Strong Version: If \(n+1, n+2, \ldots , n+k\) are consecutive composite numbers, then there are \(k\) distinct primes \(p_i\) such that \(p_i\) divides \(n+i\) for \(1 \leq i \leq k\).
Weaker Version: If there is no prime number in the interval \( \left[ n+1, n+k \right] \), then $$\prod_{1 \leq x \leq k}{(n+x)}$$ contains at least \(k\) prime divisors.
| Author | Date Published | |
| Carl Albert Grimm | 1969 |
[1] Grimm, C. A. (1969). "A conjecture on consecutive composite numbers". The American Mathematical Monthly. 76 (10): 1126–1128. doi:10.2307/2317188. JSTOR 2317188.
[2] Erdös, P.; Selfridge, J. L. (1971). "Some problems on the prime factors of consecutive integers II" (PDF). Proceedings of the Washington State University Conference on Number Theory: 13–21.
[3] Guy, R. K. "Grimm's Conjecture." §B32 in Unsolved Problems in Number Theory, 3rd ed., Springer Science+Business Media, pp. 133–134, 2004. ISBN 0-387-20860-7
[4] Laishram, Shanta; Murty, M. Ram (2012). "Grimm's conjecture and smooth numbers". The Michigan Mathematical Journal. 61 (1): 151–160. arXiv:1306.0765. doi:10.1307/mmj/1331222852.
[5] Ramachandra, K. T.; Shorey, T. N.; Tijdeman, R. (1975). "On Grimm's problem relating to factorisation of a block of consecutive integers". Journal für die reine und angewandte Mathematik. 273: 109–124. doi:10.1515/crll.1975.273.109.
[6] Sukthankar, Neela S. (1977). "On Grimm's conjecture in algebraic number fields-III". Indagationes Mathematicae (Proceedings). 80 (4): 342–348. doi:10.1016/1385-7258(77)90030-0.
consecutive composite numbers; prime divisors