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.

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

Formal statement

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

Weaker version

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

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

has at least $k$ distinct prime divisors.

Consequences

If Grimm's conjecture is true, then

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

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

See also

• Prime gap

References

• Erdős, 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.

• Grimm, C. A. (1969). "A conjecture on consecutive composite numbers". 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". 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.

External links

• Prime Puzzles #430