Problem 962

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claims/: The 3 claim pages of Problem 962, one per claimant's result; the problem's standing derives from them.

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Statement. Let $k(n)$ be the maximal $k$ such that there exists $m\leq n$ such that each of the integers

$$m+1,\ldots,m+k$$

are divisible by at least one prime $>k$. Estimate $k(n)$ - in particular, is it true that

$$\log k(n) \leq (\log n)^{1/2+o(1)}?$$

Formulation. The site's wording (page last edited 3 April 2026). $k(n)$ is Erdős's 1965 function (item 5 of his 1965 survey). Erdős's 1976 paper works with the inverse: $n_k$ is the least $n$ such that each of $n+1,\ldots,n+k$ has a prime factor greater than $k$; since some $m\le n$ works for $k$ exactly when $n_k\le n$, one has $k(n)=\max\{k:n_k\le n\}$, and the bounds below are translated between the two notations by that inverse (an observation made here). Two questions: the estimate of $k(n)$, to which the page-level status attaches, and the displayed question, which is the inverse form of Erdős's 1976 conjecture (7), $n_k>\exp((\log k)^{2-\epsilon})$ for every $\epsilon>0$ and large $k$. The site's source keys are [Er65] and [Er76e, p. 273].

Status. Open, for both questions. In hand: lower bounds $\log k(n)\ge(\log n)^{1/2-o(1)}$ (Erdős 1965, asserted as "not hard to prove" without proof) and, from Erdős's 1976 display (6), $n_k<k^{\log k/\log\log k}$, proved by a short smooth-number count, the bound $\log k(n)\ge(1/\sqrt2-o(1))\sqrt{\log n\log\log n}$ (a substitution made here; the site prints $\log k(n)\gg\sqrt{\log n\log\log n}$), which a forum note of December 2025 (Tang) also proves directly with the same constant; upper bounds $k(n)\le(1+o(1))n^{1/2}$ (a forum argument of October 2025 by Tao, accepted into the site's commentary and checked here) and, reported by Erdős in 1976 without proof, $n_k>k^2\exp((\log k)^c)$, that is $k(n)\le n^{1/2}\exp(-(\log n)^{c'})$; Erdős could not show $k(n)\le n^{1/2-c}$, "a ridiculously weak result". Nothing approaches the displayed question. Erdős's 1976 lower bound, his reported upper bound and Tang's dated note have claim pages: Erdős 1976 (accepted, refereed), Erdős 1976, upper bound (claimed) and Tang 2025 (claimed). Tao's argument is a thread post, not a dated manuscript, so it has no claim page and is recorded below as progress. No later source 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/962, accessed 2026-09-18: the problem page (labeled OPEN, with the site's note that no finite computation can settle it; last edited 3 April 2026; source keys [Er65], [Er76e, p. 273]; a thanks line crediting Terence Tao and Quanyu Tang), its five-comment discussion thread (12 October 2025 to 2 May 2026) and its empty proof-claim tab. Cite as: T. F. Bloom, Erdős Problem #962, https://www.erdosproblems.com/962, accessed 2026-09-18.

References.

• [Er65] Erdős, P., Extremal problems in number theory. Proc. Sympos. Pure Math. VIII (Theory of Numbers), Amer. Math. Soc. (1965), 181--189, DOI 10.1090/pspum/008/0174539; item 5, printed p. 183. Library home: erdos_1965_extremal_problems_number_theory; result page item 5.
• [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; the $n_k$ passage with displays (6) and (7), printed p. 273; the definitions, pp. 271--272. Library home: erdos_1976_problems_results_consecutive_integers; result page inequality (6).
• [Ta25] Tang, Q., An improved lower bound on Erdős Problem #962. Two-page note (pdfTeX, dated 28 December 2025 (UTC) in its metadata) in the repository QuanyuTang/erdos-problem-962 (commit of 2025-12-28, linked from the claim page); Theorems 1.2 and 2.1. A forum note, not a refereed source; not filed.
• [OEIS] Schoenfield, J. E., Sequence A327909, The On-Line Encyclopedia of Integer Sequences (2019; entry last modified 28 October 2025, server time): the least start of a run of $n$ or more integers each with a prime factor greater than $n$, for $n\le999$ (b-file); accessed 2026-09-18.

