Math Department | Westfield State UniversityDepartment of Mathematics

GAPs We Have Found

We believe that all of these are firsts, except for maybe the Smallest 2x2x2 GAP. If that claim is incorrect for any of these, please let us know.

Here, we use Granville's definition that GAPs are ordered by the largest prime in the progression. In the event that there is more than one with the same largest prime, they should be ordered by their initial prime, then second largest, etc.

See below for a legend which adds detail about the information in these charts.

Size	Equation	Largest Prime	Notes
3x3	5+12i+42j	113	Smallest possible example.
3x3	29+12i+30j	113	Same largest prime as smallest possible example.
3x4	11+36i+90j	353	Smallest possible example.
3x5	23+24j+210j	911	Smallest possible example.
3x6	83+2778i+2310j	17,189	Smallest possible example.
3x7	83+570i+9660j	59,183	Smallest possible example.
3x8	16,223+173,478i+11,550j	444,029	Smallest possible example.
3x9	2,892,419 + 169,062i + 4,633,860j	40,301,423	Smallest possible example.
4x4	503+1218i+360j	5237	Smallest possible example.
4x5	607+2310i+1140j	12,097	Smallest possible example.
4x6	355,763+325,608i+90,090j	1,783,037	Smallest possible example.
4x7	unknown	unknown	Lower bound is 18,000,000; all largest primes up to that point have been tested.
5x5	219,767+145,860i+768,810j	3,878,447
2x2x2	5+6i+8j+18k	37	Smallest possible example.
2x2x2	5+2i+6j+96k	109	Smallest possible example composed entirely of twin prime pairs.
2x2x2	11+2i+6j+90k	109	Also composed only of twin prime pairs; same largest prime as smallest possible example.
3x3x3	929 + 2904i + 3150j + 7440k	27,917	Smallest possible example; See also our Introduction to GAPs
3x3x3	42,197 + 11,970i + 22,680j + 43,230k	197,957	Our first GAP: The elusive example that answered the challenge from Granville's paper.
2x2x2x2	11+6i+20j+30k+42m	109	Smallest possible example.
2x2x2x2	n/a	< 1,000	All 2x2x2x2 GAPs with largest prime less that 1,000 (There are 8,099 of them.)
2x2x2x2	n/a	< 10,000	We have computed all 2x2x2x2 GAPs with largest prime under 10,000. There are 17,957,358 of them. A copy of this file, or any portion thereof, is available on request.
2x2x2x2	29+2i+30j+42k+78m	181	Smallest possible example composed only of twin prime pairs.
2x2x2x2	n/a	< 1,000	All 2x2x2x2 GAPS composed only of twin primes with largest prime less than 1,000 (There are 23 of them.)
2x2x2x2	n/a	< 10,000	All 2x2x2x2 GAPs composed only of twin primes with largest prime less than 10,000 (There are 11,919 of them.)
2x2x2x2x2	43+66i+120j+270k+340m+378n	1217	Smallest possible example.
2x2x2x2x2	239+2i+570j+1062k+1428m+2550n	5851	Smallest possible example composed only of twin prime pairs.
2x2x2x2x2	n/a	< 12,000	All 2x2x2x2x2 GAPs with largest prime less than 12,000 (There are 14,950 of them.)
2x2x2x2x2	n/a	< 12,000	All 2x2x2x2x2 GAPs composed only of twin primes with largest prime less than 12,000 (There are 6 of them.)

Legend

Size:
This indicates the dimensions of the array of primes. For example, 3x3x3 indicates that there are 27 primes, arranged in an three-dimensional array so that there are three primes in each direction. The difference between any two primes in one of these three dimensions is always the same, as specified by the equation.
Equation:
This is a formula for finding all of the primes in the array. Each of the variables will vary from 0 to n-1, where n=the length of that dimension in the GAP. So, for example, the 2x2 array of primes:

3 5

11 13

can be described using the equation: 3+2i+8j. Each of the variables i & j vary from 0 to 1, since the dimensions are both of length 2. So, the four entries in our example correspond to:

i=0, j=0 i=1, j=0

i=0, j=1 i=1, j=1

In the 3x9 example 2,892,419 + 169,062i + 4,633,860j, this means that i varies from 0 to 2, and j varies from 0 to 8, and so the largest prime is:

2,892,419 + 2(169,062) + 8(4,633,860) = 40,301,423
Largest Prime:
This is the largest prime in array, corresponding to the entry where all of the variables are at their maximum value. This is particularly important, because GAPs are ordered by their largest prime, so when we say "Smallest possible example", we mean that the largest prime is as small as possible. One reason for this is that one can say with certainty that one has found the smallest possible example -- if GAPs were ordered by their initial prime, there is always the possibility of a larger common difference that could yield a GAP.
If we have given an example, without specifying that it is the smallest possible, then that example serves as an upper bound for the size of the smallest example.
Notes:
We have been very careful to indicate in the Notes which examples are the smallest possible, and which are just a known example, and therefore are only an upper bound for the smallest possible example.

Math Department

Westfield State University

i=0, j=0	i=1, j=0
i=0, j=1	i=1, j=1

GAPs We Have Found

Legend

Size:

Equation:

Largest Prime:

Notes: