I am happy to announce the first known BIMAGIC square of primes. And its order is also prime: 11. See this page by Boyer.

Reminder. A bimagic square of order n is a nxn magic square remaining magic after each of its n² numbers have been squared. 

137	131	317	47	5	457	541	359	467	353	683
401	277	239	647	23	421	229	181	7	419	653
463	269	701	59	157	257	563	557	179	191	101
593	311	379	503	197	83	53	521	149	619	89
307	617	397	241	571	661	109	107	79	127	281
373	443	29	587	383	61	19	409	631	389	173
73	11	607	433	613	577	263	97	227	313	283
43	599	151	199	509	487	223	163	293	691	139
673	37	113	271	193	31	601	431	331	337	479
67	233	103	439	499	251	547	659	491	41	167
367	569	461	71	347	211	349	13	643	17	449

Characteristics:

-121 distinct prime integers, and more precisely
 121 consecutive prime integers <= 701, only excluding 2, 3, 523, 641, 677
-bimagic square (11 rows, 11 columns, 2 diagonals) with magic sums S1=3497, S2=1578251
It is the first solved problem on the 10 open problems published in my Math Intelligencer article, Spring 2005.

But it means also that 9 problems are still unsolved. A lot of work remains to be done, including "small" unsolved problems: 

- 3x3 magic square of squares (only semi-magic are known)
- 4x4 magic square of cubes (only semi-magic are known)
- 5x5 bimagic square (only semi-bimagic are known)

...questions 1 and 2 of puzzle 287 are still unsolved:
-nobody knows a better 4x4 example
-nobody knows a 4x4, 5x5,… 10x10 bimagic square of primes. 

However, you will see here a 10x10 nearly bimagic square of primes.

***

Jaroslaw Wroblewski wrote on 11/04/2014

Dear Carlos and Christian,
Below is an example of 8x8 bimagic square of primes.

Best regadrs,
Jarek

===================================================

Bimagic 8x8 square of primes, Jaroslaw Wroblewski, April 11, 2014

Related to Q2. of Puzzle 287

General form:
Row 1 :
k + a*t
c + k + b*s + b*t
d + k + a*s + b*s
c + d + k + a*s + a*t + b*t
c + k + a*s + b*s + a*t
k + a*s + b*t
c + d + k
d + k + b*s + a*t + b*t
Row 2 :
k + a*s + b*s
c + k + a*s + a*t + b*t
d + k + a*t
c + d + k + b*s + b*t
c + k
k + b*s + a*t + b*t
c + d + k + a*s + b*s + a*t
d + k + a*s + b*t
Row 3 :
c + d + k + a*s + a*t
d + k + a*s + b*s + b*t
c + k + b*s
k + a*t + b*t
d + k + b*s + a*t
c + d + k + b*t
k + a*s
c + k + a*s + b*s + a*t + b*t
Row 4 :
c + d + k + b*s
d + k + a*t + b*t
c + k + a*s + a*t
k + a*s + b*s + b*t
d + k + a*s
c + d + k + a*s + b*s + a*t + b*t
k + b*s + a*t
c + k + b*t
Row 5 :
d + k + b*t
c + d + k + b*s + a*t
k + a*s + b*s + a*t + b*t
c + k + a*s
c + d + k + a*s + b*s + b*t
d + k + a*s + a*t
c + k + a*t + b*t
k + b*s
Row 6 :
d + k + a*s + b*s + a*t + b*t
c + d + k + a*s
k + b*t
c + k + b*s + a*t
c + d + k + a*t + b*t
d + k + b*s
c + k + a*s + b*s + b*t
k + a*s + a*t
Row 7 :
c + k + a*s + b*t
k + a*s + b*s + a*t
c + d + k + b*s + a*t + b*t
d + k
k + b*s + b*t
c + k + a*t
d + k + a*s + a*t + b*t
c + d + k + a*s + b*s
Row 8 :
c + k + b*s + a*t + b*t
k
c + d + k + a*s + b*t
d + k + a*s + b*s + a*t
k + a*s + a*t + b*t
c + k + a*s + b*s
d + k + b*s + b*t
c + d + k + a*t

We set:
a = 93439
b = 76751
s = 8720021310
t = 21174423270
k = 8434585665512663
c = 1823940260813120
d = 54939573183077448

We get:
Row 1 :
10413102601438193
12552954442285363
64858219275339011
69616564276909621
13721103289000213
10874533897093523
65198099109403231
67647104300475221
Row 2 :
9918646092261563
14676991093832173
65352675784515641
67492527625362811
10258525926325783
12707531117397773
68660676472077661
65814107080170971
Row 3 :
67991406116513851
66483377435734781
10927796281889593
12038260761833963
66021946140079451
66823257269799001
9249375736697753
15346261449395983
Row 4 :
65867369464967041
66977833944911411
13051832933436403
11543804252657333
64188948919775201
70285834632473431
11082372957002003
11883684086721553
Row 5 :
64999317008985881
67845886400892571
13522321188582863
11073315997510873
68307317696547901
66167465855700731
13862201022647083
9103856021076473
Row 6 :
68461894371660311
66012889180588321
10059743825908433
12906313217815123
68801774205724531
64043429204153921
13367744513470453
11227892672623283
Row 7 :
12698474157906643
11897163028187093
69471044561288341
63374158848590111
10729014181472243
12237042862251313
67792624016096501
66682159536152131
Row 8 :
14531471378210893
8434585665512663
67638047340984091
66836736211264541
12853050833019053
11742586353074683
65668587364549691
67176616045328761

The 64 entires of the square are given by the formula:
8434585665512663 + 1823940260813120 x1 + 54939573183077448 x2 +

>   814790071185090 x3 + 1978516935925530 x4 + 669270355563810 x5 +

>   1625158160395770 x6
with xi = 0,1

Number of distinct primes: 64
Set of 32 magic/pandiagonal sums: {314881681191944376}
Set of 18 bimagic sums: {18452511010972604328133473014803880}

Later same day he wrote:

In the meantime I found another solution, which has smaller max term
and smaller bimagic sum, but larger magic sum.

The program will run till tomorrow and since the number of expected
solutions increases as 4-th power of run time, I hope to find more
than 10 solutions.

Below a,b,s,t, are common for all solutions I am searching. The new
solution comes first as it has smaller max term.

Jarek
 

a = 93439
b = 76751
s = 8720021310
t = 21174423270
 

=============================================================
k = 36626754475106729
c = 1065633391001958
d = 14137229751986844
Max term:
56917353141165731 = 5.691735314116573*10^16
Magic sum:
374176430465089840 = 3.741764304650899*10^17
Bimagic sum:
17918328766580454033921060846127400 = 1.791832876658045*10^34
=============================================================
 

k = 8434585665512663
c = 1823940260813120
d = 54939573183077448
 

Max term:
70285834632473431 = 7.028583463247344*10^16
Magic sum:
314881681191944376 = 3.148816811919444*10^17
Bimagic sum:
18452511010972604328133473014803880 = 1.845251101097261*10^34