Problem 375

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 $1\leq i\leq k$ ?

Source: erdosproblems.com/375

Number theory
Primes

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Open

No claim settles this problem.

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.

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.
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.

Problem 375

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