Magic squares of 6th powers
See also the Magic squares of cubes general page

At least one magic square of 6th powers has been known since 2003: the 4096x4096 hexamagic square of Pan Fengchu, directly raising its integers to the sixth power.

But is it possible to construct smaller squares of 6th powers? Yes! Allowing non consecutive integers (but still distinct), and using methods similar to Morgenstern's 6x6 method of semi-magic squares of nth powers.

In June 2011, Jaroslaw Wroblewski worked on 36x36 semi-magic squares. He did not try directly to obtain two taxicab(6, 6, 6) numbers, but computed solutions of a^6 + b^6 + c^6 + d^6 + e^6 + f^6 = N * k^6. With N=76, he obtained these 13 rational decompositions of 6th powers:

1. k =13 {25/13, 20/13, 18/13, 17/13, 9/13, 5/13}
2. k = 41 {2, 61/41, 1, 30/41, 11/41, 3/41} *
3. k = 109 {207/109, 187/109, 134/109, 75/109, 61/109, 44/109}
4. k = 143 {291/143, 185/143, 115/143, 75/143, 62/143, 32/143} *
5. k = 167 {291/167, 290/167, 274/167, 159/167, 139/167, 5/167} *
6. k = 173 {355/173, 174/173, 135/173, 113/173, 47/173, 20/173} *
7. k = 221 {414/221, 373/221, 300/221, 271/221, 103/221, 71/221} *
8. k = 325 {654/325, 421/325, 421/325, 237/325, 68/325, 47/325}
9. k = 331 {657/331, 493/331, 415/331, 185/331, 162/331, 82/331} *
10. k = 347 {695/347, 473/347, 409/347, 400/347, 120/347, 93/347}
11. k = 359 {731/359, 449/359, 313/359, 306/359, 233/359, 30/359}
12. k = 397 {776/397, 655/397, 246/397, 67/397, 43/397, 27/397}
13. k = 467 {892/467, 803/467, 460/467, 435/467, 255/467, 29/467}

And with N=1300, he obtained these 7 rational decompositions of 6th powers:

1. k = 17 {49/17, 45/17, 41/17, 40/17, 27/17, 8/17} *
2. k = 29 {93/29, 61/29, 57/29, 56/29, 46/29, 23/29}
3. k = 59 {191/59, 119/59, 115/59, 102/59, 40/59, 27/59} *
4. k = 115 {361/115, 55/23, 252/115, 201/115, 34/23, 157/115} *
5. k = 211 {661/211, 523/211, 438/211, 357/211, 346/211, 29/211} *
6. k = 283 {825/283, 725/283, 652/283, 630/283, 589/283, 545/283} *
7. k = 295 {926/295, 744/295, 593/295, 97/59, 49/59, 33/295} *

Selecting the six decompositions marked * of N=76, and the six marked * of N=1300, this generates 1296 distinct products, a necessary condition for a 36x36 square.

The magic sum should be as big as 76 * (41*143*167*173*221*331)^6 * 1300 * (17*59*115*211*283*295)^6 = 2.52e+157, but in fact is slightly lower, taking the least common multiples of denominators: 76 * 953145898991^6 * 1300 * 6887595985^6 = 7.91e+135.

Here is a partial view of this fascinating square, with its impressive numbers totaling to this colossal magic sum! This square can be downloaded: see at the end of this page.

June 2011: top left corner of the 36x36 semi-magic square of 6th powers, by Jaroslaw Wroblewski
S6 ~7.91e+135, MaxNb = 43610458567903181050525^6
S6 = 7908957311616866591404736287729521652955367340513765346608535651854720549216887690309083816367020250742635942617903195625061972393750000

378446246451143212241906	316194551196896097304156	206080267477432643613896	150481617249798758995506	...
353628771034516937655756	171300128401347566353256	126247766673262060598756	85343707845906149538756	...
275891614713955130469156	222205294798306166132506	236028170397959247868566	129747606852536950949106	...
316971611395908553370006	114150640611038368451406	89539142363059177052556	72551291696308623648456	...
...	...	...	...	...

The above square is semi-magic. Because the integers used are very big, it is very probable that no magic diagonal is possible by rearranging its cells.

