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740 - 749

740

The smallest squares containing k 740's :
674041 = 8212,
27401174089 = 1655332,
297740374074025 = 172551552.

7402 = 547600, 5 * 4 * 7 + 600 = 740.

Komachi equation: 7402 = 12 * 22 + 342 * 52 * 62 - 782 * 92.

430k + 740k + 1470k + 1585k are squares for k = 1,2,3 (652, 23252, 874252).

Page of Squares : First Upload September 19, 2005 ; Last Revised March 23, 2011
by Yoshio Mimura, Kobe, Japan

741

The smallest squares containing k 741's :
741321 = 8612,
327417417616 = 5722042,
7410741374149569 = 860856632.

7412 = 549081 is a zigzag square with different digits.

7412 = 549081, 54 + 9 + 0 + 81 = 122,
7412 = 549081, 54 + 90 + 81 = 152,
7412 = 549081, 5 + 490 + 81 = 242.

7412 = 43 + 263 + 813.

7412 + 7422 + 7432 + ... + 7602 = 7612 + 7622 + 7632 + ... + 7792.

The square root of 741 is 27. 2 21 3 15 1 7 ..., and 272 = 22 + 212 + 32 + 152 + 12 + 72,
the square root of 741 is 27. 22 13 1 5 1 7 ..., and 272 = 222 + 132 + 12 + 52 + 12 + 72.

7412 = 549081, 5 * 4 + 90 * 8 + 1 = 741.

20748k + 134121k + 164502k + 229710k are squares for k = 1,2,3 (7412, 3134432, 1378193312).

3-by-3 magic squares consisting of different squares with constant 7412:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(9, 270, 690, 390, 585, 234, 630, 366, 135),(16, 157, 724, 472, 556, 131, 571, 464, 88),
(39, 234, 702, 486, 522, 201, 558, 471, 126),(44, 131, 728, 376, 632, 91, 637, 364, 104),
(44, 284, 683, 332, 619, 236, 661, 292, 164),(44, 427, 604, 469, 484, 308, 572, 364, 299),
(52, 149, 724, 299, 668, 116, 676, 284, 107),(52, 299, 676, 404, 556, 277, 619, 388, 124),
(52, 299, 676, 416, 572, 221, 611, 364, 208),(52, 416, 611, 509, 424, 332, 536, 443, 256),
(54, 423, 606, 474, 486, 297, 567, 366, 306),(61, 124, 728, 196, 707, 104, 712, 184, 91),
(61, 452, 584, 508, 451, 296, 536, 376, 347),(67, 296, 676, 424, 571, 208, 604, 368, 221),
(68, 221, 704, 451, 572, 136, 584, 416, 187),(100, 416, 605, 445, 520, 284, 584, 325, 320),
(101, 208, 704, 256, 676, 163, 688, 221, 164),(104, 364, 637, 448, 541, 236, 581, 352, 296),
(112, 404, 611, 496, 413, 364, 539, 464, 208),(116, 355, 640, 515, 500, 184, 520, 416, 325)

7412 = 549081 appears in the decimal expression of e:
  e = 2.71828•••549081••• (from the 101577th digit).

Page of Squares : First Upload September 19, 2005 ; Last Revised September 7, 2011
by Yoshio Mimura, Kobe, Japan

742

The smallest squares containing k 742's :
17424 = 1322,
2742407424 = 523682,
489742742337424 = 221301322.

7422 = 550564, 5 + 5 + 0 + 5 + 6 + 4 = 52.

Page of Squares : First Upload September 19, 2005 ; Last Revised September 7, 2006
by Yoshio Mimura, Kobe, Japan

743

The smallest squares containing k 743's :
743044 = 8622,
17436674304 = 1320482,
1958743743743809 = 442576972.

7432 = 552049, 5 + 5 + 2 + 0 + 4 + 9 = 52.

7432 is the 4th square which is the sum of 8 sixth powers :
26 + 26 + 46 + 46 + 46 + 46 + 46 + 96.

The square root of 743 is 27. 25 8 0 2 6 ..., and 272 = 252 + 82 + 02 + 22 + 62.

