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 p_i such that p_i 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 [math]\displaystyle{ [n+1, n+k] }[/math], then [math]\displaystyle{ \prod_{1\le x\le k}(n+x) }[/math] has at least k distinct prime divisors. The weaker version of this conjecture is equivalent to the statement that [math]\displaystyle{ k! }[/math] has at least [math]\displaystyle{ k }[/math] primes that divide it, since a product of [math]\displaystyle{ k }[/math] consecutive integers is always divisible by [math]\displaystyle{ k! }[/math] and [math]\displaystyle{ \prod_{1\le x\le k}(n+x) }[/math] is a product of [math]\displaystyle{ k }[/math] consecutive integers.

See also

• Prime gap

References

• Erdös, P.; Selfridge, J. L. (1971). "Some problems on the prime factors of consecutive integers II". Proceedings of the Washington State University Conference on Number Theory: 13–21. https://old.renyi.hu/~p_erdos/1971-24.pdf. 
• Grimm, C. A. (1969). "A conjecture on consecutive composite numbers". The American Mathematical Monthly 76 (10): 1126–1128. doi:10.2307/2317188. 
• 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. doi:10.1307/mmj/1331222852. https://projecteuclid.org/euclid.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. http://www.worldscientific.com.ezp3.lib.umn.edu/doi/abs/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". http://mathworld.wolfram.com/GrimmsConjecture.html. 

External links

• Prime Puzzles #430