Problems & Puzzles: Puzzles

Puzzle 80.- Twin primes - magic squares

Days ago Luis Rodríguez from Venezuela mentioned in an email (19/01/2000) the "twin primes - 3x3 magic squares" (the corresponding cells contain twin primes). According to his note he knew since many years ago one example, other was sent to him by Lee Sallows and other 10 by Mutsumi Suzuki.

As a matter of fact, they are not hard to get. According to a code I made on purpose, the smallest ever possible is this couple:

 239 17 191 241 19 193 101 149 197 Twins 103 151 199 107 281 59 Smallest 109 283 61

after which there comes hundreds of them...

Some conspicuous I produced are ending with the same digit, by example:

 7457 227 4787 7459 229 4789 1487 4157 6827 Twins 1489 4159 6829 3527 8087 857 same ending digit 3529 8089 859

My largest one (inside the capabilities of the prm function of Ubasic) is this couple :

 161879 96221 145007 161881 96223 145009 117497 134369 151241 Twins 117499 134371 151243 123731 172519 106859 my largest 123733 172519 106861

Other curios couple of magic squares that I searched, for is this one:

 219829 19309 136849 Twins +10 type 219839 19319 136859 42349 125329 208309 Consecutives 42359 125339 208319 113809 231349 30829 all ending in "9" 113819 231359 30839

Questions:

a) Can you get a twin primes - 3x3 magic square having the least prime K digits, for K= 8, 10 & 12?
b) Can you get a twin M x M magic square for M = 4, 5, 6, ...?

John E. Everett, from Waynesboro, VA has sent (24/03/2000) the following twin 4x4 example:

(the lower twin)

 101 4229 8837 1229 3461 5477 3581 1877 4637 3389 1949 4421 6197 1301 29 6869

***

John has also sent the remarkable 4x4 prime matrix A, such that B =A+2 (his prime upper twin) & C =A+B+1 is also a prime 4x4 magic square!!!... here is this formidable A magic square:

 197 4157 7559 2549 4799 6359 3167 137 4049 3329 1607 5477 5417 617 2129 6299

***

Sudipta Das sent (3/12/2001) the following interesting result

A
209977   353011   326539
413071   296509   179947
266479   240007   383041
B
209987   353021   326549
413081   296519   179957
266489   240017   383051
C
210019   353053   326581
413113   296551   179989
266527   240049   383083

B = A + 10
C = B + 32

All the elements of A , B and C are primes .
Also , the middle numbers , i.e. 296509 , 296519 , 296551 are consecutive
primes .

***

Sudipta Das also found (23/1/2002)one solution to question a) K=8:

The smallest 3 X 3 twin magic squares whose least prime is 8 digits long :

( the lower twin )

10308929       10063721       10188467

10066577       10187039       10307501

10185611       10310357       10065149

and also this one:

The smallest 3 X 3 twin magic squares whose least prime is 10 digits long :

( the lower twin )

1001994911      1000075187      1001367329

1000518227      1001145809      1001773391

1000924289      1002216431      1000296707

***

Radko Nachev wrote on April 2013:

Smallest Twins of Primes

3x3 +/-1

240 18 192
102 150 198
108 282 60

(this is the same shown above by me)

4x4+/-1

30 180 228 1062
1050 240 108 102
150 60 1092 198
270 1020 72 138

***

Natalia Makarova wrote on November 26, 2014:

I found three solutions to the puzzle # 80.

n=4 (minimal)

17  11  419  137

269  227  59  29

107  197  101  179

191  149  5  239

S=584

n=5 (not minimal ?

5  11  617  179  281

71  311  101  191  419

239  461  149  17  227

347  41  29  569  107

431  269  197  137  59

S=1093

n=6 (not minimal ?)

5 431 59 2081 197 641

239 419 1487 149 1019 101

1151 659 569 617 191 227

41 1289 809 281 137 857

1667 17 461 179 821 269

311 599 29 107 1049 1319

S=3414

***

Later, on Monday, 1 Dec, 2014 she sent this:

I found a minimal solution for n = 6 in the puzzle # 80

n = 6 (minimal)

5  17  1049  11  857  419

29  881  41  59  521  827

149  659  569  227  137  617

347  431  281  641  461  197

1019  101  239  821  71  107

809  269  179  599  311  191

S=2358

***

On Dec. 5, 2014 she added:

I found that this solution is minimal 5x5

5  11  617  179  281

71  311  101  191  419

239  461  149  17  227

347  41  29  569  107

431  269  197  137  59

S=1093

I also found the following solutions:

n=8 (minimal)

179 419 1277 239 1721 1451 71 1997

1229 821 599 1667 191 1319 1301 227

107 2141 41 2027 281 809 1931 17

29 1289 1487 101 641 1787 149 1871

1949 461 1619 857 1091 311 5 1061

2111 827 431 569 521 1607 1151 137

1481 1049 881 197 1031 11 2087 617

269 347 1019 1697 1877 59 659 1427

S=7354

n=9 (not minimal ?)

1319 5 107 71 3389 2027 2081 3251 29

11 2129 1229 521 3119 1289 881 311 2789

101 17 1049 2999 179 3371 2339 2087 137

1787 617 431 1931 239 197 2729 1019 3329

1277 2381 1481 191 2267 1667 1301 1487 227

3167 461 3299 1451 281 1061 821 1697 41

3257 2711 1877 809 659 641 347 827 1151

1091 3539 857 1619 149 1427 59 569 2969

269 419 1949 2687 1997 599 1721 1031 1607

S=12279

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

***

On Dec 8, she wrote again:

I found a minimal solution for n = 7 to a puzzle # 80.

n=7 (minimal)

419  1061  881  71  569  107  1301
17  641  821  179  1031  1289  431
1427  41  269  1151  191  521  809
1229  857  461  659  827  137  239
599  1607  347  1319  281  29  227
101  5  1481  11  1451  1049  311
617  197  149  1019  59  1277  1091

S=4409

I wrote about the puzzle here:

***
On Dec 9 she wrote this:

I found the best solution for n = 9 in the puzzle # 80.

n=9 (possible minimal ?)

1619 1487 179 2027 617 827 1949 2657 11

1151 2549 191 1061 2687 599 1697 1289 149

821 269 239 2339 857 29 1319 2789 2711

1931 1277 2801 1091 641 2111 1019 461 41

2267 1427 809 431 1301 1871 659 521 2087

1787 1229 1877 2309 311 2081 17 281 1481

1031 59 2729 101 227 1997 881 3299 1049

569 107 2129 1667 2141 137 2381 5 2237

197 2969 419 347 2591 1721 1451 71 1607

S=11373

There are only two potential magic constants, which I can not find the solution: 11353 and 11371.

I can imagine that such solutions do not exist.

***

On October 31, 2018 Paul Cleary wrote:

I have found a 12 digit twin prime magic square, I’m pretty sure its the smallest but not 100% certain, anyway here it is.

100006078651              100004158927              100005451069

100004601967              100005229549              100005857131

100005008029              100006300171              100004380447

The magic constant is 300015688647 and using the smallest of the twins.

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