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Gap Cycle Gap Length Summation Tallies

These results are related to the paper "On Polignac's Conjecture" which can be found here.

Software used to compute these results can be found here

A data visualization of this data can be found here

Comments
File Format
File Link
Gap Cycles G(3#) to G(37#)
Max Gap Length of 500

This Excel Workbook includes all the below as well as formatting not available in the below CSV files.
Zipped Excel 2007 Workbook
Download
2.62 MB
Gap Cycle G(3#)
Zipped Text File
(CSV)
Download
1 KB
Gap Cycle G(5#)
Zipped Text File
(CSV)
Download
4 KB
Gap Cycle G(7#)
Zipped Text File
(CSV)
Download
6 KB
Gap Cycle G(11#)
Zipped Text File
(CSV)
Download
29 KB
Gap Cycle G(13#)
Zipped Text File
(CSV)
Download
42 KB
Gap Cycle G(17#)
Zipped Text File
(CSV)
Download
62 KB
Gap Cycle G(19#)
Zipped Text File
(CSV)
Download
93 KB
Gap Cycle G(23#)
Zipped Text File
(CSV)
Download
135 KB
Gap Cycle G(29#)
Zipped Text File
(CSV)
Download
189 KB
Gap Cycle G(31#)
Zipped Text File
(CSV)
Download
256 KB
Gap Cycle G(37#)
Zipped Text File
(CSV)
Download
334 KB

 

 

Simple estimates

Estimates are computed with 100 decimal precision, then rounded to the nearest integer.

Constellations
Number of Data Records
Comments
File Format
File Link


  2
  6
  8
  242
  2,10,2
  2,10,2,10,2
  66
  666
  2 (Hardy Littlewood)
  242 (Hardy Littlewood)

78,927
Estimate calculations, rounded to nearest integer up to
p = 1,005,761
Zipped text file (CSV)
Download
2.96 MB


  246
  624
  2424 
  2466
  2462
  24246
  2462642

78,927
Estimate calculations, rounded to nearest integer up to
p = 1,005,761
Zipped text file (CSV)
Download
1.39 MB

 

 

Log adjusted estimates

Log adjusted estimates are computed with 100 decimal precision, then rounded to the nearest integer.

Constellations
Number of Data Records
Comments
File Format
File Link


  2
  6
  8
  242
  2,10,2
  2,10,2,10,2
  66
  666
  2 (Hardy Littlewood)
  242 (Hardy Littlewood)

78,927
Estimate calculations, log adjusted and rounded to nearest integer up to
p = 1,005,761
Zipped text file (CSV)
Download
2.91 MB


  246
  624
  2424 
  2466
  2462
  24246
  2462642

78,927
Estimate calculations, log adjusted and rounded to nearest integer up to
p = 1,005,761
Zipped text file (CSV)
Download
1.33 MB

 

 

Actual constellation counts

Constellations counts are computed using a modified Eratosthenes sieve

Constellations
Number of Data Records
Comments
File Format
File Link


  2
  6
  8
  242
  2,10,2
  2,10,2,10,2
  66
  666

78,927
Sieve up to 1,011,555,189,121
Giving a limit of
p = 1,005,761
Zipped text file (CSV)
Download
2.38 MB


  246
  624
  2424 
  2466
  2462
  24246
  2462642
  30
  30,30,30,30

78,927
Sieve up to 1,011,555,189,121
Giving a limit of
p = 1,005,761
Zipped text file (CSV)
Download
1.84 MB

 

 

Gap cycle data

Gap cycles are computed using the recursion on the cycle of gaps

Larger gap cycles are presented as links to files for download, gap cyles grow very quickly, it is not practical to include beyond G(19#) on this website. Gaps with more than one digit are wrapped in commas.

Gap Cycle
Data
G(2#)
2
G(3#)
24
G(5#)
64242462
G(7#)
10,242462642466264264684242486462462664246264242,10,2
G(11#)
Download 0.3 KB Zipped text file
G(13#)
Download 1.7 KB Zipped text file
G(17#)
Download 28 KB Zipped text file
G(19#)
Download 561 KB Zipped text file

 

 

The computation of constellations between p and p2

Counting the number of constellations in the range [p,p2] is achieved using an algorithm designed to track gap patterns. There are three main parts to this process:

  1. A prime number sieve used to identify prime numbers
  2. A process of matching gap sequences to the constellation patterns being analyzed
  3. Storage of the running total of constellation counts at p and p2

The actual implementation also includes:

  1. The ability to compute constellation counts for multiple constellations of varied length concurrently
  2. The ability to pause and continue computations such that progress backups can be taken and the computer running the computation can periodically be used for other purposes

Our prime number sieve is currently a reasonably simple segmented sieve of Eratosthenes.

Multiple constellations are defined in a two dimensional array. Since we are tracking multiple constellations of varied length, a structure for the definition of the constellations is necessary. Below is a one dimensional representation of the definition for the constellation 2,10,2

2,10,2 = [3,2,10,2,0,2]

The first value, 3, is the number of elements in the constellation 2,10,2.

The next three values define the actual gap sequence of the constellation… 2,10,2

The value 0, after the constellation gap sequence, is a marker to define the end of the constellation gap sequence.

A one dimensional array of constellation match cursors is used to track match positions of the various constellations being analyzed, when a gap between primes matches a constellation sequence then that constellation match cursor is incremented. If the next gap in the constellation definition array is defined as a 0 then a constellation match has occurred and the total count for that constellation is incremented and the match cursor is reset, ready to match the next instance of that constellation.

The last value, 2, in our example defines the location in the constellation pattern where the match cursor should re-start at after a match occurs. This is required for constellations where overlaps are possible, for example if there is a gap sequence in the primes of 2,10,2,10,2 then we have two constellations of 2,10,2 overlapping on the middle 2. For the constellation 2,10,2 we define the reset of the constellation match cursor to start at the third element (a zero based array means that the third element in our constellation definition array is indexed by the value 2)... when we match the constellation 2,10,2 we have already matched the first 2 in a possible next instance of 2,10,2.

The total number of constellations between p and p2 is a subtraction of the total count of each constellation at p from the total count of those constellations when we reach p2. However, we need to be mindful of the extremities in the range [p, p2] and consider that the total number of constellations at p and p2 may include a constellation that starts before and ends after the extremities of the range and hence should not be included in our constellation counts. In our algorithm we need only consider this possibility at p and not p2 and of course only for multi element constellations.

We store constellation counts in a two dimensional array, two dimensional because we are analyzing multiple constellations concurrently. We use Legendres' estimate, pi(x) = x /(log x - 1.08366) to size the width of the array, this basic estimate is suitable for the ranges we are currently analyzing. The array contains as many rows as there are constellations being computed.

Our prime number sieve loops through all odd integers.

If an odd integer is prime then our algorithm enters the constellation match process and we also store the square of each prime found in an array for future reference.

If an odd integer is not prime then we look to see if we are at a prime square, if so then we perform the final calculation of subtracting p from p2, interrogating the boundary at p for a possible instance of a multi element constellation crossing at p.