Magic squares of 6th powers

Order	Date	Author	S6	MaxNb
90	Nov. 2011	François Labelle	1.00e+40	4271863^6
64	March 2012	Jaroslaw Wroblewski	1.63e+96	10245064958712675^6
	April 2013	Toshihiro Shirakawa	2.02e+40	4933788^6

In November 2011, François Labelle created a magic square of 6th powers, i.e. this time with two magic diagonals. He used a clever new method to obtain the two magic diagonals, using supplemental properties of two taxicab numbers.

This taxicab(6, 10, 9) ~ 10^20:

• T1 = 100000000000258843390
  = 32^6 + 1847^6 + 359^6 + 654^6 + 723^6 + 967^6 + 1149^6 + 1181^6 + 1324^6 + 1912^6
  = 431^6 + 508^6 + 339^6 + 422^6 + 499^6 + 514^6 + 1094^6 + 1597^6 + 1773^6 + 1923^6
  = 446^6 + 626^6 + 661^6 + 821^6 + 827^6 + 892^6 + 1107^6 + 1249^6 + 1707^6 + 2022^6
  = 634^6 + 158^6 + 8^6 + 51^6 + 141^6 + 307^6 + 769^6 + 831^6 + 1268^6 + 2137^6
  = 914^6 + 1801^6 + 298^6 + 717^6 + 787^6 + 879^6 + 1229^6 + 1412^6 + 1692^6 + 1759^6
  = 977^6 + 194^6 + 128^6 + 227^6 + 591^6 + 633^6 + 941^6 + 958^6 + 1789^6 + 2004^6
  = 1282^6 + 619^6 + 383^6 + 934^6 + 1074^6 + 1171^6 + 1467^6 + 1552^6 + 1629^6 + 1907^6
  = 1861^6 + 386^6 + 981^6 + 996^6 + 1213^6 + 1277^6 + 1328^6 + 1354^6 + 1497^6 + 1723^6
  = 1868^6 + 1588^6 + 83^6 + 291^6 + 447^6 + 554^6 + 1047^6 + 1193^6 + 1595^6 + 1658^6

is constructed with a supplemental property: each of its first two columns have a sum = T1*(9/10).

And this taxicab(6, 9, 10) ~ 10^20:

• T2 = 100000000000003308921
  = 1^6 + 67^6 + 451^6 + 649^6 + 1073^6 + 1283^6 + 1553^6 + 1711^6 + 1949^6
  = 5^6 + 551^6 + 935^6 + 937^6 + 1087^6 + 1271^6 + 1487^6 + 1621^6 + 1999^6
  = 841^6 + 19^6 + 347^6 + 463^6 + 529^6 + 739^6 + 1031^6 + 1823^6 + 1987^6
  = 1013^6 + 281^6 + 425^6 + 629^6 + 737^6 + 863^6 + 1247^6 + 1867^6 + 1933^6
  = 1117^6 + 433^6 + 449^6 + 505^6 + 1021^6 + 1415^6 + 1439^6 + 1661^6 + 1973^6
  = 1151^6 + 131^6 + 295^6 + 419^6 + 671^6 + 1481^6 + 1523^6 + 1751^6 + 1891^6
  = 1303^6 + 29^6 + 61^6 + 311^6 + 659^6 + 839^6 + 1423^6 + 1787^6 + 1943^6
  = 1615^6 + 101^6 + 509^6 + 1255^6 + 1265^6 + 1375^6 + 1493^6 + 1709^6 + 1777^6
  = 1811^6 + 65^6 + 73^6 + 691^6 + 989^6 + 1301^6 + 1339^6 + 1601^6 + 1819^6
  = 1903^6 + 355^6 + 523^6 + 535^6 + 877^6 + 1291^6 + 1429^6 + 1531^6 + 1721^6

is constructed with a supplemental property: the first column has a sum = T2*(10/9).

Using these two taxicab numbers for generating 90x90 = 8100 distinct integers, the magic sum of the generated square is S6 = T1*T2 = 10000000000026215231100000856492328882190 ~ 10^40. The two diagonals use the products of the first two columns of the first taxicab with the first column of the second taxicab, the sum of these two diagonals being T1*(9/10) * T2*(10/9) = T1*T2 = the correct magic sum! MaxNb = (2137*1999)^6 = 4271863^6.

