Bimagic squares of primes

In 1900, Henry E. Dudeney was probably the first to study magic squares of primes: see below 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.

Oct. 2004: CB16. The smallest possible 4×4 semi-bimagic square of prime numbers.
S1 = 1190, S2 = 549100.

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.

Nov. 2006: first known bimagic square of primes,
order 11, S1=3497, S2=1578251

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:

1. 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.
2. Select carefully a good set of 121 prime numbers, using some modulo reasoning.
3. With this set, compute the S1 and S2 sums.
4. Compute the list of bimagic series of 11 primes, from the selected set, having the good S1 and S2 sums.
5. Using these series, with combinatorics algorithms, to find 22 series that we can cross and correctly organize, producing a semi-bimagic square.
6. 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.

Because the open problem 4 is now 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?

Nearly bimagic square of primes, order 10,
S1=2582, S2=934474

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

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".

1900: by Dudeney

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).

1918: by Sayles

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:

1914: pandiagonal by Ch. Shuldham

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 from 1916. For example, he gives this interesting method using sums of squares.

1925: by André Gérardin

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.

1979-80: pandiagonal by Allan Johnson

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.

1988: using consecutive primes, by Harry Nelson

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.

For more on prime magic squares, see:

• http://www.magic-squares.net/primesqr.htm
• http://mathworld.wolfram.com/PrimeMagicSquare.html

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