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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.
 
==Formal statement==
 
Suppose ''n''&nbsp;+&nbsp;1, ''n''&nbsp;+&nbsp;2, &hellip;, ''n''&nbsp;+&nbsp;''k'' are all [[composite numbers]], then there are ''k'' distinct primes ''p''<sub>''i''</sub> such that ''p''<sub>''i''</sub> [[Divisor|divides]] ''n''&nbsp;+&nbsp;''i'' for 1&nbsp;&le;&nbsp;''i''&nbsp;&le;&nbsp;''k''.
 
==Weaker version==
 
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.
 
==See also==
*[[Prime gap]]
 
==References==
*{{mathworld|urlname=GrimmsConjecture|title=Grimm's Conjecture}}
*[[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
 
[[Category:Conjectures about prime numbers]]

Revision as of 17:27, 28 September 2013

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 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 [n+1,n+k], then ∏x≤k(n+x) has at least k distinct prime divisors.

See also

References

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  • 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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