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, 2011by 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:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
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).
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, 2006by 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:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
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, 2013by 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).
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:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
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).
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:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
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, 2011by 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:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
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