Bimagic squares of primes
In 1900, Henry E. Dudeney was probably the first to study magic squares of primes: below see a brief history. In 2005, among the 10 open problems of my Mathematical Intelligencer article, I asked this one:
Open problem 4. Construct a bimagic square using distinct prime numbers.
Bimagic means a magic square remaining magic after each of its numbers have been squared. As a first step, in the Supplement of this article, I gave this CB16 square which was previously constructed in October 2004, as mentioned in the Puzzle 287 of Carlos Rivera asking the same problem. It is only a "semi"bimagic square, meaning that the 2 diagonals were not bimagic.
29 
293 
641 
227 
277 
659 
73 
181 
643 
101 
337 
109 
241 
137 
139 
673 
In November 2006, I was happy to construct the first known bimagic square of primes. Fully bimagic, with its two diagonals bimagic. The Open problem 4 is the first solved problem on the ten! In this 11thorder square, 121 consecutive prime integers <= 701 are used, only excluding 2, 3, 523, 641, 677.
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 
Here are the 6 main steps of the method used:
Thanks to Jaroslaw Wroblewski: he is the fastest person to publish my square. It was in his paper titled "Kwadraty bimagicze" published in the Polish magazine MMM = Magazyn Milosnikow Matematyki, issue 18.
Since the open problem 4 is solved, I propose now a new challenge, the goal being to find smaller squares:
Construct bimagic squares of primes, for each order < 11.
We have above a 4x4 semibimagic square. Here is a 10x10 semibimagic square, using 100 consecutive primes <= 571, only excluding 2, 3, 5, 547 and 563. A very nearly bimagic square: 10 bimagic rows, 10 bimagic columns, 1 bimagic diagonal, the other diagonal being "only" magic. Who will construct a fully bimagic square of order 10, or of a smaller order?
277 
7 
283 
383 
541 
103 
167 
293 
71 
457 
449 
19 
127 
229 
389 
151 
503 
31 
311 
373 
269 
173 
367 
571 
233 
467 
53 
23 
263 
163 
61 
199 
569 
463 
139 
313 
347 
239 
11 
241 
131 
379 
83 
271 
487 
47 
401 
461 
41 
281 
521 
439 
37 
79 
181 
227 
101 
211 
479 
307 
149 
509 
359 
191 
331 
13 
107 
419 
431 
73 
97 
409 
337 
257 
43 
443 
157 
89 
193 
557 
137 
197 
353 
29 
179 
397 
523 
317 
433 
17 
491 
251 
67 
109 
59 
421 
223 
499 
349 
113 
June 11th, 2014, Jaroslaw Wroblewski constructed this incredible bimagic square of order 8. This is also a pandiagonal magic square, as Tarry's square.
10413102601438193 
12552954442285363 
64858219275339011 
69616564276909621 
13721103289000213 
10874533897093523 
65198099109403231 
67647104300475221 
9918646092261563 
14676991093832173 
65352675784515641 
67492527625362811 
10258525926325783 
12707531117397773 
68660676472077661 
65814107080170971 
67991406116513851 
66483377435734781 
10927796281889593 
12038260761833963 
66021946140079451 
66823257269799001 
9249375736697753 
15346261449395983 
65867369464967041 
66977833944911411 
13051832933436403 
11543804252657333 
64188948919775201 
70285834632473431 
11082372957002003 
11883684086721553 
64999317008985881 
67845886400892571 
13522321188582863 
11073315997510873 
68307317696547901 
66167465855700731 
13862201022647083 
9103856021076473 
68461894371660311 
66012889180588321 
10059743825908433 
12906313217815123 
68801774205724531 
64043429204153921 
13367744513470453 
11227892672623283 
12698474157906643 
11897163028187093 
69471044561288341 
63374158848590111 
10729014181472243 
12237042862251313 
67792624016096501 
66682159536152131 
14531471378210893 
8434585665512663 
67638047340984091 
66836736211264541 
12853050833019053 
11742586353074683 
65668587364549691 
67176616045328761 
He used this construction method:
k + at 
c + k + bs + bt 
d + k + as + bs 
c + d + k + as + at + bt 
c + k + as + bs + at 
k + as + bt 
c + d + k 
d + k + bs + at + bt 
k + as + bs 
c + k + as + at + bt 
d + k + at 
c + d + k + bs + bt 
c + k 
k + bs + at + bt 
c + d + k + as + bs + at 
d + k + as + bt 
c + d + k + as + at 
d + k + as + bs + bt 
c + k + bs 
k + at + bt 
d + k + bs + at 
c + d + k + bt 
k + as 
c + k + as + bs + at + bt 
c + d + k + bs 
d + k + at + bt 
c + k + as + at 
k + as + bs + bt 
d + k + as 
c + d + k + as + bs + at + bt 
k + bs + at 
c + k + bt 
d + k + bt 
c + d + k + bs + at 
k + as + bs + at + bt 
c + k + as 
c + d + k + as + bs + bt 
d + k + as + at 
c + k + at + bt 
k + bs 
d + k + as + bs + at + bt 
c + d + k + as 
k + bt 
c + k + bs + at 
c + d + k + at + bt 
d + k + bs 
c + k + as + bs + bt 
k + as + at 
c + k + as + bt 
k + as + bs + at 
c + d + k + bs + at + bt 
d + k 
k + bs + bt 
c + k + at 
d + k + as + at + bt 
c + d + k + as + bs 
c + k + bs + at + bt 
k 
c + d + k + as + bt 
d + k + as + bs + at 
k + as + at + bt 
c + k + as + bs 
d + k + bs + bt 
c + d + k + at 
with these parameters:
The 64 integers of the square are given by this formula, where x_{i} = 0 or 1:
With the same set of (a, b, s, t), but with various (k, c, d), he later found 14 other solutions. Here is the list of Wroblewski's 15 bimagic squares of primes, order 8, sorted by their MaxNb. The above square is solution #8 in this list.
