ODLK2025

[[File:{{#setmainimage:Odlk2025.jpg|409x180px}}|alt=logo image|center|frameless]]ODLK2025 is a BOINC based volunteer computing project that needs your help to search for Symmetric k-Tuples of consecutive Primes.

Why ODLK2025?

ODLK2025 is a subproject of the BOINC project Symmetric Prime Tuples (SPT)

Goal

ODLK2025 solves the problem of finding symmetric tuples of consecutive prime numbers, which cannot be found in the BOINC project SPT due to the search range limitation to 2^64.

In particular, the problem of finding symmetric tuples of length 17 of consecutive prime numbers according to the following pattern

0, 6, 24, 36, 66, 84, 90, 114, 120, 126, 150, 156, 174, 204, 216, 234, 240

The existence of such tuples is a necessary condition for the existence of a symmetric tuple of length 19 of consecutive prime numbers with a minimum diameter of 252.

Currently, this subproblem is being solved in a non-BOINC project

https://boinc.termit.me/adsl/forum_thread.php?id=79

https://boinc.progger.info/odlk/forum_thread.php?id=293

The problem of finding a symmetrical 19-tuplet with a minimum diameter of 252 is being solved by a group of participants of the dxdy.ru forum.

See the topic "Symmetric tuples of consecutive prime numbers" https://dxdy.ru/topic100750.html

Methods

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.

https://www.primepuzzles.net/problems/prob_062.htm

Project team / Sponsors

• Nataliya Makarova, Project scientist
• termit, Project administrator

Results Repository

• https://boinc.mak.termit.me:5000/