Grimm's conjecture

In number theory, Grimm's conjecture (named after Carl Albert Grimm, 1 April 1926 – 2 January 2018) states that to each element of a set of consecutive composite numbers one can assign a distinct prime that divides it. It was first published in American Mathematical Monthly, 76(1969) 1126-1128.

Formal statement
If n + 1, n + 2, ..., n + k are all composite numbers, then there are k distinct primes pi such that pi divides n + i for 1 &le; i &le; k.

Weaker version
A weaker, though still unproven, version of this conjecture states: If there is no prime in the interval $$[n+1, n+k]$$, then $$\prod_{1\le x\le k}(n+x)$$ has at least k distinct prime divisors.