Green-Tao theorem: Difference between revisions
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==History== | ==History== | ||
This result is a special case of the [[Erdős conjecture on arithmetic progressions]], which states that any [[large set]] contains | This result is a special case of the [[Erdős conjecture on arithmetic progressions]], which states that any [[large set]] contains aribtrarily long arithmetic progression. (It is a special case because the [[set of primes is large]]). | ||
==Statement== | ==Statement== | ||
This states that for any positive integer <math>k</math>, there exists a [[prime arithmetic progression]] of length <math>k</math>, i.e., an arithmetic progression of length <math>k</math> all of whose members | This states that for any positive integer <math>k</math>, there exists a [[fact about::prime arithmetic progression]] of length <math>k</math>, i.e., an [[fact about::arithmetic progression]] of length <math>k</math> all of whose members are [[prime number]]s. | ||
==Relation with other facts/conjectures== | ==Relation with other facts/conjectures== | ||
===Stronger facts=== | ===Stronger facts/conjectures=== | ||
* [[Weaker than::Erdős conjecture on arithmetic progressions]] | * [[Weaker than::Erdős conjecture on arithmetic progressions]] | ||
* [[Weaker than::Tao-Ziegler theorem on primes in polynomial progressions]] | * [[Weaker than::Tao-Ziegler theorem on primes in polynomial progressions]] | ||
* [[Weaker than::Zhou's theorem on arbitrarily large arithmetic progressions of Chen primes]] | |||
Latest revision as of 02:07, 2 May 2010
History
This result is a special case of the Erdős conjecture on arithmetic progressions, which states that any large set contains aribtrarily long arithmetic progression. (It is a special case because the set of primes is large).
Statement
This states that for any positive integer , there exists a prime arithmetic progression of length , i.e., an arithmetic progression of length all of whose members are prime numbers.