\le268$ (b-file from Najman's paper, not consulted here); JSON record accessed.

Formalization. Statement only. The file ErdosProblems/961.lean of formal-conjectures at the linked commit (the head of main when fetched, 2026-09-18T10:30Z) defines Erdos961Prop k n (every block of n consecutive integers starting above k contains a non-(k+1)-smooth integer) and f k as the least such n, and declares erdos_961 : answer(sorry) ↔ ∃ C : ℕ, ∀ᶠ k : ℕ in atTop, f k < log k ^ C under category research open with proof sorry: the file encodes the site's closing expectation $f(k)\ll(\log k)^{O(1)}$, not the open-ended estimate. Four research solved items record the Sylvester--Schur bound f k ≤ k (erdos_961.sylvester_schur, as Erdos961Prop k k) and the existence of f (erdos_961.variants.well_defined, proved in the file from the Sylvester--Schur item; the other three have sorry bodies), erdos_961.variants.erdos_upper_bound : ∀ᶠ k in atTop, f k < 3 * k / log k (credited to [Er55d]) and erdos_961.variants.jutila_ramachandra_shorey_upper_bound, the $O$-form of display (1); one category test item proves the case $k=n=1$. No formal_proof attribute. The community database (teorth/erdosproblems, at the linked commit of 2026-09-18) records the problem open (last changed 31 August 2025), the statement formalized since 30 December 2025, formal_status unformalized and no formal-proof URL; the site's indicator shows the statement formalized. Nothing was built.

Current assessment

The question (site formulation of 2026-09-18). The statement above; OPEN, with the site's note that no finite computation can settle it, last edited 3 April 2026. The commentary, in this page's words: it reads the question as the longest possible run of consecutive $k$-smooth integers; it attributes $f(k)\le k$ to Sylvester and Schur, citing [Er34], the bound $f(k)<3k/\log k$ to [Er55d], and the iterated-logarithm bound $f(k)\ll k\log\log\log k/(\log k\log\log k)$ to [Ju74] and [RaSh73]; it expects $f(k)$ to be bounded by a power of $\log k$; and it calls the problem essentially equivalent to Problem 683. The thread has one comment (29 January 2026), asking which bound of Problem 683 the last sentence refers to, while writing the statement for the formal-conjectures collection; it records no mathematics. The proof-claim tab is empty.

The origins. [Er55d], printed p. 124, opens with the Sylvester--Schur theorem, that for every $k$ and $n>k$ the product $n(n+1)\cdots(n+k-1)$ has a prime divisor $p>k$, restated as the fact that $k$ consecutive integers above $k$ always include a multiple of a prime above $k$, and then defines the function: "Define now $f(k)$ as the least integer so that the product of $f(k)$ consecutive integers, each greater than $k$ always contains a prime greater than $k$", so that the theorem reads $f(k)\le k$. [Er76e], printed p. 271, defines $f(k)$ the same way (as "the smallest integer so that the product of $f(k)$ consecutive integers greater than $k$ always contain [sic] a prime greater than $k$"), recalls the Sylvester--Schur bound $f(k)\le k$ and Erdős's own $f(k)<3k/\log k$ [4], and attributes to Jutila, Ramachandra and Shorey [15], as much stronger recent results improving earlier ones of Tijdeman, the display (1) $f(k)<c_1k\log\log\log k/(\log k\log\log k)$. Erdős adds that (1) "is certainly very far from the 'truth'", that "It seems sure that $f(k)=o(k^\epsilon)$ and probably $f(k)<c_1(\log k)^{c_2}$" (the constants absolute, and not necessarily equal when they share an index), and that these conjectures are out of reach, with nothing he can contribute toward them. The reference list (p. 282) resolves [4] to [Er55d] and [15] to Jutila's paper (to appear in Indian J. Math.), [RaSh73] and Shorey's 1974 sequel.

