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 .
If are all composite numbers, then there is a sequence of distinct prime numbers such that divides for .
A weaker, though still unproven, version of this conjecture states that if there is no prime in the interval , then
has at least distinct prime divisors.
If Grimm's conjecture is true, then
for all consecutive primes and . This goes well beyond what the Riemann hypothesis would imply about gaps between prime numbers: the Riemann hypothesis only implies an upper bound of .