Thanks to all people that have partecipate.

The winner is Jarek (see Ranking)

The result is in this table:

[insert_php]

include_once “pmg/pmg.php”;

echo pmg_db_get_result(4);

[/insert_php]

Here are the best solution arrived after the ending of the competition:

[insert_php]

echo pmg_db_get_result_extra(4);

[/insert_php]

]]>[insert_php]

include_once “pmg/pmg.php”;

echo pmg_db_get_classif(4);

[/insert_php]

]]>This competition is organized by Макарова Наталия (Natalia Makarova)

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.

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

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]

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

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]

Example: symmetric 5-tuple

18713: 0, 6, 18, 30, 36

The

diameterd 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.

A

pandiagonal magic squareis 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.

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

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

In tasks 1 and 2 k <= 50.

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

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*.

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.

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

[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

Thanks to all people that have partecipate.

The result is in this table:

[insert_php]

include_once “pmg/pmg.php”;

echo pmg_db_get_result(3);

[/insert_php]

Here are the best solution arrived after the ending of the competition:

[insert_php]

echo pmg_db_get_result_extra(3);

[/insert_php]

]]>[insert_php]

include_once “pmg/pmg.php”;

echo pmg_db_get_classif(3);

[/insert_php]

]]>This competition is organized by Макарова Наталия (Natalia Makarova)

Magic square is called ultra magic square, if it is both associative (center symmetric) and pandiagonal.

Ultra magic squares exist for orders n > 4.

In the contest is required to built ultra magic squares of order 7 – 16 of distinct primes.

Example

n = 5 113 1151 1229 911 101 839 521 41 1013 1091 941 953 701 449 461 311 389 1361 881 563 1301 491 173 251 1289 S=3505

This is the minimal solution for n = 5.

Known as a minimal solution for n = 6 (author M. Alekseyev)

103 59 163 233 139 293 229 257 307 131 13 53 283 17 67 173 181 269 61 149 157 263 313 47 277 317 199 23 73 101 37 191 97 167 271 227 S=990

For each order n = 7 – 16 you can imagine several solutions with magic constants S1 <S2 <S3 …

Known solutions:

n = 7, S = 4613 (author N. Makarova) n = 8, S = 2640 (author N. Makarova) n = 9, S = 24237 (author A. Chernov)

Contestant shall not be deemed a winner, if he would submit only solutions with known magic constants

The solution is represented in the form:

n: a (1), a (2), a (3), …, a (n^2)

Example:

5:113,1151,1229,911,101,839,521,41,1013,1091,941,953,701,449,461,311,389,1361,881,563,1301,491,173,251,1289

Contestant receives for every *n* the score: *Smin/S*, where

*Smin* – the minimal magic constant of solution in the contest;

*S* – the minimal magic constant of solution by contestant.

Winner receives a prize of 3000 rubles.

If the winner is not from Russia, the prize will be paid in $USA at the official rate on the day of end the contest.

Organizers *N. Makarova* and *S. Tognon* can participate in the contest, but does not receive the prize in case of winning.

]]>

Remember that you can still try to find new solution for the problems (without price) and in that cases the result will be presented here.

Thanks

]]>[insert_php]

include_once “pmg/pmg.php”;

echo pmg_db_get_classif(2);

[/insert_php]

]]>This competition is organized by Макарова Наталия (Natalia Makarova)

Magic square is called pandiagonal, if the sum of the numbers in each of the broken diagonal is the magic constant of the square.

For example – pandiagonal square of order 8 of prime numbers:

5 13 463 293 443 283 53 31 313 379 71 73 89 79 191 389 23 211 167 331 199 353 149 151 449 239 41 97 59 127 349 223 19 47 439 269 457 317 29 7 241 383 109 103 17 83 229 419 101 139 181 311 277 281 163 131 433 173 113 107 43 61 421 233 Magic constant of the square S = 1584

In the contest is required to construct pandiagonal squares of consecutive primes.

It is necessary to solve the problem for the orders n = 4 – 10.

Basic rule: solutions with known magic constants are not accepted.

n=4: Required to find solutions with magic constant S > 682775764735680

n=6: Required to find solutions with magic constant S > 930

- n = 4 (minimal, author Max Alekseyev)
170693941183817+ 0 116 132 164 162 134 30 86 74 42 206 90 176 120 44 72 S=682775764735680

See

http://oeis.org/A245721

http://dxdy.ru/post891839.html#p891839 - n=4 (authors J. Wroblewski and J. K. Andersen)
320572022166380833+ 0 88 16 84 76 24 60 28 78 10 94 6 34 66 18 70 S = 1282288088665523520

See

http://dxdy.ru/post751928.html#p751928

http://www.primepuzzles.net/conjectures/conj_042.htmRequired to find solutions with magic constant S > 682775764735680.

- n = 6 (minimal)
67+ 0 126 4 184 42 172 72 166 46 114 90 40 174 30 124 22 96 82 6 100 64 162 84 112 132 36 160 34 60 106 144 70 130 12 156 16 S=930

Required to find solutions with magic constant S > 930.

- For the orders n = 5, 7 – 10 unknown solutions.

Solutions should be introduced in a normalized form, plus dimension.

The first line must contains the dimension of problem (4, 5..10)

The second the normalized form of solution.

For example:

6 67: 0,126,4,184,42,172,72,166,46,114,90,40,174,30,124,22,96,82,6,100,64,162,84,112,132,36,160,34,60,106,144,70,130,12,156,16

Contestant receives one point for each new decision.

Solutions with the same magic constant are considered equal, even if they are not isomorphic.

~~Instituted a price to the participant who has won first place – $ 100 USA.~~

~~ In cases of more people will have the same final score for the first positons, the price goes to the one that makes that score before the other.~~

~~ If the winner will be the contestant from Russia, he will receive a prize in rubles at the official exchange rate on the last day of the competition.~~

The competition was taken from 23/09/2014 to 23/12/2014 but due to no winners (no one introduce a valid solution), it is now extended just for fun and not for price.

- https://ru.wikipedia.org/wiki/Магический_квадрат
- https://en.wikipedia.org/wiki/Magic_square
- B. Rosser and R. J. Walker. The algebraic theory of diabolic magic squares

http://yadi.sk/d/tl-_Ab-o5AYhS - N. Makarova. Unconventional pandiagonal squares of primes

http://www.natalimak1.narod.ru/panpr.htm

http://www.natalimak1.narod.ru/pannetr.htm

http://www.natalimak1.narod.ru/pannetr2.htm - Contest “Pandiagonal Magic Squares of Prime Numbers” (from Al Zimmermann)

http://www.azspcs.net/Contest/PandiagonalMagicSquares/FinalReport - The smallest magic constant for any n x n magic square made from consecutive primes

http://oeis.org/A073520 - The smallest magic constant of pan-diagonal magic squares which consist of distinct prime numbers

http://oeis.org/A179440

Thanks to all people that have partecipate.

The winner is Natalia, but price goes to Dmitry

The result is in this table:

[insert_php]

include_once “pmg/pmg.php”;

echo pmg_db_get_result(1);

[/insert_php]

Here are the best solution arrived after the ending of the competition:

[insert_php]

echo pmg_db_get_result_extra(1);

[/insert_php]

]]>