Grimm's conjecture


In mathematics, and in particular number theory, Grimm's conjecture 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 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 ≤ i ≤ k.

Weaker version

A weaker, though still unproven, version of this conjecture goes: If there is no prime in the interval, then has at least k distinct prime divisors. The weaker version of this conjecture is equivalent to the statement that has at least primes that divide it, since a product of consecutive integers is always divisible by and is a product of consecutive integers.