Problems & Puzzles: Puzzles

Puzzle 287.  Multimagic prime squares

For sure you know already the interesting and nice site of Christian Boyer about the multimagic squares. If not, maybe is a good idea you review it before going on with this puzzle.

In any case the very basic concept is this one:

A magic square is bimagic (or 2-multimagic) if it remains magic after each of its numbers have been squared. By extension, a square is P-multimagic if it remains magic after each of its numbers have been replaced by their k-th power (for k=1, 2, ..., to P).

Some basic facts about P-multimagic squares are:

  • The first bimagic historically produced is an 8x8 square constructed in 1890 by G. Pfefferman.
  • No bimagic is possible (using consecutive numbers) for a nxn square for n = 3, 4, 5 & 6. For n=7 the impossibility hasn't been completely established, but is believed to be impossible.
  • The smallest orders n currently known of P-multimagic squares are:
P n
2 8, 9, 10, 11,16
3 12, 32, 64, 128
4 256, 512
5 729, 1024
6 4096

Well, this week Christian and me crossed emails about Mr. Chebrakov (See our Puzzle 79). In some moment during this interchange, I asked him (13/10/04):

One quick question: Is possible to get multimagic prime-squares?

He responded (13/10/04):

Hmmmm, I do not know. A very VERY difficult question. Imagine that, today, nobody has succeeded to construct a bimagic square of distinct numbers which is smaller than the well-known 8x8 bimagic squares of consecutive numbers. And "distinct" is far less limited than "prime"!

A 3x3 bimagic square using distinct integers (or of course prime integers) is proved to be impossible. My 4x4 best result is this semi-bimagic square:

 

9 55 105 36
69 100 21 15
28 49 19 109
99 1 60 45



4 bimagic rows, 4 bimagic columns, but only one magic diagonal S1=205, S2=15427.

You can find in my site a 6x6 square constructed by Pfeffermann in 1894:
6 bimagic rows, 6 bimagic columns, two magic diagonals (but not bimagic).

I replied (13/10/04):

What I wanted to know is if there is some theoretical impossibility in order to get such kind of objects (prime-bimagic squares). May I suppose from your response that you are not aware of such impossibility?

In  his turn Christian responded (14/10/04):

Yes, I can't see any theoretical impossibility of a nxn bimagic square of prime numbers, with n>=4. But impossible for n<4.

Thinking about this problem, I have created this morning this nice (and probably first) semi-bimagic square of prime numbers:

 

29 293 641 227
277 659 73 181
643 101 337 109
241 137 139 673



You can check that:

1) The 16 used numbers are prime integers, and are distinct.
2) The 4 rows and 4 columns have the same sum S1=1190.
3) And when these 16 prime numbers are squared, the 4 rows and 4 columns have the same sum S2=549100.
4) But the 2 diagonals are not magic or bimagic.

Who will be the first to create a bimagic square of prime numbers, including two bimagic diagonals?

It is a VERY difficult problem. Remember that nobody has succeeded to solve the "easier" problem to create a bimagic square of distinct integers, the square being smaller than 8x8.

Enough!...

Questions:

1. Improve the Boyer 4x4 prime semi-bimagic square (at least one bimagic diagonal).

2. Get a (full) prime-bimagic square (what is the minimal nxn order for this target?)


Preliminary results related to question 2 came from Luke Pebody and J. C. Rosa.

Both proved that is impossible to construct 4x4 bimagic squares of distinct numbers.

Here is the J. C. Rosa's demonstration:

Square  I
 
   a    b    c    d
   e    f     g    h
   i     j     k    l
  m    n    o    p 
 
           Square  II
  a^2   b^2   c^2   d^2
  e^2   f^2    g^2   h^2
  i^2    j^2    k^2   l^2
  m^2  n^2   o^2   p^2
 
If the square I is magic we have (*), by example, the equality:

       a+m=h+l   (1)

If the square II is magic we have the same thing :

      a^2+m^2=h^2+l^2    (2)

The equalities (1) and (2) give : a*m=h*l   (3)

(remark: if a,m,h,l are prime numbers the equality (3) is impossible )

The equalities (1) and (3) give the following equation :

   m^2-(h+l)*m+h*l=0

This equation has two solutions: m=h or m=l. Therefore a 4x4 bimagic square with 16 distinct numbers is impossible.

_________
(*) This supposes - for example - the validity of the Bergholt's formula for 4x4 magic squares; otherwise (1) needs a demonstration.

The Luke's proof is this one:

There are no bimagic squares of distinct integers for 4x4.

I do not use the lines through E,H,I or L in the diagram below, but I do use both bimagic diagonals.

Proof:

Let

ABCD
EFGH
IJKL
MNOP

be a magic square.

Then:
A+B+C+D=S1
M+N+O+P=S1
A+F+K+P=S1

D+G+J+M=S1
B+F+J+N=S1
C+G+K+O=S1

Adding the top 3, and subtracting the bottom 3:
2(A+P)=2(G+J).

