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| In [[mathematics]], and in particular [[number theory]], '''Grimm's conjecture''' (named after C. A. Grimm) states that to each element of a set of consecutive [[composite number]]s one can assign a distinct prime that divides it. It was first published in ''[[American Mathematical Monthly]]'', 76(1969) 1126-1128.
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| ==Formal statement==
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| Suppose ''n'' + 1, ''n'' + 2, …, ''n'' + ''k'' are all [[composite numbers]], then there are ''k'' distinct primes ''p''<sub>''i''</sub> such that ''p''<sub>''i''</sub> [[Divisor|divides]] ''n'' + ''i'' for 1 ≤ ''i'' ≤ ''k''.
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| ==Weaker version==
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| A weaker, though still unproven, version of this conjecture goes: If there is no prime in the interval <math>[n+1, n+k]</math>, then <math>\prod_{x\le k}(n+x)</math> has at least k distinct [[prime divisor]]s.
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| ==See also==
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| *[[Prime gap]]
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| ==References==
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| *{{mathworld|urlname=GrimmsConjecture|title=Grimm's Conjecture}}
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| *[[Richard K. Guy|Guy, R. K.]] "Grimm's Conjecture." §B32 in ''Unsolved Problems in Number Theory'', 3rd ed., [[Springer Science+Business Media]], pp. 133-134, 2004. ISBN 0-387-20860-7
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| [[Category:Conjectures about prime numbers]]
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Latest revision as of 16:13, 10 September 2014
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