[seqfan] A020497 name needs improvement
davidwwilson at comcast.net
Tue Jun 4 07:28:46 CEST 2013
I think the description of A020497 is incorrect as it is conjectural.
I believe that A020497(n) gives smallest k such that k consecutive integers
admits a permissible prime pattern of size n.
The first Hardly-Littlewood conjecture (the k-tuples conjecture) implies
that an admissible prime pattern is satisfied by an infinite number of prime
constellations (which implies there are an infinite number of values x with
pi(x+a(n)) - p(x) = n). However, this is still conjectural, and has not been
proved even for the admissible pattern (0,2), which is to say, we have not
yet proved the twin prime conjecture. For sufficiently large k (where
sufficiently large is not very large), we can neither demonstrate nor prove
that a single prime constellation satisfies a maximal admissible prime
pattern of size n.
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