Nicolas Rouanet, France, is an engineer working at LATMOS, optical department. This "Laboratoire ATmosphères, Milieux, Observations Spatiales" (= Atmospheres, Environments, Space Observations Laboratory) is a joint research unit of CNRS + University of Versailles + Sorbonne University. From January to November 2018, he worked on bimagic squares of primes, orders from 5 to 25, and obtained these excellent results:
Here are two of these squares:
17 
787 
199 
503 
751 
617 
379 
211 
947 
103 
809 
131 
263 
227 
827 
641 
823 
607 
59 
127 
173 
137 
977 
521 
449 
673 
653 
61 
277 
317 
89 
601 
449 
71 
149 
383 
641 
487 
569 
137 
683 
541 
271 
257 
613 
499 
163 
37 
739 
677 
7 
379 
431 
761 
313 
359 
193 
503 
719 
127 
107 
331 
179 
691 
463 
139 
397 
631 
709 
97 
643 
211 
293 
167 
523 
509 
239 
41 
743 
617 
281 
349 
401 
773 
103 
587 
421 
467 
19 
From November 2018 to April 2020, independently of Nicolas Rouanet, Huang Jianchao constructed five bimagic squares of consecutive primes: orders 24, 25, 26, 27, 28. Huang Jianchao is a Chinese mathematics teacher, Nanyang Center School of China, Xiaoshan District, Hangzhou, Zhe Jiang Province.
Because Nicolas Rouanet and Huang Jianchao worked on two same orders, 24 and 25, we can compare characteristics of their squares:

Nicolas Rouanet's bimagic squares #1 
Nicolas Rouanet's bimagic squares #2 
Huang Jianchao's bimagic squares 

Order 
Set of primes 
S1 
S2 
Set of primes 
S1 
S2 
Set of primes 
S1 
S2 
24 
5 to 4241 
47 018 
129 551 040 
5 to 4253 
47 016 
129 534 552 
283 to 4721 
57 490 
177 744 912 
25 
5 to 4657 
53 823 
162 896 257 
5 to 4679 
53819 
162 859 729 
1609 to 6827 
103 855 
488 411 113 
Magic squares of primes, a brief history
As stated in his book Amusements in Mathematics, page 125, it seems that the first to discuss the problem of constructing magic squares with prime numbers was Henry Ernest Dudeney. It was in The Weekly Dispatch, 22nd July and 5th August 1900. Unfortunately, at that time, "1" was considered as a prime number. The magic sum 111 of his 3x3 square is the lowest possible, allowing "1".
67 
1 
43 
13 
37 
61 
31 
73 
7 
Harry A. Sayles also studied magic squares with prime numbers, and was probably the first to publish the correct 3x3 square, not using 1, with the smallest possible magic sum 177. It was in 1918, in The Monist, page 142. Rudolph Ondrejka later found the same square, reported by Joseph M. Madachy in 1966 in Mathematics on Vacation, page 95 (book later republished with another title, Madachy's Mathematical Recreations).
71 
5 
101 
89 
59 
29 
17 
113 
47 
In 1914, Charles D. Shuldham published, in The Monist, three interesting papers on magic squares of prime numbers with numerous examples of various orders, for example, this pandiagonal magic square, page 608:
73 
41 
13 
113 
23 
103 
83 
31 
107 
7 
47 
79 
37 
89 
97 
17 
André Gérardin, in SphinxOedipe, published various magic squares of primes beginning in 1916. For example, he gives this interesting method using sums of squares.
3² 
5² 
7² 
37² 
+ 
2² 
8² 
52² 
58² 
= 
13 
89 
2753 
4733 
37² 
7² 
5² 
3² 
52² 
58² 
2² 
8² 
4073 
3413 
29 
73 

5² 
3² 
37² 
7² 
58² 
52² 
8² 
2² 
3389 
2713 
1433 
53 

7² 
37² 
3² 
5² 
8² 
2² 
58² 
52² 
113 
1373 
3373 
2729 
We know now numerous magic squares of various orders, using only prime numbers. I particularly mention the numerous squares and cubes of Allan W. Johnson, Jr. published in different issues of the Journal of Recreational Mathematics, for example, this pandiagonal magic square.
