# Competition #4

This competition is organized by Макарова Наталия (Natalia Makarova)

## Definitions

### 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.

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

### Definition 4:

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

Example: 8-tuple

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

It has a diameter d = 26.

### Definition 5:

A **pandiagonal magic square** is a magic square with the additional property that the broken diagonals, i.e. the diagonals that wrap round at the edges of the square, also add up to the magic constant.

## The Contest

In the contest is required to compete for those tasks:

**Task 1**

Required to find **k-tuples** with the minimal value p:

for an even** k > 24**; for odd **k > 15**.

Example

15-tuple, p=3945769040698829 (minimal)
3945769040698829: 0, 12, 18, 42, 102, 138, 180, 210, 240, 282, 318, 378, 402, 408, 420

**Task 2**

Required to find **k-tuples** with a **minimal diameter** *d*:

for an even** k > 10**; for odd **k > 13**.

Example

8-tuple with a minimal diameter d = 26

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

**Task 3**

Required to find the **16-tuple**, the elements of which it is possible to make **pandiagonal magic square** of order 4 with magic constant S as: 94615738903617540 < S < 29643562211780078520
Example 16-tuple

23653934725904299: 0, 12, 22, 34, 48, 60, 70, 82, 90, 102, 112, 124, 138, 150, 160, 172

pandiagonal magic square

23653934725904299+
0 160 60 124
82 102 22 138
112 48 172 12
150 34 90 70
S=94615738903617540

### Rule:

Prime numbers can contain **no more than 100 digits**.

In tasks 1 and 2 k <= 50.

## Format of solution

For every task the first line is the number of task you are entering: so 1, 2 or 3. After there are those lines according to the tasks you are inserting.

In tasks 1 and 2

k-tuple is represented as

p: a1, a2, a3, …, ak

Example

18713: 0, 6, 18, 30, 36

In task 3

it is 16-tuple and pandiagonal magic square of order 4, composed of the elements of the tuple.

Example

23653934725904299: 0, 12, 22, 34, 48, 60, 70, 82, 90, 102, 112, 124, 138, 150, 160, 172
0, 160, 60, 124, 82, 102, 22, 138, 112, 48, 172, 12, 150, 34, 90, 70

## Scoring

The contestant receives one point for every solution to tasks 1 and 3.

For each* k* in task 2 one point counted towards only those participants, who will have a minimum diameter *d*.

## The Prize

If two or more contestants have equal number of points, the winner will be the entrant who submitted solutions ahead of other contestants.

The winner will receive a prize of **5,000 rubles**.

If the winner is not from Russia, the prize will be paid in US dollars at the official exchange rate on the day ending of the contest.

## Thanks

We thanks **Wolfram Alpha** as we use their **API** for testing the primality of the given big numbers.

## Links

[1] https://en.wikipedia.org/wiki/Prime_k-tuple

[2] https://en.wikipedia.org/wiki/Pandiagonal_magic_square

[3] http://oeis.org/A256234

[4] http://oeis.org/A081235

[5] http://oeis.org/A055380

[6] http://oeis.org/A055382

[7] http://oeis.org/A175309

[8] http://www.primepuzzles.net/problems/prob_060.htm

[9] http://dxdy.ru/topic93581.html

[10] http://dxdy.ru/topic87170.html