## Pandiagonal Squares of Consecutive Primes

# Competition #2

This competition is organized by Макарова Наталия (Natalia Makarova)

## Definitions

### Definition 1: Pandiagonal magic squares

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

## The Contest

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.

### Rule:

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

### Known solutions

- 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.

## Format of solution

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

## Scoring

Contestant receives one point for each new decision.

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

## The Prize

~~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.

## Help links

- 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