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.

If n + 1, n + 2, ..., n + k are all composite numbers, then there are k distinct primes p_i such that p_i divides n + i for 1 ≤ i ≤ k.

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

• Prime gap

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• Prime Puzzles #430