90... why this strange order? He chose to work on a k*(k+1) order because the difficulty obtaining taxicab(6, k, k+1) and taxicab(6, k+1, k) numbers is similar. He estimated that the probability of success with 7*8 and 8*9 was too low, and chose 9*10 which seemed possible. We can add that 8*9 also has another problem: the taxicab number with two good columns needs to be a taxicab(6, even, k) number. And he estimated that the probability of success with taxicab numbers <= 10^19 was too low, which is why he searched for numbers > 10^20.

Compared to the 4096x4096 hexamagic square of Pan Fengchu, all the characteristics of Labelle's square are smaller: its order, its magic sum S6, and its MaxNb.

November 2011: top left corner of the 90x90 magic square of 6th powers, by François Labelle
S6 = 10000000000026215231100000856492328882190 ~10^40, MaxNb = 4271863^6

326	3596	7236	11496	...
23977646	579526	6501496	13093536	...
14971476	17251726	416966	4677776	...
8075916	12834336	14789086	357446	...
...	...	...	...	...

In March 2012, Jaroslaw Wroblewski created a smaller square of 6th powers than François Labelle's: 64x64 instead of 90x90.

He used the same trick as François Labelle, but with rational numbers:

T1 = 179924108
  {{14, 11, 654/29, 521/29, 382/29, 225/29, 138/29, 19/29},
  {15, 10, 441/19, 284/19, 101/19, 74/19, 45/19, 2/19},
  {22, 8, 930/47, 594/47, 487/47, 465/47, 205/47, 175/47},
  {1, 12, 834/37, 698/37, 339/37, 7, 25/37, 20/37},
  {2, 21, 381/19, 307/19, 262/19, 243/19, 128/19, 46/19},
  {6, 20, 93/5, 92/5, 89/5, 59/5, 51/5, 28/5},
  {9, 3, 541/25, 96/5, 388/25, 76/5, 197/25, 154/25},
  {19, 17, 939/43, 332/43, 259/43, 254/43, 108/43, 24/43}}

with the first two columns 14^6 + 15^6 + 22^6 + 1^6 + 2^6 + 6^6 + 9^6 + 19^6 = 11^6 + 10^6 + 8^6 + 12^6 + 21^6 + 20^6 + 3^6 + 17^6 = T1

T2 = 1384448692172
  {{11, 1668/17, 1371/17, 1301/17, 794/17, 665/17, 614/17, 54/17},
  {41, 1939/19, 1396/19, 1163/19, 1041/19, 918/19, 498/19, 334/19},
  {42, 1752/19, 1699/19, 1317/19, 1307/19, 1087/19, 550/19, 526/19},
  {53, 1253/13, 1146/13, 830/13, 636/13, 601/13, 192/13, 13},
  {62, 101, 922/13, 834/13, 823/13, 531/13, 287/13, 162/13},
  {69, 1677/19, 1637/19, 1621/19, 858/19, 542/19, 340/19, 98/19},
  {86, 1556/17, 1385/17, 1091/17, 906/17, 696/17, 553/17, 519/17},
  {96, 1426/17, 1225/17, 1132/17, 915/17, 591/17, 353/17, 218/17}}

with the first column 11^6 + 41^6 + 42^6 + 53^6 + 62^6 + 69^6 + 86^6 + 96^6 = T2

These two sets become, when multiplied by the common denominators:

{{14420744450, 11330584925, 23229475050, 18505438075, 13568286650, 7991791875, 4901632350, 674862425},
  {15450797625, 10300531750, 23908076325, 15396584300, 5475545825, 4011786050, 2439599625, 108426650},
  {22661169850, 8240425400, 20381903250, 13018118850, 10673104175, 10190951625, 4492785125, 3835304375},
  {1030053175, 12360638100, 23217955350, 19431813950, 9437514225, 7210372225, 695981875, 556785500},
  {2060106350, 21631116675, 20655276825, 16643490775, 14203891150, 13173837975, 6939305600, 2493812950},
  {6180319050, 20601063500, 19158989055, 18952978420, 18334946515, 12154627465, 10506542385, 5768297780},
  {9270478575, 3090159525, 22290350707, 19777020960, 15986425276, 15656808260, 8116819019, 6345127558},
  {19571010325, 17510903975, 22493486775, 7952968700, 6204273775, 6084500150, 2587110300, 574913400}}

