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 × 10 10 {\displaystyle n<1.9\times 10^{10}} .
Formal statement If n + 1 , n + 2 , … , n + k {\displaystyle n+1,n+2,\dots ,n+k} are all composite numbers, then there is a sequence of distinct prime numbers ( p i ) i = 1 k {\displaystyle \left(p_{i}\right)_{i=1}^{k}} such that p i {\displaystyle p_{i}} divides n + i {\displaystyle n+i} for 1 ≤ i ≤ k {\displaystyle 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 ] {\displaystyle [n+1,n+k]} , then
∏ 1 ≤ x ≤ k ( n + x ) {\displaystyle \prod _{1\,\leq \,x\,\leq \,k}(n+x)}
has at least k {\displaystyle k} distinct prime divisors.
Consequences If Grimm's conjecture is true, then
p i + 1 − p i ≪ ( p i log p i ) 1 / 2 {\displaystyle p_{i+1}-p_{i}\ll {\Big (}{\frac {p_{i}}{\log p_{i}}}{\Big )}^{1/2}}
for all consecutive primes p i {\displaystyle p_{i}} and p i + 1 {\displaystyle 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 ( log p i ) ) {\displaystyle O({\sqrt {p_{i}}}(\log p_{i}))} .
See also Prime gap
Notes
References
Further reading
External links Prime Puzzles #430