Upper bounds in hand. The Sylvester--Schur theorem ([Er34], p. 282, claims checked): if $n>k$ then one of $n,n+1,\ldots,n+k-1$ has a prime divisor greater than $k$; this is $f(k)\le k$. Erdős's proof (pp. 283--288) works with the equivalent form that $\binom nk$ has a prime divisor greater than $k$ when $n\ge2k$, from the lemma that a prime power dividing $\binom nk$ is at most $n$ (p. 283, where the case $8\le k\le\sqrt n$ is settled); pp. 284--288 are mapped on the result page, not verified. Theorem 1 of [Er55d] (p. 124, claims checked): "There is a constant $c_1>1$ so that $f(k)\le c_1\frac{k}{\log k}$ (1). In other words the sequence $u+1,u+2,\ldots,u+t$, $t=[c_1\frac{k}{\log k}]$, $u\ge k$ has at least one prime $>k$." The proof begins on p. 125 with the Hoheisel--Ingham prime-counting estimate, which settles the range $u\le k^{3/2}$; its remainder (p. 126) is a binomial-coefficient argument, sketched on the result page and not verified. The constant $3$ of the site's "$f(k)<3k/\log k$" is not in the 1955 paper, whose Theorem 1 leaves $c_1$ unspecified; it is Erdős's 1976 restatement, which cites the 1955 paper for it. A site-versus-source note, not a status matter. Display (1) of [Er76e] (result page, p. 271, claims checked), $f(k)<c_1k\log\log\log k/(\log k\log\log k)$, is the current record; Erdős attributes it to Jutila, Ramachandra and Shorey jointly, and the site to [Ju74] and [RaSh73]. Neither paper is held, so the bound, its hypotheses and its exact form rest on this attestation and the site's; the papers' own statements are not recorded here (the routes are under Search scope). The three upper bounds are accepted partial claims on their journal publications: Erdős 1934, Erdős 1955 and Jutila, Ramachandra and Shorey.

Lower bound in hand. Display (3) of [Er55d] (p. 124, claims checked): from Rankin's theorem on prime gaps there are consecutive primes $k<p_r<p_{r+1}<2k$ with $p_{r+1}-p_r>c_2\log k\log\log k\log\log\log\log k/(\log\log\log k)^2$ (display (2)); "Clearly all prime factors of the product $(p_r+1)\ldots(p_{r+1}-1)$ are less than $k$. Thus $f(k)>c_3\frac{\log k\log\log k\log\log\log\log k}{(\log\log\log k)^2}$ (3)." The argument is Erdős's: a run of composites between consecutive primes of $(k,2k)$ is a run of $k$-smooth integers above $k$, so $f(k)$ exceeds every such run's length. No later source is cited by the site for the lower bound; no source recorded here applies a later prime-gap theorem inside $(k,2k)$. On p. 125 Erdős writes that "The gap between (1) and (3) is extremely large", that $f(k)$ is probably not much larger than the largest prime gap in $(k,2k)$, hence the guess (4) $f(k)=(1+o(1))(\log k)^2$ on Cramér's conjecture, "there is of course no real evidence that (4) is true"; that $f(k)$ can be determined in finitely many steps by a theorem of Pólya and Störmer, but no effective bound is known; that he cannot prove $f$ is nondecreasing; and the values $f(2)=2$, $f(3)=f(4)=3$, $f(5)=f(6)=4$, "It seems likely that $f(7)=f(8)=f(9)=f(10)=4$, but $f(13)\ge6$". The OEIS entry A213253 (accessed; not recomputed here) lists $f(n)=1,2,3,3,4,4,4,4,4,4,4,4,6,6,\ldots$ for $n=1,2,\ldots$, which agrees with these values and guesses, and records the heuristic that $f(n)$ should be of order $(\log n)^2$.

