Problem 375

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claims/: The 3 claim pages of Problem 375, one per claimant's result; the problem's standing derives from them.

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Statement. Is it true that for any $n,k\geq 1$, if $n+1,\ldots,n+k$ are all composite then there are distinct primes $p_1,\ldots,p_k$ such that $p_i\mid n+i$ for $1\leq i\leq k$?

Status. Falsifiable: the site's label (page last edited 24 January 2026, accessed 2026-10-07); the question is open, with no proof and no counterexample recorded, and the label is explained as a body note under Current assessment.

Source. erdosproblems.com/375, accessed 2026-09-04 and 2026-10-07. Cite as: T. F. Bloom, Erdős Problem #375, https://www.erdosproblems.com/375.

References.

• [Gr69] Grimm, C. A., A conjecture on consecutive composite numbers. Amer. Math. Monthly (1969), 1126-1128.
• [Gu04] Guy, Richard K., Unsolved problems in number theory, 3rd ed. Problem Books in Mathematics, Springer (2004), xviii+437 pp. B32 "Grimm's conjecture", printed p. 133: the conjecture as stated above, two worked examples, and the report that Ramachandra, Shorey and Tijdeman proved finitely many exceptions under a hypothesis of Schinzel; no proofs. Library home: guy_2004_unsolved_problems_number_theory.
• [LaSh06] Laishram, Shanta and Shorey, T. N., Grimm's conjecture on consecutive integers. Int. J. Number Theory (2006), 207-211.
• [RST75] Ramachandra, K. and Shorey, T. N. and Tijdeman, R., On Grimm's problem relating to factorisation of a block of consecutive integers. J. Reine Angew. Math. (1975), 109-124.

Formalization. Statement in formal-conjectures.

Current assessment

Scope. The section draws on the site's page, its discussion thread (seven comments as of 2026-10-07), the two sources with library cards named under References, [Gu04] and [LaSh06], and the papers of the claim pages; no wider literature search has been made. The site's label is a body note, not a claim: the problem is marked falsifiable because a counterexample, a run of composites $n+1,\ldots,n+k$ whose terms admit no system of distinct prime divisors, would be a finite check, while a proof would not; no counterexample and no proof is recorded. Three refereed partial results have claim pages, each accepted: Grimm (1969), for runs of length up to $\log n/(2\log\log n)$ once $n$ is large; Ramachandra, Shorey and Tijdeman (1975), for runs of length up to $\alpha_3(\log n/\log\log n)^3$; and Laishram and Shorey (2006), for every run after $n\leq19236701629$. None settles the question, so the problem stays open. Erdős and Selfridge's range $k\leq(1+o(1))\log n$ appeared in the proceedings of the 1971 Washington State University conference on number theory (Pullman), 13--21, a volume with no record of refereeing, and for all large $n$ it lies inside the range of Ramachandra, Shorey and Tijdeman, so it has no claim page. The thread's verifications to $10^{12}$ and $10^{13}$ are thread posts with code archives, not dated manuscripts, so they have no claim pages.

What the sources record. The site's commentary (accessed 2026-10-07): the statement is Grimm's conjecture of 1969 and is trivial for $k\le2$; it implies a prime-gap bound $p_{n+1}-p_n<p_n^{1/2-c}$ for some $c>0$, which the site says would in particular resolve Legendre's conjecture, and which is why the site calls it very difficult (the bound gives a prime between consecutive squares for all large $n$, the asymptotic form of Legendre's conjecture, as the thread comment of 16 January 2026 notes); Grimm proved it for $k\ll\log n/\log\log n$, Erdős and Selfridge for $k\le(1+o(1))\log n$, and Ramachandra, Shorey and Tijdeman [RST75] for $k\ll(\log n/\log\log n)^3$; Laishram and Shorey [LaSh06] verified it for every $k$ and all $n\le1.9\times10^{10}$; Guy's B32 discusses it, and Problem 860 is related. The two sources: Guy's B32 (printed p. 133) states the conjecture with two worked examples and reports the conditional result of Ramachandra, Shorey and Tijdeman that there are finitely many exceptions under a hypothesis of Schinzel; the Laishram--Shorey paper, digested on its card, reduces the verification to the runs of composites between consecutive primes $p_N$ with $N\le8.5\times10^8$ and reports a computation covering all $n\le p_{N_0}=19236701629$. The thread: two posts report computational verifications beyond [LaSh06], for every maximal run of composites whose closing prime is below $10^{12}$ (27 August 2026, longest run $k=539$) and below $10^{13}$ (1 October 2026, longest run $k=673$), each describing itself as a finite verification and not a proof, with code and logs linked from the posts; a comment of 16 January 2026 says that the provenance of the prime-gap implication is unclear, since the 1971 paper of Erdős and Selfridge proves only $p_{n+1}-p_n\ll p_n^{1/2}/(\log p_n)^{1/2}$ and asserts the power saving without details. The formal-conjectures statement file, at the commit linked, states the problem with a sorry body and records as variants the prime-gap bound, also with a sorry body, the asymptotic form of Legendre's conjecture derived from it, a proof of the cases $k\le2$, and the Ramachandra--Shorey--Tijdeman range $k\ll(\log n/\log\log n)^3$ of [RST75], also with a sorry body.

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.

• laishram_2006_grimm_s_conjecture_consecutive_integers
• laishram_2006_grimm_s_conjecture_consecutive_integers / theorem_1
• laishram_2006_grimm_s_conjecture_consecutive_integers / theorem_2
• guy_2004_unsolved_problems_number_theory
• ramachandra_1976_grimm_s_problem_relating_factorisation_block
• ramachandra_1976_grimm_s_problem_relating_factorisation_block / theorem_1