# Competition #5

This competition is organized by Макарова Наталия (Natalia Makarova)

## Introduction

Basic definitions of k-tuples primes can be found in [1] and [3].

### Definition 1:

A

prime k-tupleis a finite collection of values(p + a1, p + a2, p + a3, …, p + ak), wherep,p + a1,p + a2,p + a3, …,p + akare 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.

### Definition 2:

The

diameterd of k-tuple is the difference of its largest and smallest elements.

p + a1, p + a2, p + a3, …, p + ak

**Definition 2**and

**Definition 3**in [3].

### Definition 3:

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)forkeven, is calledsymmetric, if the following condition is satisfied:

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

### Definition 4:

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)forkodd calledsymmetric, 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]

We write the **k-tuple pattern** this way

0, p_2, p_3, …, p_k

where *p_k* are even numbers.

For example, a pattern for 17-tuple

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

We write the **k-tuple** this way

X: 0, p_2, p_3, …, p_k

where X is a prime number.

For example, 17-tuple (author J. Wroblewski)

1006882292528806742267: 0, 6, 24, 36, 66, 84, 90, 114, 120, 126, 150, 156, 174, 204, 216, 234, 240

The most difficult task is to find the **minimum k-tuple** with the **minimum diameter d**.

In such a k-tuple, for the minimum value of d, the value of X is the minimum.

For example, a minimum 17-tuple with a minimum diameter of 240 (author J. Wroblewski)

258406392900394343851: 0, 12, 30, 42, 60, 72, 78, 102, 120, 138, 162, 168, 180, 198, 210, 228, 240

d = 240 (minimum)

X = 258406392900394343851 (minimum)

For the competition there are 7 tasks that you can compete.

# Tasks

## Task #1

Search 17-tuple with pattern

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

Currently there are 8 known solutions (authors J. Wroblewski and D. Petukhov)

1006882292528806742267: 0 6 24 36 66 84 90 114 120 126 150 156 174 204 216 234 240 3954328349097827424397: 0 6 24 36 66 84 90 114 120 126 150 156 174 204 216 234 240 4896552110116770789773: 0 6 24 36 66 84 90 114 120 126 150 156 174 204 216 234 240 6751407944109046348063: 0 6 24 36 66 84 90 114 120 126 150 156 174 204 216 234 240 7768326730875185894807: 0 6 24 36 66 84 90 114 120 126 150 156 174 204 216 234 240 19252814175273852997757: 0 6 24 36 66 84 90 114 120 126 150 156 174 204 216 234 240 154787380396512840656507: 0 6 24 36 66 84 90 114 120 126 150 156 174 204 216 234 240 901985248981556228168767: 0 6 24 36 66 84 90 114 120 126 150 156 174 204 216 234 240

We do not guarantee that there are no missing solutions in this list.

It is required to** find missing solutions**, if they exist.

And you also need to continue the list of solutions, that is, find solutions for **X > 901985248981556228168767**.

See [7].

## Task #2

Currently two 19-tuples are known

6919940122097246303: 0, 48, 78, 138, 198, 204, 210, 264, 288, 294, 300, 324, 378, 384, 390, 450, 510, 540, 588 7325015925425379457: 0, 6, 30, 90, 126, 132, 150, 162, 216, 246, 276, 330, 342, 360, 366, 402, 462, 486, 492

Solutions found in the BOINC project SPT. [9]

It is not known whether the first solution is minimal.

We need to** find the minimum solution**, if it exists.

The **diameter can be any**.

In other words, we need to find a solution

**Х: 0, p_2, p_3, …, p_17, p_18, p_19**

for** X < 6919940122097246303**

if such a solution exists.

See [6].

## Task #3

You need to find a** 19-tuple with a minimum diameter d = 252**

###### Х: 0, 6, 12, 30, 42, 72, 90, 96, 120, 126, 132, 156, 162, 180, 210, 222, 240, 246, 252

If you guarantee that X is minimal, that’s good, but that condition doesn’t have to be true.

So far we do not know any solution for this task.

Task #1 can help you complete this task.

See [7] and [10].

## Task #4

Two 18-tuples with a minimum diameter d = 82 are known (author J. Wroblewski)

824871967574850703732309: 0, 4, 10, 12, 18, 22, 28, 30, 40, 42, 52, 54, 60, 64, 70, 72, 78, 82 2124773992554613163708029: 0, 4, 10, 12, 18, 22, 28, 30, 40, 42, 52, 54, 60, 64, 70, 72, 78, 82

It is not known whether the first solution is minimal.

Need to **find 18-tuple**

Х: 0, 4, 10, 12, 18, 22, 28, 30, 40, 42, 52, 54, 60, 64, 70, 72, 78, 82

for **X < 824871967574850703732309**, if such a solution exists.

See [11] and [12].

## Task #5

Known 20-tuple with a minimum diameter d = 94 (authors N. Makarova and J. Wroblewski)

824871967574850703732303: 0, 6, 10, 16, 18, 24, 28, 34, 36, 46, 48, 58, 60, 66, 70, 76, 78, 84, 88, 94

It is unknown whether this solution is minimal.

We need to **find the minimum solution**, if it exists.

