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.

Formal statement

If are all composite numbers, then there is a sequence of distinct prime numbers such that divides for.

Weaker version

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.

Consequences

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.