850
The smallest squares containing k 850's :
850084 = 9222,
88502085049 = 2974932,
1850985068501776 = 430230762.
Komachi square sum : 8502 = 82 + 162 + 3922 + 7542.
178k + 242k + 494k + 850k are squares for k = 1,2,3 (422, 10282, 274682).
8502 = (12 + 1)(22 + 1)(72 + 1)(382 + 1) = (22 + 1)(42 + 1)(72 + 1)(132 + 1)
= (32 + 1)(72 + 1)(382 + 1) = (82 + 4)(92 + 4)(112 + 4) = (112 + 4)(762 + 4)
= (12 + 9)(52 + 9)(462 + 9) = (42 + 9)(52 + 9)(292 + 9).
8502 + 8512 + 8522 + ... + 26362 = 768412,
8502 + 8512 + 8522 + ... + 2660742 = 792399602.
by Yoshio Mimura, Kobe, Japan
851
The smallest squares containing k 851's :
285156 = 5342,
2185188516 = 467462,
85185128517241 = 92295792.
Komachi fraction : 324/6517809 = (6/851)2.
8512 = 724201, 7 + 2 + 4 + 2 + 0 + 1 = 42.
3-by-3 magic squares consisting of different squares with constant 8512:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(6, 273, 806, 481, 666, 222, 702, 454, 159), | (6, 463, 714, 562, 534, 351, 639, 474, 302), |
(33, 86, 846, 166, 831, 78, 834, 162, 49), | (33, 246, 814, 474, 671, 222, 706, 462, 111), |
(50, 426, 735, 474, 625, 330, 705, 390, 274), | (58, 159, 834, 321, 778, 126, 786, 306, 113), |
(63, 426, 734, 526, 558, 369, 666, 481, 222), | (86, 369, 762, 534, 618, 239, 657, 454, 294), |
(111, 222, 814, 274, 786, 177, 798, 239, 174), | (111, 474, 698, 518, 594, 321, 666, 383, 366), |
(126, 342, 769, 526, 639, 198, 657, 446, 306), | (162, 366, 751, 399, 706, 258, 734, 303, 306), |
(166, 474, 687, 561, 582, 266, 618, 401, 426), | (174, 495, 670, 545, 450, 474, 630, 526, 225) |
8512 = (1 + 2 + 3 + ... + 23)2 + (24 + 25 + 26 + ... + 46)2.
8512 = 724201, 723 + 43 + 23 + 03 + 13 = 6112.
Page of Squares : First Upload December 5, 2006 ; Last Revised September 25, 2009by Yoshio Mimura, Kobe, Japan
852
The smallest squares containing k 852's :
85264 = 2922,
85285225 = 92352,
2664852668528521 = 516222112.
8522 = 725904, a square with different digits.
8522 = 4! + 5! + 9! + 9!.
143452 = 1312 + 1322 + 1332 + ... + 8522.
8522 = 725904, 7 + 2590 + 4 = 512.
Page of Squares : First Upload December 5, 2006 ; Last Revised September 25, 2006by Yoshio Mimura, Kobe, Japan
853
The smallest squares containing k 853's :
853776 = 9242,
85349285316 = 2921462,
8530853388539161 = 923626192.
3-by-3 magic squares consisting of different squares with constant 8532:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(12, 371, 768, 408, 672, 331, 749, 372, 168), | (20, 147, 840, 525, 660, 128, 672, 520, 75), |
(24, 128, 843, 452, 717, 96, 723, 444, 88), | (51, 528, 668, 592, 459, 408, 612, 488, 339), |
(52, 243, 816, 576, 592, 213, 627, 564, 128), | (56, 252, 813, 408, 723, 196, 747, 376, 168), |
(84, 187, 828, 372, 756, 133, 763, 348, 156), | (96, 243, 812, 488, 684, 147, 693, 448, 216), |
(123, 516, 668, 564, 452, 453, 628, 507, 276), | (164, 333, 768, 432, 704, 213, 717, 348, 304) |
8532 = 727609, 7 + 27 + 6 + 0 + 9 = 72,
8532 = 727609, 7 + 2 + 7 + 609 = 252,
8532 = 727609, 72 + 760 + 9 = 292.
by Yoshio Mimura, Kobe, Japan
854
The smallest squares containing k 854's :
788544 = 8882,
118548544 = 108882,
585473854585489 = 241965672.
8542 = 729316, a square with different digits.
Komachi equation : 8542 = 729316 (Cf. 567).
Komachi square sums : 8542 = 12 + 72 + 92 + 5842 + 6232 = 42 + 92 + 232 + 672 + 8512.