7432 = 2222 + 3012 + 6422 : 2462 + 1032 + 2222 = 3472.

3-by-3 magic squares consisting of different squares with constant 7432:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(3, 86, 738, 486, 558, 67, 562, 483, 54),(3, 158, 726, 374, 627, 138, 642, 366, 77),
(18, 157, 726, 194, 702, 147, 717, 186, 58),(18, 275, 690, 310, 630, 243, 675, 282, 130),
(18, 354, 653, 387, 562, 294, 634, 333, 198),(42, 114, 733, 166, 717, 102, 723, 158, 66),
(42, 301, 678, 518, 498, 189, 531, 462, 238),(45, 318, 670, 410, 570, 243, 618, 355, 210),
(51, 238, 702, 438, 558, 221, 598, 429, 102),(58, 318, 669, 462, 509, 282, 579, 438, 158),
(77, 294, 678, 498, 483, 266, 546, 482, 147),(102, 387, 626, 474, 518, 243, 563, 366, 318),
(158, 339, 642, 381, 598, 222, 618, 282, 301) 

7432 = 24 + 174 + 224 + 224.

Page of Squares : First Upload September 19, 2005 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan

744

The smallest squares containing k 744's :
7744 = 882,
7447441 = 27292,
167445877447744 = 129400882.

744 = (12 + 22 + 32 + ... + 312) / (12 + 22 + 32).

7442 = 553536, a square with odd digits except the last digit 6.

The squares which begin with 744 and end in 744 are
74480959744 = 2729122,   744058057744 = 8625882,   744617119744 = 8629122,
744920895744 = 8630882,   7441503879744 = 27279122,...

7442 = 556516 is a square consisting of just 3 kinds of digits.

7442 = 553536, 5 + 5 + 35 + 36 = 92,
7442 = 553536, 5 + 535 + 36 = 242,
7442 = 553536, 553 + 536 = 332.

the cubic polynomial :
(X + 5762)(X + 7442)(X + 13932) = X3 + 16812X2 + 1378962 + 5969617922.

7442 = 553536 appears in the decimal expression of e:
  e = 2.71828•••553536••• (from the 94403rd digit).

Page of Squares : First Upload September 19, 2005 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan

745

The smallest squares containing k 745's :
74529 = 2732,
17454637456 = 1321162,
677450745517456 = 260278842.

7452 = 555025, a sqqure consisting of just 3 kinds of digits.

The square root of 745 is 27. 2 9 4 6 8 8 12 7 9 12 3 6 1 ...,
and 272 = 22 + 92 + 42 + 62 + 82 + 82 + 122 + 72 + 92 + 122 + 32 + 62 + 12.

3-by-3 magic squares consisting of different squares with constant 7452:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(0, 255, 700, 447, 560, 204, 596, 420, 153),(12, 191, 720, 295, 660, 180, 684, 288, 65),
(20, 120, 735, 231, 700, 108, 708, 225, 56),(20, 120, 735, 420, 609, 88, 615, 412, 84),
(20, 240, 705, 345, 628, 204, 660, 321, 128),(20, 240, 705, 516, 513, 160, 537, 484, 180),
(20, 321, 672, 420, 560, 255, 615, 372, 196),(20, 345, 660, 420, 540, 295, 615, 380, 180),
(48, 236, 705, 380, 615, 180, 639, 348, 160),(48, 264, 695, 439, 552, 240, 600, 425, 120),
(48, 385, 636, 511, 480, 252, 540, 420, 295),(56, 183, 720, 480, 560, 105, 567, 456, 160),
(159, 412, 600, 488, 516, 225, 540, 345, 380),(180, 351, 632, 380, 600, 225, 615, 268, 324)

Page of Squares : First Upload September 19, 2005 ; Last Revised August 25, 2009
by Yoshio Mimura, Kobe, Japan

746

The smallest squares containing k 746's :
746496 = 8642,
11746874689 = 1083832,
746746674675556 = 273266662.

7462 = 556516, a square consisting of just 3 kinds of digits.

7462± 3 are primes.