Theoretical patterns for 20-tuple with minimum diameter

0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 60, 64, 66, 70, 76, 78, 84, 88, 90, 94 0, 4, 6, 10, 16, 18, 24, 28, 34, 36, 58, 60, 66, 70, 76, 78, 84, 88, 90, 94 0, 4, 6, 10, 16, 18, 24, 28, 36, 46, 48, 58, 66, 70, 76, 78, 84, 88, 90, 94 0, 4, 6, 10, 16, 18, 24, 30, 34, 46, 48, 60, 64, 70, 76, 78, 84, 88, 90, 94 0, 4, 6, 10, 16, 18, 24, 34, 36, 46, 48, 58, 60, 70, 76, 78, 84, 88, 90, 94 0, 6, 10, 16, 18, 24, 28, 34, 36, 46, 48, 58, 60, 66, 70, 76, 78, 84, 88, 94

See [12], [13] and [15].

## Task #6

Currently two 26-tuples are known

5179852391836338871: 0, 12, 18, 28, 46, 76, 78, 120, 186, 210, 226, 232, 238, 300, 306, 312, 328, 352, 418, 460, 462, 492, 510, 520, 526, 538 7331618973503379271: 0, 22, 30, 36, 40, 52, 58, 72, 120, 160, 190, 192, 220, 348, 376, 378, 408, 448, 496, 510, 516, 528, 532, 538, 546, 568

Solutions found in the BOINC project SPT. [9]

It is not known whether the first solution is minimal.

Need to **find 26-tuple**

**Х: 0, p_2, p_3, …, p_24, p_25, p_26**

for **X < 5179852391836338871**, if such a solution exists.

See [9].

## Task #7

Need to **find 21-tuple**

**Х: 0, p_2, p_3, …, p_19, p_20, p_21**

for any X and for any diameter d.

For minimum d = 324 there are two theoretical patterns

0, 12, 30, 42, 54, 60, 72, 84, 114, 120, 162, 204, 210, 240, 252, 264, 270, 282, 294, 312, 324 0, 12, 30, 42, 54, 60, 84, 114, 120, 144, 162, 180, 204, 210, 240, 264, 270, 282, 294, 312, 324

See [14] and [15].

## Rules

You can complete one or more of the seven suggested tasks.

You can use the algorithms and programs that you will find at the links below.

You are allowed to use any computer technology, including clusters and supercomputers.

When you send a solution in the submit entry page, to help as in verify it, please enter a tuple in this format:

T#X: 0, p_2, p_3, …, p_k

Where T is the task number and the rest is the k-tuple in standard format in a new line.

## Organizational matters

The competition begins on January 4, 2024 and will last six months (ending on July 4, 2024).

Solutions found during the competition will not be published until the end of the competition.

Solutions must be entered in the specified format only (see Rules).

Solutions will be checked by the competition organizers.

The winner of the competition will **receive a prize of 10,000 RUR.**

For a foreign winner, the amount will be converted into US dollars at the Russian exchange rate on the day the competition ends.

The winner will be determined by the competition organizers based on the totality of submitted solutions.

### Organizers

Natalia Makarova

contact **[email protected]**

Stefano Tognon

contact **[email protected]**

### Links

- Prime k-tuple

https://en.wikipedia.org/wiki/Prime_k-tuple - Problem 60. Symmetric primes on each side.

http://www.primepuzzles.net/problems/prob_060.htm - Problem 62. Symmetric k-tuples of consecutive primes

http://www.primepuzzles.net/problems/prob_062.htm - Smallest prime starting a sequence of 2n consecutive odd primes with symmetrical gaps about the center.

https://oeis.org/A055382 - Symmetric tuples of sequential primes in OEIS

https://boinc.progger.info/odlk/forum_thread.php?id=259 - Central prime p in the smallest (2n+1)-tuple of consecutive primes that are symmetric with respect to p.

https://oeis.org/A055380 - Development of a new algorithm

https://boinc.progger.info/odlk/forum_thread.php?id=268 - Symmetric tuples of sequential primes

https://dxdy.ru/topic100750.html - BOINC-project SPT

https://boinc.termit.me/adsl/ - Symmetric tuple of length 19 with minimum diameter of consecutive primes

https://boinc.progger.info/odlk/forum_thread.php?id=269 - Minimization problem 18-tuple with minimum diameter

https://boinc.progger.info/odlk/forum_thread.php?id=266 - Smallest prime starting a (nonsingular) symmetric n-tuplet of the shortest span (=A266511(n)).

https://oeis.org/A266512 - Minimization problem 20-tuple with minimum diameter

https://boinc.progger.info/odlk/forum_thread.php?id=267 - Symmetric tuple of length 21 with minimum diameter of consecutive primes

https://boinc.progger.info/odlk/forum_thread.php?id=270 - Natalia Makarova and Vladimir Chirkov, Theoretical patterns with a minimal diameter for a(2) – a(50)

https://oeis.org/A266512/a266512_1.txt

So will it be thru the Latin Squares Boinc Project or manually?

This is manually: every partecipant must find their way to obtain solution (by using whatever software they have/built or by apply mathematical property).

ODLK1 is still finding latin square.

Hello!

You can use algorithms and programs that you will find at the links provided in the competition description.

For example, for task #1 there is a program published here

https://boinc.progger.info/odlk/forum_thread.php?id=268&postid=13287

Contact me to determine your search range.