Grimm's conjecture

In mathematics, specifically in number theory, Grimm's conjecture states that, for every set of consecutive composite numbers, there is an equally sized set of prime numbers, and a bijection that maps each composite in the former set to a prime in the latter set that it is divisible by. It was first proposed by Carl Albert Grimm in 1969.^[1]

Though still unproven, the conjecture has been verified for all ${\displaystyle n<1.9\times 10^{10}}$.^[2]

If ${\displaystyle n+1,n+2,\dots ,n+k}$ are all composite numbers, then there is a sequence of distinct prime numbers ${\displaystyle \left(p_{i}\right)_{i=1}^{k}}$ such that ${\displaystyle p_{i}}$ divides ${\displaystyle n+i}$ for ${\displaystyle 1\leq i\leq k}$.

A weaker, though still unproven, version of this conjecture states that if there is no prime in the interval ${\displaystyle [n+1,n+k]}$, then

${\displaystyle \prod _{1\,\leq \,x\,\leq \,k}(n+x)}$

has at least ${\displaystyle k}$ distinct prime divisors.^[3]

If Grimm's conjecture is true, then

${\displaystyle p_{i+1}-p_{i}\ll {\Big (}{\frac {p_{i}}{\log p_{i}}}{\Big )}^{1/2}}$

for all consecutive primes ${\displaystyle p_{i}}$ and ${\displaystyle p_{i+1}}$.^[3] This goes well beyond what the Riemann hypothesis would imply about gaps between prime numbers: the Riemann hypothesis only implies an upper bound of ${\displaystyle O(p_{i}^{1/2}(\log p_{i}))}$.^[4]

• Prime gap

• 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.

• Prime Puzzles #430