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.

 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.

 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.

 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:

 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:

 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.