17934k + 43554k + 269010k + 398818k are squares for k = 1,2,3 (8542, 4833642, 2880798202).
8542 + 8552 + 8562 + ... + 8642 = 28492.
The 4-by-4 magic square consisting of different squares with constant 854:
|
8542 = 729316, 72 + 9 + 3 + 16 = 102.
8542 = 729316 appears in the decimal expression of π:
π = 3.14159•••729316••• (from the 47149th digit).
by Yoshio Mimura, Kobe, Japan
855
The smallest squares containing k 855's :
685584 = 8282,
855855025 = 292552,
3855855218554089 = 620955332.
8552 = 731025, a square with different digits.
13 - 23 + 33 - 43 + 53 - 63 + ... + 1133 = 8552.
8552 = 523 + 573 + 743.
141272 = 3022 + 3032 + 3042 + ... + 8552.
8552 + 8562 + 8572 + ... + 46212 = 1808162.
(1)(2 + 3 + ... + 28)(29 + 30 + ... + 66) = 8552,
(1 + 2 + ... + 9)(10 + 11 + ... + 180) = 8552.
The square root of 855 is 29. 2 4 0 3, and 29 = 22 + 42 + 02 + 32.
3-by-3 magic squares consisting of different squares with constant 8552:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(2, 430, 739, 595, 530, 310, 614, 515, 298), | (10, 130, 845, 403, 746, 110, 754, 397, 70), |
(10, 130, 845, 598, 605, 86, 611, 590, 98), | (10, 205, 830, 317, 770, 194, 794, 310, 67), |
(10, 205, 830, 445, 710, 170, 730, 430, 115), | (10, 334, 787 445, 670, 290, 730, 413, 166), |
(10, 445, 730, 541, 562, 350, 662, 466, 275), | (24, 375, 768, 543, 600, 276, 660, 480, 255), |
(33, 156, 840, 420, 735, 120, 744, 408, 105), | (35, 166, 838, 530, 662, 109, 670, 515, 130), |
(35, 218, 826, 370, 749, 182, 770, 350, 125), | (35, 370, 770, 530, 595, 310, 670, 490, 205), |
(35, 394, 758, 530, 605, 290, 670, 458, 269), | (45, 270, 810, 486, 675, 198, 702, 450, 189), |
(46, 122, 845, 178, 829, 110, 835, 170, 70), | (46, 290, 803, 550, 605, 250, 653, 530, 154), |
(60, 345, 780, 480, 660, 255, 705, 420, 240), | (60, 417, 744, 480, 600, 375, 705, 444, 192), |
(61, 290, 802, 490, 670, 205, 698, 445, 214), | (70, 131, 842, 170, 830, 115, 835, 158, 94), |
(70, 365, 770, 419, 658, 350, 742, 406, 125), | (70, 365, 770, 595, 574, 218, 610, 518, 301), |
(70, 566, 637, 595, 490, 370, 610, 413, 434), | (77, 250, 814, 590, 605, 130, 614, 550, 227), |
(94, 430, 733, 467, 590, 406, 710, 445, 170), | (110, 355, 770, 451, 682, 250, 718, 374, 275), |
(115, 362, 766, 430, 691, 262, 730, 350, 275), | (115, 430, 730, 562, 590, 259, 634, 445, 362), |
(120, 336, 777, 420, 705, 240, 735, 348, 264), | (130, 515, 670, 590, 430, 445, 605, 530, 290), |
(131, 458, 710, 530, 515, 430, 658, 506, 205), | (158, 275, 794, 310, 770, 205, 781, 250, 242), |
(170, 445, 710, 590, 578, 221, 595, 446, 422), | (170, 557, 626, 590, 374, 493, 595, 530, 310), |
(205, 430, 710, 490, 653, 254, 670, 346, 403) |
8552 = 731025, 7 + 3 + 1 + 0 + 25 = 62,
8552 = 731025, 73 + 1 + 0 + 2 + 5 = 92,
8552 = 731025, 7 + 310 + 2 + 5 = 182.
by Yoshio Mimura, Kobe, Japan
856
The smallest squares containing k 856's :
28561 = 1692,
685601856 = 261842,
85623856995856 = 92533162.
The squares which begin with 856 and end in 856 are
85663923856 = 2926842, 856209699856 = 9253162, 856890867856 = 9256842,
8560399265856 = 29258162, 8562552801856 = 29261842,...
8562 + 8572 + 8582 + ... + 8792 = 42502.
The square root of 856 is 29.25, and 29 = 22 + 52.
