Problems & Puzzles: Puzzles

Puzzle 81.- Sophie Germain  primes - magic squares

A very natural extension of the Puzzle 80 is to ask for couples of magic squares such that the corresponding cells contain couples of Sophie Germain primes.

For the magic squares 3x3, here are what I think are the smallest examples, that I have found:

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

 106121 179 55733 212243 359 111467 3623 54011 104399 2p+1 7247 108023 208799 52289 107843 1901 104579 215687 3803

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

 8521 727 5479 17041 1453 10957 1867 4909 7951 2p-1 3733 9817 15901 4339 9091 1297 8677 18181 2593

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

Felice Russo has found smaller examples of SG 3x3 squares 1st and 2nd order:

1st Order

1481 1889 2063         2963 3779 4127
2393 1811 1229         4787 3623 2459
1559 1733 2141         3119 3467 4283

2nd Order:

3391 3697 7639      6781  7393  15277
9157 4909   661    18313  9817    1321
2179 6121 6427      4357 12241 12853

4339 1867 8521     8677  3733   17041
9091 4909   727   18181  9817    1453
1297 7951 5479     2593 15901  10957

***
I have received (8/3/2000) by snail-mail a 4x4 prime magic square Sophie-Germain 1st order type, sent by John E. Everett, form Waynesboro, VA:

 23 719 1229 1031 47 1439 2459 2063 1049 1019 281 653 2099 2039 563 1307 491 251 1451 809 2P+1 983 503 2903 1619 1439 1013 41 509 2879 2027 83 1019

He has sent also some ideas for other puzzles with prime-magic squares that soon I'll post in future puzzles pages.

***

John E. Everett sent (24/03/2000) the following 4x4 prime SG 2nd order type:

(the lower one)

 139 2179 4339 547 499 3319 1759 1627 3067 1399 727 2011 3499 307 379 3019

***

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.

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