Formalization. Statement only. The file ErdosProblems/962.lean of formal-conjectures at the linked commit (the head of main on 2026-09-18) defines Erdos962Prop n k and k n := Nat.findGreatest (fun k => Erdos962Prop n k) n and declares erdos_962 : answer(sorry) ↔ ∃ ε : ℕ → ℝ, (∀ δ > 0, ∀ᶠ n in atTop, |ε n| < δ) ∧ ∀ᶠ n : ℕ in atTop, log (k n : ℝ) ≤ rpow (log n) ((1 : ℝ) / 2 + ε n) under category research open with proof sorry: the displayed question only, not the estimate. Two research solved variants with sorry bodies and no formal_proof attribute record tang_lower_bound ($(1/\sqrt2-\varepsilon(n))\sqrt{\log n\log\log n}\le\log k(n)$, citing the GitHub note) and tao_upper_bound ($k(n)\le(1+\varepsilon(n))\sqrt n$, citing the forum thread). The community database (teorth/erdosproblems, accessed 2026-09-18) lists the problem open as of its last update, of 31 August 2025, the statement formalized since 3 May 2026, formal_status unformalized, OEIS A327909 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, labeled OPEN with the note that no finite computation can settle it, last edited 3 April 2026. The commentary, in summary: it records Erdős's 1965 remarks that the lower bound $\log k(n)\ge(\log n)^{1/2-o(1)}$ is easy and that $k(n)=o(n^\epsilon)$ is likely, with no nontrivial upper bound then known; it credits his 1976 paper with an argument giving $\log k(n)\gg\sqrt{\log n\log\log n}$, a bound he considered close to sharp; it records Tao's simple argument from the thread for $k(n)\le(1+o(1))n^{1/2}$; it reports Erdős's 1976 statement that he could prove $k(n)\le\exp(-(\log n)^c)n^{1/2}$ for some $c>0$ but not $k(n)\le n^{1/2-c}$ for any $c>0$, a target he found ridiculously weak; and it records Tang's lower bound $\log k(n)\ge(\frac1{\sqrt2}-o(1))\sqrt{\log n\log\log n}$. The thread, oldest first: 12 October 2025 (the account TerenceTao), the $(1+o(1))n^{1/2}$ argument (below; the site was updated); 28 October 2025 (TerenceTao), a pointer to initial numerics in an issue of the community database's repository; 28 December 2025 (the account Quanyu Tang), the note [Ta25] (the site was updated); 3 April 2026 (the account Thomas Bloom), the finding that [Er76e] proves $\log k(n)\gg\sqrt{\log n\log\log n}$, the same order as Tang's bound, with the constant Erdős's argument gives left unchecked there, and a report of the upper-bound claims; 2 May 2026 (the account deezel, signing as Darin Dimitroff), a long conditional reduction (below). The proof-claim tab is empty.

