Problems & Puzzles: Puzzles

Puzzle 81.- Sophie Germain  primes - magic squares

Couple of Magic Squares 3x3, Sophie Germain 1st order

 Couple of Magic Squares 3x3, Sophie Germain 2nd order

Questions:

a) Can you find smaller examples of these 3x3 Magic Squares?
b) Can you find examples of these kind of Magic Squares for higher orders: 4x4, 5x5, ..., etcetera?
c) Can you find the least triplet of 3x3 magic squares with SG primes p, 2p+1, 4p+3 (1st order) or p, 2p-1, 4p-3 (2nd order)?

Solution

On June 22, 2019, Adam Stinchcombe wrote:

I have discovered a 6x6 magic square of Sophie Germain primes of the first order (2p+1 type).  I was not searching specifically for the smallest sum, so my square only puts an upper bound on the smallest sum possible.  The ease with which it was found (about two days on a 10 year old desktop) suggests to me that there are plenty magic squares with Sophie Germain primes out there.  The smaller primes are (by row):

53, 2039, 683, 1931, 1499, 3539
3593, 1031, 359, 2399, 911, 1451
1229, 2459, 2543, 3389, 113, 11
3329, 431, 1583, 131, 3779, 491
1511, 293, 3623, 5, 3023, 1289
29, 3491, 953, 1889, 419, 2963

with the 2p+1 primes being of course:

107, 4079, 1367, 3863, 2999, 7079
7187, 2063, 719, 4799, 1823, 2903
2459, 4919, 5087, 6779, 227, 23
6659, 863, 3167, 263, 7559, 983
3023, 587, 7247, 11, 6047, 2579
59, 6983, 1907, 3779, 839, 5927

I took the set of 100 smallest Sophie Germain primes, started taking random subsets of size 36, figured out all sub-subsets of size 6 that add to the magic sum, then sought collection of six 6-sets that could fill in possible rows and columns (disjoint) and, basically, waited for a "hit."

Further directions to go are: the smallest 6x6 magic sum, larger size squares, and the (2p-1) type.  The first two seem like daunting tasks.  TBA.

...

Later, on July 25, 2019, Adam wrote again:

Finally found the 6x6  magic square of Sophie Germain primes of 2p-1 type.


                   1399, 1237, 547, 229, 3709, 3169
                   2851, 1759, 4111, 967, 271, 331
                    691, 4621, 1171, 3061, 19, 727
                  2791, 1657, 1531, 1867, 307, 2137
                   379, 79, 2719, 4027, 1627, 1459
                   2179, 937, 211, 139, 4357, 2467

It surprised me that this took so much longer than the 2p+1 type.  Probably just the randomizer.

***

On Set 20, 2020, Jos Luyendijk wrote:

You might like attached examples of Couples of Sophie Germain Squares (1st order):

 

4 x 4 Simple (Smallest Magic Sum)

5 x 5 Simple, 5 x 5  Associated,  5 x 5 Pan Magic

6 x 6 Symmetrical Main Diagonals (Small Magic Sum)

7 x 7 Concentric

Área de archivos adjuntos

Order 4 (Simple):
MC4a = 734 (p) MC4b = 1472 (2p+1)
281 113 89 251 563 227 179 503
29 359 53 293 59 719 107 587
233 239 83 179 467 479 167 359
191 23 509 11 383 47 1019 23

Order 5 (Simple):
MC5a=3805 (p) MC5b = 7615 (2p+1)
1511 293 41 1451 509 3023 587 83 2903 1019
53 1409 1031 83 1229 107 2819 2063 167 2459
419 23 761 1499 1103 839 47 1523 2999 2207
809 1439 491 113 953 1619 2879 983 227 1907
1013 641 1481 659 11 2027 1283 2963 1319 23

Order 5 (Associated):
MC5a=9055 (p) MC5b=18115 (2p+1)
3593 3491 1229 83 659 7187 6983 2459 167 1319
683 173 1481 3329 3389 1367 347 2963 6659 6779
1583 1559 1811 2063 2039 3167 3119 3623 4127 4079
233 293 2141 3449 2939 467 587 4283 6899 5879
2963 3539 2393 131 29 5927 7079 4787 263 59

Order 5 (Pan Magic):
MC5a=10729 (p) MC5b = 21463 (2p+1)
11 41 593 2963 7121 23 83 1187 5927 14243
2903 6323 1031 29 443 5807 12647 2063 59 887
1049 431 2753 6263 233 2099 863 5507 12527 467
6113 173 251 1451 2741 12227 347 503 2903 5483
653 3761 6101 23 191 1307 7523 12203 47 383

Order 6 (Symmetrical Main Diagonals):
MC6a=6726 (p) MC6b=13458 (2p+1)
1811 131 1583 653 419 2129 3623 263 3167 1307 839 4259
719 1733 1103 659 1559 953 1439 3467 2207 1319 3119 1907
1439 1019 1601 1481 173 1013 2879 2039 3203 2963 347 2027
743 1229 761 641 2063 1289 1487 2459 1523 1283 4127 2579
1901 683 1499 1223 509 911 3803 1367 2999 2447 1019 1823
113 1931 179 2069 2003 431 227 3863 359 4139 4007 863

Order 7 (Bordered):
MC7a=80297 (p)
18461 12329 3623 16811 16421 9629 3023
9029 21713 21383 3779 4211 6269 13913
8693 13649 7151 1601 15401 19553 14249
8243 1931 2549 11471 20393 21011 14699
7841 3389 7541 21341 15791 9293 15101
8111 16673 18731 19163 1559 1229 14831
19919 10613 19319 6131 6521 13313 4481

MC7b=160601 (2p+1)
36923 24659 7247 33623 32843 19259 6047
18059 43427 42767 7559 8423 12539 27827
17387 27299 14303 3203 30803 39107 28499
16487 3863 5099 22943 40787 42023 29399
15683 6779 15083 42683 31583 18587 30203
16223 33347 37463 38327 3119 2459 29663
39839 21227 38639 12263 13043 26627 8963

***