Second Hardy-Littlewood conjecture
- Second Hardy-Littlewood conjecture
In number theory, the second Hardy-Littlewood conjecture concerns the number of primes in intervals. If π("x") is the number of primes up to and including "x" then the conjecture states that
:π("x" + "y") ≤ π("x") + π("y")
where "x", "y" ≥ 2.
This means that the number of primes from "x" + 1 to "x" + "y" is always less than or equal to the number of primes from 1 to "y". This is probably false in general as it is inconsistent with the more likely first Hardy-Littlewood conjecture on prime "k"-tuples, but the first violation is likely to occur for very large values of "x". For example, an admissible "k"-tuple [cite web | title=Prime pages: k-tuple | url=http://primes.utm.edu/glossary/page.php?sort=ktuple | accessdate=2008-08-12] (or prime constellation) of 447 primes can be found in an interval of "y" = 3159 integers, while π(3159) = 446. If the first Hardy-Littlewood conjecture holds, then the first such "k"-tuple is expected for "x" greater than 1.5 x 10174 but less than 2.2 x 101198 [cite web | title=447-tuple calculations | url=http://www.opertech.com/primes/residues.html | accessdate=2008-08-12] .
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