SPT

[[File:{{#setmainimage:Spt.jpg}}|alt=logo image|center|frameless]] SPT is a BOINC based volunteer computing project that needs your help to research Symmetric Prime Tuples.

Goal

To continue the work of the T. Brada Experimental Grid project.

Methods

The SPT code is published here: https://github.com/DemIS-1/spt

Definition 1

A prime k-tuple is a finite collection of values (p + a1, p + a2, p + a3, …, p + ak),

where p, p + a1, p + a2, p + a3, …, p + ak  are prime numbers, (a1, a2, a3, …, ak) are pattern. Typically the first value in the pattern is 0 and the rest are distinct positive even numbers. [1]

We consider the k-tuple, where p + a1, p + a2, p + a3, ..., p + ak are consecutive primes.

Definition 2

k-tuple (p + a1, p + a2, p + a3, ..., p + a [k / 2], p + a [k / 2+1], ..., p + a [k-2], p + a [k-1], p + ak) for k even, is called symmetric, if the following condition is satisfied:

a1 + ak = a2 + a[k-1] =  a3 + a[k-2] = … = a[k/2] + a[k/2+1]

Example

symmetric 8-tuple

(17 + 0, 17 + 2, 17 + 6, 17 + 12, 17 + 14, 17 + 20, 17 + 24, 17 + 26)

Shortened we write this:

17: 0, 2, 6, 12, 14, 20, 24, 26

Definition 3

k-tuple (p + a1, p + a2, p + a3, ..., p + a [(k-1) / 2], p + a [(k-1) / 2 + 1], p + a [(k-1) / 2 + 2], ..., p + a [k-2], p + a [k-1], p + ak) for k odd called symmetric, if the following condition is satisfied:

a1 + ak = a2 + a[k-1] = a3 +a [k-2] =…= a[(k-1)/2] + a[(k-1)/2+2] = 2 a[(k-1)/2+1]

Example

symmetric 5-tuple

18713: 0, 6, 18, 30, 36

(See in [2])

Definition 4

The diameter d of k-tuple is the difference of its largest and smallest elements. [1]

Example

8-tuple

17: 0, 2, 6, 12, 14, 20, 24, 26

It has a diameter d = 26.

Known solutions with a minimal diameter and a minimal value of p for k = 2, 4, 6, 8 see in [1]

Further,

k=5 ( d, p minimal)

18713: 0, 6, 18, 30, 36

k=7 (d, p minimal)

12003179: 0, 12, 18, 30, 42, 48, 60

k=9 (d, p minimal)

1480028129: 0, 12, 24, 30, 42, 54, 60, 72, 84

k=10 (d, p minimal)

13: 0, 4, 6, 10, 16, 18, 24, 28, 30, 34

k=11 (possible minimal ?)

660287401247651: 0, 6, 30, 42, 60, 66, 72, 90, 102, 126, 132

k=12 (not minimal)

137: 0, 2, 12, 14, 20, 26, 30, 36, 42, 44, 54, 56

k=13 (not minimal)

5348080416833699: 0, 12, 30, 42, 48, 72, 90, 108, 132, 138, 150, 168, 180

k=14 (not minimal)

19636011281690651: 0, 2, 8, 12, 18, 26, 30, 38, 42, 50, 56, 60, 66, 68

k=15 (not minimal)

5348080416833681: 0, 18, 30, 48, 60, 66, 90, 108, 126, 150, 156, 168, 186, 198, 216

k=16 (not minimal)

19636011281690647: 0, 4, 6, 12, 16, 22, 30, 34, 42, 46, 54, 60, 64, 70, 72, 76

k=18 (not minimal)

49549273441123: 0, 4, 24, 40, 46, 54, 58, 66, 70, 84, 88, 96, 100, 108, 114, 130, 150, 154

k=20 (not minimal)

11785542108641839: 0, 4, 10, 18, 24, 30, 52, 70, 72, 84, 118, 130, 132, 150, 172, 178, 184, 192, 198, 202

k=22 (not minimal)

18620445306703861: 0, 10, 36, 46, 66, 76, 82, 96, 102, 130, 136, 162, 168, 196, 202, 216, 222, 232, 252, 262, 288, 298

k=24 (not minimal)

22930603692243271: 0, 70, 76, 118, 136, 156, 160, 178, 202, 222, 238, 250, 378, 390, 406, 426, 450, 468, 472, 492, 510, 552, 558, 628

(See [3])

For k = 17, 19, 21, 23 solutions no found.

Questions

1. Find solutions with a minimal diameter and a minimal value of p for 10 < k < 17, k = 18, 20, 22, 24.

2. Find solutions for the remaining k, minimal or not minimal.

Problem 62. Symmetric k-tuples of consecutive primes.

Project team

Natalia Makarova. Alex Belyshev. Tomáš Brada.

Scientific results

Results of this project are available in the Prime Tuple Database.