Grimm's conjecture

In mathematics, and in particular number theory, Grimm's conjecture (named after Carl Albert Grimm, 1 April 1926 – 2 January 2018) states that to each element of a set of consecutive composite numbers one can assign a distinct prime that divides it. It was first published in American Mathematical Monthly, 76(1969) 1126-1128.

Formal statement

If n + 1, n + 2, …, n + k are all composite numbers, then there are k distinct primes pi such that pi divides n + i for 1 ≤ i ≤ k.

Weaker version

A weaker, though still unproven, version of this conjecture goes: If there is no prime in the interval , then has at least k distinct prime divisors. The weaker version of this conjecture is equivalent to the statement that has at least primes that divide it, since a product of consecutive integers is always divisible by and is a product of consecutive integers.

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See also

References
  • 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.
  • Grimm, C. A. (1969). "A conjecture on consecutive composite numbers". The American Mathematical Monthly. 76 (10): 1126–1128. doi:10.2307/2317188. JSTOR 2317188.
  • 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
  • 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.
  • Laishram, Shanta; Shorey, T. N. (2006). "Grimm's conjecture on consecutive integers". International Journal of Number Theory. 2 (2): 207–211. doi:10.1142/S1793042106000498.
  • 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.
  • Ramachandra, K. T.; Shorey, T. N.; Tijdeman, R. (1976). "On Grimm's problem relating to factorisation of a block of consecutive integers. II". Journal für die reine und angewandte Mathematik. 288: 192–201. doi:10.1515/crll.1976.288.192.
  • Sukthankar, Neela S. (1973). "On Grimm's conjecture in algebraic number fields". Indagationes Mathematicae (Proceedings). 76 (5): 475–484. doi:10.1016/1385-7258(73)90073-5.
  • Sukthankar, Neela S. (1975). "On Grimm's conjecture in algebraic number fields. II". Indagationes Mathematicae (Proceedings). 78 (1): 13–25. doi:10.1016/1385-7258(75)90009-8.
  • 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.
  • Weisstein, Eric W. "Grimm's Conjecture". MathWorld.
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