The bounds map. With $\log_j$ the $j$-fold iterated logarithm, $c\,\log k\log_2k\log_4k/(\log_3k)^2<f(k)<c'\,k\log_3k/(\log k\log_2k)$, the lower bound attested in [Er55d] and the upper bound second-hand; the expected order is a power of $\log k$, and $f(k)=o(k^\epsilon)$ is not proved. The related function $g(k)$ of [Er55d], the least number of integers with a prime factor $>k$ among $k$ consecutive integers above $k$, satisfies $g(k)=(1+o(1))k/\log k$ (its Theorem 2, statement only); it is context, not this problem. The adjacent 2026 result of van Doorn and Tang ([vDTa26], Theorem 1.1) bounds the least $n>2k$ whose preceding $k$ integers avoid the primes of $(k,2k)$; its cutoff is the interval $(k,2k)$, not "a prime $>k$", so it neither bounds nor is bounded by $f(k)$, and it is recorded on Problem 451, not as progress here.

Search scope (2026-09-18 UTC). None of the routes below found a bound on $f(k)$ beyond those above, a copy of [Ju74] or [RaSh73], or a proof claim.

• The site: problem page, discussion thread and proof-claim tab; formal-conjectures 961.lean at the pinned commit; the community database at the commit linked under Formalization.
• arXiv: the API queries abs:"consecutive integers" AND abs:"prime factor" AND (abs:Erdos OR abs:Erdős) (two records, 1904.05096 and 1612.05438, neither on $f(k)$), abs:"consecutive integers" AND abs:smooth AND (abs:Jutila OR abs:Ramachandra OR abs:"large prime factor") and abs:"large prime factor" AND abs:"consecutive integers" (no records); the API searches titles and abstracts only, so these zeros are weak.
• Crossref: bibliographic records of [Er34], [Er76e] and [RaSh73] (with Shorey's 1974 sequel); a query for [Ju74] returned no record of the paper.
• Open-archive routes: the EuDML record 205214 for [RaSh73] (HTTP 200), whose ICM digital-library PDF link answered HTTP 403 with a bot-check page; the Indian Mathematical Society journal's web host for [Ju74] (the .com host failed at the TLS certificate check without reaching a server; the .co.in host answered HTTP 403 with a challenge page). Neither paper was obtained.
• OEIS: the JSON record of A213253.
• The primary sources: [Er34] pp. 282--283, [Er55d] pp. 124--125 and [Er76e] pp. 271 and 282; [vDTa26] pp. 1--2.

Not searched: MathSciNet, zbMATH, Google Scholar, X. Not held: [Ju74], [RaSh73], Shorey 1974, Tijdeman's earlier bounds (named in [Er76e] without a reference), Rankin 1938 (cited by [Er55d]).

Remaining gaps. (1) The record upper bound rests on two papers not held; their exact statements and hypotheses are not compared here, and the bound is quoted from Erdős's 1976 attestation and the site. Reopening condition: a readable copy of [RaSh73] (the ICM library or its PLDML record) or of [Ju74]. (2) The lower bound is the 1955 Rankin-type bound; no source was found that inserts a later prime-gap theorem into Erdős's argument, and this page does not do so. (3) The site's equivalence with Problem 683 is recorded as the site's remark; the equivalence is not derived here. (4) Proof coverage is at statement level: the proofs of [Er34] and of Theorem 1 of [Er55d] are not checked here beyond their structure. (5) The constant $3$ attributed by the site to [Er55d] is Erdős's 1976 restatement (above).

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_1934_theorem_sylvester_schur
• erdos_1934_theorem_sylvester_schur / lemma_p283
• erdos_1934_theorem_sylvester_schur / theorem
• erdos_1955_consecutive_integers
• erdos_1955_consecutive_integers / inequality_3
• erdos_1955_consecutive_integers / theorem_1
• erdos_1955_consecutive_integers / theorem_2
• doorn_2026_consecutive_integers_free_certain_prime_factors
• doorn_2026_consecutive_integers_free_certain_prime_factors / theorem_1_1
• erdos_1976_problems_results_consecutive_integers
• erdos_1976_problems_results_consecutive_integers / inequality_1