Tables and information concerning the Hardy-Littlewood conjecture that
p(x+y) -  p(x) <= p (y)

This conjecture fails if the k-tuple conjecture is true with a value of y = 3159.
An admissible k-tuple of 447 primes can be created in an interval of 3159 integers,
while p(3159) = 446.

Exhaustive searching has verified the Hardy-Littlewood conjecture is true for intervals up to 2529.
Exhaustive searching has identified all maximum density k-tuple patterns in intervals of 3 to 2331.

2529 < y <= 3159

lower bound of  2529 < y (Apr 09)
upper bound of  y <= 3159 (Feb 05)

Verified as maximum density
p(x+1417) - p(x) = p(1417) = 223
p(x+2529) - p(x) = p(2529) = 369
p(x+2655) - p(x) = p(2655) = 384

Patterns exist - not verified as maximum
p(x+3153) - p(x) = p(3153) = 446
p(x+3157) - p(x) = p(3157) = 446

List of residues for the 447-tuples of width 3159.
Calculations on the first occurrence of 447-tuples of width 3159.

Plot of k-p(w) versus width
Plot of k/p(w/2) versus ln(w)
Trophy Case - Table of first known widths for super-dense constellations
Linked Summary of k-tuple information up to width of 42741
Also, Summary as an entire text document  (734k)

Other K-tuple information:

Dense Admissible Sets by  Daniel M. Gordon and Gene Rodemich
Admissible Prime Constellations by  Tomas Oliveira e Silva
Primes in Tuples I by D.A. Goldston, J. Pintz, and Y. Yildirim
Prime Constellation Records by Jens Kruse Anderen
Definition of  k-tuple entry from Prime Pages.
Definition of  k-tuple entry from World of Mathematics.

Sequence ID A020497 from On-Line Encyclopedia of Integer Sequences.

Problem #47 on the Prime Puzzle and Problems page.
List of residues for 50-tuples, 100-tuples, 150-tuples, and 200-tuples.

Using an exhaustive search program, written in assembler, ALL minimum
width k-tuple variations have been identified for k-values of 1 through 342.
These k-values correspond to widths of 3 through 2331.
For k-values of 343 and larger, one or more variations of the listed k-tuple exist.

© 2008 Thomas J Engelsma