The origins. Item 5 of [Er65], printed p. 183: "What is the largest $k=k(n)$ for which there is an $m\le n$ so that each of the integers $m+i$, $1\le i\le k$, are divisible by at least one prime $>k$? It is not hard to prove that $k(n)>\exp(\log n)^{1/2-\varepsilon}$. It seems likely that $k(n)=o(n^\varepsilon)$, but I have not been able to obtain any non-trivial upper bound for $k(n)$." The display is printed as $\exp(\log n)^{1/2-\varepsilon}$; the exponent is $(\log n)^{1/2-\varepsilon}$, as the site reads it. [[../library/primes/erdos_1976_problems_results_consecutive_integers/inequality_6|The $n_k$ passage of [Er76e]]], printed p. 273: "Denote by $n_k$ the smallest integer with $f(n_k,k)=k$" (so that all of $n_k+1,\ldots,n_k+k$ have a prime factor $>k$); the Chinese remainder theorem gives $n_k<\prod_{i=0}^{k-1}p_{s+i}$ over the consecutive primes above $k$; counting the $k$-smooth integers below $n_k$ against de Bruijn's asymptotic $U(k^\alpha,k)=(c_\alpha+o(1))k^\alpha$ gives "(6) $n_k<k^{\log k/\log\log k}$" for $k>k_0$; "I think (6) is fairly sharp. I feel sure that for every $\varepsilon>0$ and $k>k_0(\varepsilon)$ (7) $n_k>\exp((\log k)^{2-\varepsilon})$. I am very far from being able to prove (7), in fact can not even show $n_k>k^{2+\varepsilon}$ which seems a ridiculously weak result. The best that I can show is $n_k>k^2\exp((\log k)^c)$ for a certain $c>0$." Claims checked for both passages; the 1965 bound and the 1976 reported bound carry no proof in the sources.

Lower bounds. Under the inverse $k(n)=\max\{k:n_k\le n\}$, display (6) gives $\log k(n)\ge(1/\sqrt2-o(1))\sqrt{\log n\log\log n}$: with $\log k=c\sqrt{\log n\log\log n}$ the exponent $(\log k)^2/\log\log k$ of (6) equals $(2c^2+o(1))\log n$, which is at most $\log n$ for large $n$ when $c<1/\sqrt2$, so $n_k\le n$ and $k(n)\ge k$ (a substitution made here, named as such; the site prints the $\gg$ form). Tang's note [Ta25] proves the same constant directly: Theorem 2.1, for every fixed $0<c<1/\sqrt2$ and $n\ge n_0(c)$, $k(n)\ge\lfloor\exp(c\sqrt{\log n\log\log n})\rfloor$, by a pigeonhole lemma (if fewer than $\lfloor n/y\rfloor$ integers up to $n$ are $y$-smooth, some block of $y$ consecutive integers below $n$ has no $y$-smooth member) and the de Bruijn--Hildebrand estimate $\Psi(n,y)=n\rho(u)(1+o(1))$ with $\rho(u)=u^{-u+o(u)}$; its Theorem 1.2 is the $(1/\sqrt2-o(1))$ form the site prints. The argument is the 1976 argument with constants and was not checked step by step; the note has no refereed publication or independent review (claimed, on its claim page). The 1965 bound $\exp((\log n)^{1/2-\epsilon})$ is weaker and unproved in print.

Upper bounds. The forum argument of 12 October 2025 (Tao), as the site accepts it: if $k\ge(1+\varepsilon)n^{1/2}$ and $n$ is large, take a prime $p$ with $n^{1/2}<p<(1+\varepsilon)n^{1/2}$ (such primes exist by the prime number theorem, and $k>p$); the first $p$ terms $m+1,\ldots,m+p$ of the block contain a multiple of $p$, and that multiple is at most $m+p\le n+p<(1+\varepsilon)n$, so it cannot also carry a prime factor $q>k$, since then it would be at least $pq>(1+\varepsilon)n$; hence $k(n)\le(1+o(1))n^{1/2}$. The multiple is taken among the first $p$ terms so that the bound $(1+\varepsilon)n$ holds whatever the size of $k$: the last term $m+k$ of the block is below $(1+\varepsilon)n$ only when $k<\varepsilon n$. The same end is reached by first reducing to $k=\lceil(1+\varepsilon)n^{1/2}\rceil$, since a sub-block of a block whose terms all have a prime factor $>k$ is such a block for the smaller length. Both repairs were made here; the inequalities were checked here; the argument is elementary and the site calls it simple. The comment adds that it is unclear what Erdős meant by the trivial bound. Erdős's 1976 report $n_k>k^2\exp((\log k)^c)$, if granted, gives $k(n)\le n^{1/2}\exp(-(\log n)^{c'})$ for some $c'>0$ (since $\log k\ge(\log n)^{1/2-o(1)}$; a substitution made here, matching the site's translation), and the unproved $n_k>k^{2+\epsilon}$ would give $k(n)\le n^{1/2-c}$; neither has a proof in print. The displayed question is the inverse of conjecture (7): $n_k>\exp((\log k)^{2-\epsilon})$ for all $\epsilon>0$ is $\log k(n)\le(\log n)^{1/(2-\epsilon)}$, that is $\log k(n)\le(\log n)^{1/2+o(1)}$. Erdős writes that (6) is "fairly sharp", so his expectation is that the lower bound above is close to the truth.

