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

Currently the SPT application takes up about 1.3 GB of RAM in the computer memory. 99% of this 1.3 GB is taken up by constructing a matrix of prime numbers.

The matrix of prime numbers is made by the code primesieve. Specific code snippet: https://github.com/kimwalisch/primesieve/tree/2b2c4a5c62f0cd9dfd9f56cca580ea47fa84dc2d

Original sources from Tomash Brada: https://github.com/tomasbrod/tbboinc/tree/primes/symprtu

Adapted by Demis for SPT project: https://github.com/DemIS-1/spt

The search code itself (in SPT), according to the constructed matrix, takes up only 5-6MB in memory. And it doesn't increase.

When the project started, the memory size for the matrix of one task was approximately 620MB. The memory size is now 1.3 GB and will slowly increase.

The longer the numbers, the more memory they require in the primesieve matrix.

One of our tasks takes, on average, from 45 minutes to 1.5 hours, although there are exceptions to this rule.

There are cruncher computers that count in 20 minutes and also computers that count in 240 minutes.[1]

The SPT application is based on Problem 62. Symmetric k-tuples of consecutive primes presented by Natalia Makarova.

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.

Project team

Demis. Natalia Makarova. Alex Belyshev. Tomáš Brada.

Scientific results

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