Problems & Puzzles: Puzzles

Puzzle 769. Magic squares and consecutive twin primes.

Due to his great work on the Puzzle 80, very recently I asked to Natalia Makarova for magic squares composed only by consecutive twin primes. A few days after she came with more than good news, and the best, with some questions related.

 

All of the following solutions are original by Natalia:
 

n=5 (minimal)

 

107  311  599  809  347

821  431  179  281  461

191  269  827  227  659

857  521  419  239  137

197  641  149  617  569

 

S=2173

 

This square is made up of a sequence of simple twin 107 - 857. The second square (+2):

 

109 313 601 811 349

823 433 181 283 463

193 271 829 229 661

859 523 421 241 139

199 643 151 619 571

 

S=2183

 

Further similarly.

 

n=6 (minimal)

 

419 281 269 1451 1619 1607

1667 521 1481 347 1319 311

809 1427 1487 431 461 1031

881 1277 599 1301 569 1019

1229 1049 659 827 1061 821

641 1091 1151 1289 617 857

 

S=5646

 

n=8 (minimal)

 

17 1619 1091 1697 1229 881 269 809

1319 179 857 101 347 1931 1451 1427

521 227 1289 2129 107 311 1997 1031

29 1301 1787 827 1061 71 1877 659

1607 1667 461 821 191 641 137 2087

1481 431 59 1019 2081 2111 149 281

1487 1949 197 599 569 1049 1721 41

1151 239 1871 419 2027 617 11 1277

 

S=7612

 

n=9 (minimal)

 

809 2591 419 1319 2999 827 821 2129 1931

1667 461 2801 311 179 2549 1487 3329 1061

599 1451 3167 1619 1871 617 1091 1049 2381

1151 659 2141 2111 3299 1787 569 431 1697

137 3251 521 2687 1289 2711 1019 281 1949

3257 227 149 881 2237 857 1481 2027 2729

2081 2969 1031 3119 347 101 2789 1301 107

1877 239 2339 1607 1427 2087 3359 641 269

2267 1997 1277 191 197 2309 1229 2657 1721

 

S=13845

 

n=10 (minimal)

 

41 2657 2129 149 1997 3539 827 2381 1787 1277

1721 3371 3851 179 3359 599 1451 137 2087 29

1619 1667 3671 71 191 2729 2267 2081 1871 617

3461 227 1301 461 17 1877 3581 11 3299 2549

1151 2801 1487 2111 1031 431 101 2969 881 3821

197 659 59 3119 3257 2339 3557 2789 569 239

3389 1481 2309 1949 1319 311 1061 1931 1427 1607

419 2999 821 2591 857 2711 1697 269 1091 3329

3767 281 347 2687 1229 2141 5 1049 3251 2027

1019 641 809 3467 3527 107 2237 3167 521 1289

 

S=16784

 

n=11 (minimal)

 

4721 2081 2591 3821 4229 2999 347 461 137 431 3371

3917 419 2657 1319 1607 4157 617 521 4259 4049 1667

4481 2237 2381 3851 3539 3527 1427 2309 227 569 641

3257 311 269 179 1301 3251 3389 3557 107 4931 4637

1721 4547 1931 1619 827 1487 197 4271 4421 2141 2027

2267 4001 1049 191 881 4649 4787 1787 281 2129 3167

857 2087 239 3671 2111 3119 3467 1019 3461 1229 3929

1151 1031 4337 1061 3299 101 4799 1697 4091 2801 821

809 4517 2549 4019 2339 659 3329 2729 2687 71 1481

1949 3359 4217 3581 1289 149 2789 1871 1277 2711 1997

59 599 2969 1877 3767 1091 41 4967 4241 4127 1451

 

S=25189

 

n=12 (possibly minimal ?)

 

2789 5009 1019 2657 1031 2801 461 3257 4721 4787 5477 2129

4217 1721 6449 2969 1277 4517 4127 2267 2027 1619 3467 1481

1997 3671 5657 659 3527 3851 419 227 3539 4001 2711 5879

3581 4271 269 881 2729 5501 3389 5867 2339 6551 521 239

1229 1151 311 4091 2309 4967 1949 5441 3917 5651 191 4931

4799 3821 3299 4259 5741 1487 3371 599 3167 4337 827 431

3119 347 3461 4649 641 4019 4421 6131 1427 1697 137 6089

5021 809 4637 2549 2237 2111 2687 3557 4241 179 6659 1451

4157 1607 3251 6197 2381 1061 1049 617 3929 4049 3359 4481

2081 6569 857 821 6269 2141 5417 1289 2087 1667 5639 1301

2999 1931 6359 6299 3767 1091 3329 1787 4547 281 1871 1877

149 5231 569 107 4229 2591 5519 5099 197 1319 5279 5849

 

S=36138

 

I have not found a solutions for n = 3, 4, 7.

 

Q. Can you find solutions, minimal or not, for n=3*, 4 & 7?

_____
* Solving the case for n=3 might be the most difficult case considering the solution for just consecutive primes. See Harvey Heinz, Prime Magic Squares.

 


Contribution came from Sergey Zorkin

***

Sergey wrote:

I have found solution of this puzzle for n=7, in co-working with Natalia Makarova.

Here it is:

431 2267 2237  347 1487 2087  419

2129 2027  569  461 1427 2141  521

857 827 1931 1721 881 1607 1451

1289 821 809 1949 1871 659 1877

1091 1667 1787 1019 1061 1031 1619

1997 617 641 2081 1229 599 2111

1481 1049 1301 1697 1319 1151 1277

Itís minimal.

***

Natalia Makarova wrote again on Nov 12, 2015:

n=3

 204860134660098317297 +

162 0 144

84 102 120

60 204 42

S=614580403980294952197

The author of solution Jaroslaw Wroblewski.

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