A conditional forum reduction (lead, not status). The comment of 2 May 2026 proposes a fourth-moment argument: with $K=X^\kappa$ and $W_1(m)=\sum_{i\le K}\omega_{>K}(m+i)$, a bad block of length $2K$ forces many $m$ with $W_1(m)\ge K$, so a bound $M_4(W_1)\ll XK^{2+o(1)}$ uniformly for $\kappa>1/e$ would give $k(n)\ll n^{1/e+o(1)}$; the fourth moment reduces to a variance bound and a level-4 sieve discrepancy (its L5 and L6), which the author supports numerically and cannot prove, stating explicitly that no theorem improving Erdős's 1976 bound $k(n)\le\exp(-(\log n)^c)n^{1/2}$ is claimed. Nothing about it was checked here, the site's commentary does not mention it, and it changes no bound.

The bounds map. $(1/\sqrt2-o(1))\sqrt{\log n\log\log n}\le\log k(n)\le\tfrac12\log n+o(1)$, the lower bound from [Er76e] (translated here) and [Ta25], the upper bound from the thread; Erdős's conjecture puts the truth at the lower end, $\log k(n)=(\log n)^{1/2+o(1)}$. The OEIS entry A327909 (accessed 2026-09-18; not recomputed) lists the least start $m+1$ of a run of $n$ or more integers each with a prime factor greater than $n$: $2,5,13,19,55,65,113,151,151,226,364,406,736,736,1057,\ldots$ for $n=1,2,\ldots$, that is $n_k+1$ in Erdős's notation; a data lead only.

Search scope. None of the routes below found a bound on $k(n)$ beyond those above, a proof of Erdős's reported upper bound, or a refereed version of the forum results.

• The site: problem page, discussion thread and proof-claim tab; formal-conjectures 962.lean at the pinned commit; the community database.
• The note [Ta25], in full.
• 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 $k(n)$), 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: the bibliographic records of [Er65] and [Er76e].
• OEIS: the JSON record of A327909.
• The primary sources: [Er65] p. 183 and [Er76e] pp. 271--273.

Not searched: MathSciNet, zbMATH, Google Scholar, X; the community database's issue tracker (the numerics comment). Not held: de Bruijn 1951 (cited by [Er76e]); Hildebrand's smooth-number estimate (cited by [Ta25]).

Remaining gaps. (1) The upper bound $k(n)\le n^{1/2}\exp(-(\log n)^{c'})$ rests on Erdős's 1976 sentence "The best that I can show", with no proof found in print; until one is found, the proved upper bound is $(1+o(1))n^{1/2}$ from the forum argument. (2) The lower bound's constant $1/\sqrt2$ rests on a forum note and on a substitution made here in Erdős's display (6); no refereed source states it. (3) The displayed question, Erdős's conjecture (7), is untouched; the 2 May 2026 comment's route is conditional on unproved sieve estimates. (4) Proof coverage is at statement level; the smooth-number computation behind (6) was not carried out here.

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_1965_extremal_problems_number_theory
• erdos_1965_extremal_problems_number_theory / item_5
• erdos_1976_problems_results_consecutive_integers
• erdos_1976_problems_results_consecutive_integers / inequality_6