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


Gap Cycle G(3#)

Zipped Text File (CSV)


Gap Cycle G(5#)

Zipped Text File (CSV)


Gap Cycle G(7#)

Zipped Text File (CSV)


Gap Cycle G(11#)

Zipped Text File (CSV)


Gap Cycle G(13#)

Zipped Text File (CSV)


Gap Cycle G(17#)

Zipped Text File (CSV)


Gap Cycle G(19#)

Zipped Text File (CSV)


Gap Cycle G(23#)

Zipped Text File (CSV)


Gap Cycle G(29#)

Zipped Text File (CSV)


Gap Cycle G(31#)

Zipped Text File (CSV)


Gap Cycle G(37#)

Zipped Text File (CSV)


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)


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)


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)


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)


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)


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)


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#)


G(13#)


G(17#)


G(19#)


The computation of constellations between p and p^{2}
Counting the number of constellations in the range [p,p^{2}] is achieved
using an algorithm designed to track gap patterns. There are three main parts to
this process:
 A prime number sieve used to identify prime numbers
 A process of matching gap sequences to the constellation patterns being analyzed
 Storage of the running total of constellation counts at p and p^{2}
The actual implementation also includes:
 The ability to compute constellation counts for multiple constellations of varied
length concurrently
 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 constellationâ€¦ 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 restart 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 p^{2} is a subtraction
of the total count of each constellation at p from the total count of those constellations
when we reach p^{2}. However, we need to be mindful of the extremities in
the range [p, p^{2}] and consider that the total number of constellations
at p and p^{2} 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 p^{2}
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 p^{2},
interrogating the boundary at p for a possible instance of a multi element constellation
crossing at p.