Dominance (economics): Difference between revisions

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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==
 
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 17:47, 4 February 2014

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