7462 = 2242 + 3422 + 6242 : 4262 + 2432 + 4222 = 6472.

7462 = 556516 appears in the decimal expression of π:
  π = 3.14159•••556516••• (from the 52213rd digit).

Page of Squares : First Upload September 19, 2005 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

747

The smallest squares containing k 747's :
174724 = 4182,
22374774724 = 1495822,
747317747174769 = 273371132.

7472 = 558009, 55 + 80 + 0 + 9 = 122.

7472 = (12 + 13 + 14 + 15 + 16 + 17)2 + (18 + 19 + 20 + 21 + 22 + 23)2 + (24 + 25 + 26 + 27 + 28 + 29)2 + ... + (60 + 61 + 62 + 63 + 64 + 65)2.

7472 = 123 + 173 + 823.

3-by-3 magic squares consisting of different squares with constant 7472:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(7, 86, 742, 346, 658, 73, 662, 343, 46),(7, 206, 718, 478, 553, 154, 574, 458, 137),
(7, 322, 674, 514, 487, 238, 542, 466, 217),(14, 473, 578, 502, 434, 343, 553, 382, 326),
(22, 137, 734, 487, 554, 118, 566, 482, 73),(24, 147, 732, 237, 696, 132, 708, 228, 69),
(38, 398, 631, 442, 521, 302, 601, 358, 262),(41, 202, 718, 302, 662, 169, 682, 281, 118),
(48, 141, 732, 204, 708, 123, 717, 192, 84),(48, 267, 696, 501, 528, 168, 552, 456, 213),
(55, 410, 622, 470, 503, 290, 578, 370, 295),(60, 372, 645, 525, 480, 228, 528, 435, 300),
(62, 409, 622, 442, 482, 361, 599, 398, 202),(71, 178, 722, 382, 631, 118, 638, 358, 151),
(94, 167, 722, 202, 706, 137, 713, 178, 134),(94, 167, 722, 377, 634, 118, 638, 358, 151),
(106, 358, 647, 503, 514, 202, 542, 407, 314),(110, 322, 665, 490, 535, 178, 553, 410, 290),
(118, 377, 634, 473, 454, 358, 566, 458, 167),(123, 372, 636, 444, 552, 237, 588, 339, 312),
(137, 262, 686, 326, 647, 182, 658, 266, 233) 

Page of Squares : First Upload September 19, 2005 ; Last Revised August 25, 2009
by Yoshio Mimura, Kobe, Japan

748

The smallest squares containing k 748's :
187489 = 4332,
5748217489 = 758172,
177487487488249 = 133224432.

Komachi Square Sum : 7482 = 22 + 32 + 72 + 4612 + 5892.

222k + 354k + 370k + 498k are squares for k = 1,2,3 (382, 7482, 151482).

Page of Squares : First Upload September 19, 2005 ; Last Revised March 23, 2011
by Yoshio Mimura, Kobe, Japan

749

The smallest squares containing k 749's :
749956 = 8662,
10749749761 = 1036812,
37490749374961 = 61229692.

3-by-3 magic squares consisting of different squares with constant 7492:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(16, 444, 603, 528, 421, 324, 531, 432, 304),(24, 83, 744, 268, 696, 69, 699, 264, 52),
(24, 205, 720, 380, 624, 165, 645, 360, 124),(24, 288, 691, 453, 556, 216, 596, 411, 192),
(29, 156, 732, 492, 556, 99, 564, 477, 124),(42, 434, 609, 511, 462, 294, 546, 399, 322),
(51, 272, 696, 456, 564, 187, 592, 411, 204),(60, 376, 645, 475, 480, 324, 576, 435, 200),
(69, 396, 632, 488, 456, 339, 564, 443, 216),(72, 339, 664, 389, 552, 324, 636, 376, 123),
(92, 291, 684, 324, 636, 227, 669, 268, 204),(93, 444, 596, 484, 492, 291, 564, 349, 348),
(96, 333, 664, 421, 576, 228, 612, 344, 261)

Page of Squares : First Upload September 19, 2005 ; Last Revised August 25, 2009
by Yoshio Mimura, Kobe, Japan