41 
109 
31 
59 
37 
53 
47 
103 
89 
61 
79 
11 
73 
17 
83 
67 
We also have to mention the famous square of Harry L. Nelson, who won the $100 prize offered by Martin Gardner: produce a 3x3 square of consecutive primes. See his paper in the Journal of Recreational Mathematics, 1988, pages 214216.
1480028201 
1480028129 
1480028183 
1480028153 
1480028171 
1480028189 
1480028159 
1480028213 
1480028141 
In 2004, I constructed the smallest and first known magic squares of squares of primes: look at the CB17 (4x4) and CB18 (5x5) squares of the Supplement. See also Puzzle 288 of Carlos Rivera.
In 2007, Jaroslaw Wroblewski and Hugo Pfoertner constructed the first known magic square of cubes of primes: look at their 42x42 magic square.
In 2008, Raanan Chermoni and Jaroslaw Wroblewski found the first known AP25, an arithmetic progression of 25 primes:
6171054912832631 + 366384*23#*n, for n = 0 to 24
(the notation 23# means 2*3*5*7*11*13*17*19*23
= 223092870)
With this AP25, we can easily arrange its 25 terms and construct this 5x5 pandiagonal magic squares of BIG primes:
6171054912832631 
6743218519407191 
7315382125981751 
7478857442145911 
8051021048720471 
7070169151735511 
7642332758310071 
7805808074474231 
6334530228996791 
6906693835571351 
7969283390638391 
6498005545160951 
6661480861325111 
7233644467899671 
7397119784063831 
6824956177489271 
6988431493653431 
7560595100227991 
8132758706802551 
6252792570914711 
7724070416392151 
7887545732556311 
6416267887078871 
6579743203243031 
7151906809817591 
On the orders 3 and 4, we can construct magic squares of still bigger primes, using AP9 and AP16; status of the current search at http://users.cybercity.dk/~dsl522332/math/aprecords.htm
For example, with the biggest known AP9, computed in 2012 by Ken Davis and Paul Underwood:
(65502205462 + 6317280828*n)*2371# + 1, for n = 0 to 8
we can construct a 3x3 magic square of primes where the biggest used integer is this prime of 1014 digits!
6528 1797806551 4223969026 9465606678 8430413319 5255367568 8539223625 1894181816 2295567702 4019319671 9333832115 0061841041 7707195791 2803140095 8352804788 0185757536 6154874648 2856691539 8087025088 4358010975 5464536678 5697944644 8414005984 6147638197 4705147888 5491138440 0220327868 7363977428 2737801931 0520928772 5273502621 1160921647 5304393339 1767016703 7815224345 8891520911 6236415078 7937941027 3482586307 9771751835 2416069216 0455877228 7165528915 4691222310 5931802968 4750663643 9321249501 1529753060 2177270003 6356499195 3633416904 8702177467 8208223768 2956050987 5838438258 6851826872 9197546602 0234999349 2497621917 2738581161 9415122224 0117864207 4840555878 0205737919 5478023001 3058669932 0674106178 5760186912 4008237951 0383731554 0581804537 9660820190 6498476783 7685256998 6181217269 0236312263 0511714243 8492521158 0685298127 3315053479 1728430089 8633989547 0948111232 3941740735 3710609506 4479145195 2658117032 6376052349 8051287376 8947506973 0311459443 4926045579 0413748638 7963946563 1310017137 7738751714 9720291376 3358218759 4500584427 7708805850 0049908410 1090288421
In 2013 and 2014, Jaroslaw Wrobleski, then Max Alekseyev, sent to Natalia Makarova (through http://dxdy.ru/post751928.html#p751928) these pandiagonal squares of consecutive primes. Max Alekseyev, previously author of a 4x4x4 multiplicative magic cube, announced that his square has the smallest possible magic sum (S=682775764735680) for such squares: http://oeis.org/A245721.
320572022166380833 
320572022166380921 
320572022166380849 
320572022166380917 
320572022166380909 
320572022166380857 
320572022166380893 
320572022166380861 
320572022166380911 
320572022166380843 
320572022166380927 
320572022166380839 
320572022166380867 
320572022166380899 
320572022166380851 
320572022166380903 
170693941183817 
170693941183933 
170693941183949 
170693941183981 
170693941183979 
170693941183951 
170693941183847 
170693941183903 
170693941183891 
170693941183859 
170693941184023 
170693941183907 
170693941183993 
170693941183937 
170693941183861 
170693941183889 
In 2018, 14 years after my magic squares of squares of prime numbers, orders 4 and 5, Nicolas Rouanet constructed magic squares of orders 6, 7, 8. Look also at his BImagic square of order 8.
101² 
73² 
37² 
107² 
61² 
131² 
113² 
127² 
29² 
139² 
7² 
11² 
149² 
17² 
97² 
31² 
109² 
67² 
41² 
53² 
79² 
71² 
83² 
163² 
47² 
157² 
59² 
103² 
89² 
19² 
13² 
5² 
167² 
43² 
137² 
23² 
For more on prime magic squares, see:
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