Therefore A+P=G+J.

Similarly, by the bimagic property, A^2+P^2=G^2+J^2. Therefore {A,P}={G,J}. QED

***

Christian Boyer sends (July 2005) the following line: "Look at Puzzle 288 where the two above proofs are mentioned in a mathematical article published in 2005"

...

Later, on Nov. 2006, he added:

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 n 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)

***

Update (April 2007)

C. Boyer wrote:

...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

One day later he wrote again:

I have finished the search. Below are 15 solutions found, ordered
according to the max term of the square.
 

Jarek

a = 93439
b = 76751
s = 8720021310
t = 21174423270
=============================================================
 
k = 346095583148521
c = 2024787890686746
d = 9570070099554256
Max term:
17028689096459723 = 1.702868909645972*10^16
Magic sum:
69499138718432976 = 6.949913871843298*10^16
Bimagic sum:
810473248712773726576302920487176 = 8.10473248712774*10^32
=============================================================
k = 6099767504417347
c = 3957401605301746
d = 6002154359538036
Max term:
21147058992327329 = 2.114705899232733*10^16
Magic sum:
108987305986978704 = 1.089873059869787*10^17
Bimagic sum:
1603487825092114929358857870857576 = 1.603487825092115*10^33
=============================================================
k = 7350940185803741
c = 12040141242762432
d = 18624273023281016
Max term:
43103089974917389 = 4.31030899749174*10^16
Magic sum:
201816120642884520 = 2.018161206428845*10^17
Bimagic sum:
6090210360462645510698932969443560 = 6.090210360462645*10^33
=============================================================
k = 21609520858438571
c = 8705903888130012
d = 9422224198962546
Max term:
44825384468601329 = 4.482538446860133*10^16
Magic sum:
265739621308159600 = 2.657396213081596*10^17
Bimagic sum:
9171670382317221116535219755260520 = 9.17167038231722*10^33
=============================================================
k = 2414839437284359
c = 17796721326461878
d = 23608608802088994
Max term:
48907905088905431 = 4.890790508890543*10^16
Magic sum:
205290978104759160 = 2.052909781047592*10^17
Bimagic sum:
7031562558446688852916655886322040 = 7.03156255844669*10^33
 
=============================================================
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 = 3082762993617539
c = 9777968198481228
d = 48973638091040780
Max term:
66922104806209747 = 6.692210480620974*10^16
Magic sum:
280019471199309144 = 2.800194711993091*10^17
Bimagic sum:
14804749759230058535719483362440360 = 1.480474975923006*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
=============================================================
 
k = 6438974210534423
c = 29914381829748660
d = 36522582673925510
Max term:
77963674237278793 = 7.79636742372788*10^16
Magic sum:
337610593791252864 = 3.376105937912528*10^17
Bimagic sum:
18720487648831701494491649969032712 = 1.87204876488317*10^34
=============================================================
k = 1894055351865647
c = 10829203760508444
d = 61089248562166436
Max term:
78900243197610727 = 7.890024319761073*10^16
Magic sum:
323177194197905496 = 3.231771941979055*10^17
Bimagic sum:
20769108192119335457332251008258216 = 2.076910819211933*10^34
=============================================================
k = 3219288530528419
c = 4460791985514414
d = 76020237353642218
Max term:
88788053392755251 = 8.87880533927553*10^16
Magic sum:
368029367693134680 = 3.680293676931347*10^17
Bimagic sum:
28543987188422290822355309204179640 = 2.854398718842229*10^34
=============================================================
k = 4610969623991377
c = 23427916097307396
d = 58797227685175420
Max term:
91923848929544393 = 9.19238489295444*10^16
Magic sum:
386139274214143080 = 3.861392742141431*10^17
Bimagic sum:
26665239806539782488872053835336232 = 2.666523980653978*10^34
=============================================================
k = 7447262888338243
c = 20326242748964340
d = 59847945888341664
Max term:
92709187048714447 = 9.27091870487145*10^16
Magic sum:
400625799748210760 = 4.006257997482108*10^17
Bimagic sum:
28067829418636045019019198371717192 = 2.806782941863604*10^34
=============================================================
k = 21417414557883031
c = 1959104035390162
d = 74349006105022086
Max term:
102813260221365479 = 1.028132602213655*10^17
Magic sum:
496922699116994040 = 4.969226991169941*10^17
Bimagic sum:
41945081655028977370061542422191480 = 4.194508165502898*10^34
=============================================================
k = 11605009505666141
c = 35951874740628720
d = 52815332971624202
Max term:
105459952740989263 = 1.054599527409893*10^17
Magic sum:
468259848986621616 = 4.682598489866217*10^17
Bimagic sum:
35587739108122460860143167766152840 = 3.558773910812246*10^34
=============================================================

***

 

 



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