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 pi such that pi 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 [ n + 1 , n + k ] {\displaystyle [n+1,n+k]} , then ∏ ∏ --> 1 ≤ ≤ --> x ≤ ≤ --> k ( n + x ) {\displaystyle \prod _{1\leq x\leq k}(n+x)} has at least k distinct prime divisors.
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