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

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