{{46189, 411996, 338637, 321347, 196118, 164255, 151658, 13338},
  {172159, 428519, 308516, 257023, 230061, 202878, 110058, 73814},
  {176358, 387192, 375479, 291057, 288847, 240227, 121550, 116246},
  {222547, 404719, 370158, 268090, 205428, 194123, 62016, 54587},
  {260338, 424099, 297806, 269382, 265829, 171513, 92701, 52326},
  {289731, 370617, 361777, 358241, 189618, 119782, 75140, 21658},
  {361114, 384332, 342095, 269477, 223782, 171912, 136591, 128193},
  {403104, 352222, 302575, 279604, 226005, 145977, 87191, 53846}}

and directly produce this magic square of 6th powers:

March 2012: top left corner of the 64x64 magic square of 6th powers, by Jaroslaw Wroblewski
S6 = 1630794890680863753807181713742296474903681088364281912477533838938712716991677267473160156250000
~1.63 * 10^96,
MaxNb = (23908076325 * 428519)^6 = 10245064958712675^6

6660797654010506	10729462230844506	6267055920768506	2264014966141506	...
8644420779813006	25432136497131006	40967037608679006	23928758970207006	...
35323406098877006	12760811627343006	37542677686241006	60475150755669006	...
83884886532057006	48996982653281006	17700480644379006	52075327113173006	...
...	...	...	...	...

In April 2013, Toshihiro Shirakawa produced another 64x64 magic square of 6th powers, but with smaller integers than Jaroslaw used.

T1 = 1000000000043382156
  {{717, 488, 901, 803, 553, 512, 298, 246}
  {58, 155, 818, 804, 787, 699, 641, 434}
  {293, 692, 951, 716, 477, 371, 290, 214}
  {856, 369, 894, 662, 446, 307, 287, 215}
  {326, 917, 797, 685, 538, 498, 388, 345}
  {698, 472, 813, 736, 723, 707, 673, 634}
  {732, 786, 881, 613, 602, 565, 461, 94}
  {889, 830, 746, 417, 316, 282, 233, 229}}

The first two columns have a sum = T2/(5^6) = 1294233624408559443.

T2 = (5^6)*1294233624408559443 = 20222400381383741296875
  {{3760, 4635, 4060, 3590, 2750, 2675, 1980, 1195}
  {3245, 4970, 3560, 3180, 2960, 2495, 855, 400}
  {3035, 4980, 4010, 1700, 1210, 1125, 170, 35}
  {1260, 4875, 4030, 3565, 2650, 2080, 1765, 1040}
  {3700, 1255, 5030, 3210, 2590, 1910, 1415, 1275}
  {1780, 5188, 2868, 2203, 1663, 748, 399, 352}
  {4495, 4616, 3396, 2963, 2114, 696, 646, 551}
  {645, 4364, 4358, 4072, 3521, 815, 472, 366}}

The first column has a sum = T1*(5^6) = 15625000000677846187500. These two sets produce this magic square of 6th powers:

April 2013: top left corner of the 64x64 magic square of 6th powers, by Toshihiro Shirakawa
S6 = T1*T2 = 20222400382261032624914648960804673562500  ~ 2.02 * 10^40,
MaxNb = (951 * 5188)^6 = 4933788^6

26959206	9249606	33877606	30192806	...
7982706	23266656	26057356	29237456	...
27345356	24371056	21760956	7466106	...
10117806	11352606	3099606	9034206	...
...	...	...	...	...

However, we can see that both S6 and MaxNb of this 64x64 square remain bigger than those of Labelle's 90x90 square.

Download squares of this webpage

• 36x36, 64x64, 90x90 squares of 6th powers (zipped Excel file of 361Kb)

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