# Grimm's conjecture

In mathematics, and in particular number theory, Grimm's conjecture (named after Karl Albert Grimm) 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

Suppose 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 ≤ i ≤ k.

## Weaker version

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