**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 11th-order 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:

- Choose the order. My idea -for the supplemental beauty- was to have a prime order. 2 and 3 are impossible. 5, nobody knows a bimagic square using any set of numbers. 7, probably yet too small to have a sufficient number of bimagic series. 11, hmmm.
- Select carefully a good set of 121 prime numbers, using some modulo reasoning.
- With this set, compute the S1 and S2 sums.
- Compute the list of bimagic series of 11 primes, from the selected set, having the good S1 and S2 sums.
- Using these series, with combinatoric algorithms, to find 22 series that we can cross and correctly organize, producing a semi-bimagic square.
- Close to being finished? No! As usual with magic squares, one of the most difficult, and sometimes impossible, steps: find TWO good diagonals by rearranging the cells of the semi-bimagic square. Most of the semi-bimagic squares cannot create a bimagic square: I needed several hundreds of squares before succeeding.

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 semi-bimagic square. Here is a 10x10 semi-bimagic 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:

- a = 93439
- b = 76751
- s = 8720021310
- t = 21174423270
- k = 8434585665512663
- c = 1823940260813120
- d = 54939573183077448

The 64 integers of the square are given by this formula, where x_{i}
= 0 or 1:

- 8434585665512663 + 1823940260813120*x
_{1}+ 54939573183077448*x_{2}+ 814790071185090*x_{3}+ 1978516935925530*x_{4}+ 669270355563810*x_{5}+ 1625158160395770*x_{6}

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 the solution #8 in this list.

**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 *Sphinx-Oedipe*, published various magic squares
of primes beginning in 1916. For example, he gives this interesting method
using sums of squares.

3˛ |
5˛ |
7˛ |
37˛ |
+ |
2˛ |
8˛ |
52˛ |
58˛ |
= |
13 |
89 |
2753 |
4733 |

37˛ |
7˛ |
5˛ |
3˛ |
52˛ |
58˛ |
2˛ |
8˛ |
4073 |
3413 |
29 |
73 |
||

5˛ |
3˛ |
37˛ |
7˛ |
58˛ |
52˛ |
8˛ |
2˛ |
3389 |
2713 |
1433 |
53 |
||

7˛ |
37˛ |
3˛ |
5˛ |
8˛ |
2˛ |
58˛ |
52˛ |
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 214-216.

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* and *CB18* 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 |

For more on prime magic squares, see:

- http://www.magic-squares.net/primesqr.htm
- http://mathworld.wolfram.com/PrimeMagicSquare.html
- http://www.azspcs.net/Contest/PandiagonalMagicSquares
with impressive results from
**Jaroslaw Wroblewski**in July-August 2013, again improved in September-October 2013 after the contest ended.

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