8562 = 732736, 7 + 3 + 273 + 6 = 172,
8562 = 732736, 73 + 2736 = 532.
8562 = 732736 appears in the decimal expression of π:
π = 3.14159•••732736••• (from the 97156th digit).
by Yoshio Mimura, Kobe, Japan
857
The smallest squares containing k 857's :
185761 = 4312,
5848578576 = 764762,
11857857538576 = 34435242.
3-by-3 magic squares consisting of different squares with constant 8572:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(9, 92, 852, 372, 768, 79, 772, 369, 48), | (15, 380, 768, 432, 660, 335, 740, 393, 180), |
(16, 288, 807, 327, 744, 272, 792, 313, 96), | (24, 273, 812, 588, 596, 183, 623, 552, 204), |
(47, 216, 828, 324, 772, 183, 792, 303, 124), | ( 57, 376, 768, 576, 552, 313, 632, 537, 216), |
(72, 537, 664, 583, 456, 432, 624, 488, 327), | (92, 336, 783, 567, 612, 196, 636, 497, 288), |
(92, 393, 756, 504, 588, 367, 687, 484, 168), | (119, 468, 708, 588, 561, 272, 612, 448, 399), |
(156, 457, 708, 488, 636, 303, 687, 348, 376), | (168, 524, 657, 564, 567, 308, 623, 372, 456) |
8572 = 734449, 73 + 4 * 4 * 49 = 857.
(13 + 23 + ... + 2973)(2983 + 2993 + ... + 8573) = 161514599402,
(13 + 23 + ... + 7593)(7603 + 7613 + ... + 8573) = 657588947402.
8572 = 734449, 7 + 344 + 49 = 202.
Page of Squares : First Upload December 5, 2006 ; Last Revised September 25, 2009by Yoshio Mimura, Kobe, Japan
858
The smallest squares containing k 858's :
85849 = 2932,
1858385881 = 431092,
6185868588985849 = 786502932.
8582 = 736164, a zigzag square.
8582 = 736164, 7 / 3 * 61 * 6 + 4 = 858.
The integral triangle of sides 401, 4097, 4290 has square area 8582.
858k + 9438k + 17394k + 27066k are squares for k = 1,2,3 (2342, 335402, 50923082).
858k + 16302k + 21450k + 43186k are squares for k = 1,2,3 (2862, 509082, 97337242).
72930k + 105534k + 247962k + 309738k are squares for k = 1,2,3 (8582, 4169882, 2156960522).
90090k + 155298k + 165594k + 325182k are squares for k = 1,2,3 (8582, 4066922, 2083344122).
The 4-by-4 magic square consisting of different squares with constant 858:
|
8582 = 736164, 7 + 3 + 6 + 16 + 4 = 62,
8582 = 736164, 7 + 3 + 6 + 1 + 64 = 92,
8582 = 736164, 7 + 3 + 61 + 6 + 4 = 92,
8582 = 736164, 73 + 6 + 1 + 64 = 122,
8582 = 736164, 73 + 61 + 6 + 4 = 122,
8582 = 736164, 736 + 164 = 302.
by Yoshio Mimura, Kobe, Japan
859
The smallest squares containing k 859's :
98596 = 3142,
18595685956 = 1363662,
2685985998597001 = 518264992.
(12 + 22 + ... + 622)(632 + 642 + ... + 2922)(2932 + 2942 + ... + 8592) = 116901697502.
3-by-3 magic squares consisting of different squares with constant 8592:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(6, 297, 806, 482, 666, 249, 711, 454, 162), | (14, 249, 822, 366, 742, 231, 777, 354, 94), |
(14, 294, 807, 402, 711, 266, 759, 382, 126), | (18, 454, 729, 489, 594, 382, 706, 423, 246), |
(39, 202, 834, 526, 654, 183, 678, 519, 94), | (57, 486, 706, 554, 519, 402, 654, 482, 279), |
(66, 311, 798, 473, 654, 294, 714, 462, 121), | (66, 391, 762, 426, 678, 311, 743, 354, 246), |
(66, 522, 679, 553, 546, 366, 654, 409, 378), | (158, 426, 729, 519, 634, 258, 666, 393, 374), |
(174, 519, 662, 554, 438, 489, 633, 526, 246) |
8592 = 737881, 7 + 3 + 78 + 81 = 132,
8592 = 737881, 73 + 7 + 8 + 81 = 132,
8592 = 737881, 73 + 7 + 88 + 1 = 132,
8592 = 737881, 73 + 7 + 881 = 312.
by Yoshio Mimura, Kobe, Japan