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850 - 859

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.

Page of Squares : First Upload December 5, 2006 ; Last Revised December 7, 2013
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:

A2B2C2
D2E2F2
G2H2K2
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, 2009
by 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, 2006
by 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:

A2B2C2
D2E2F2
G2H2K2
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.

Page of Squares : First Upload December 5, 2006 ; Last Revised September 25, 2009
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:

02 32192222
62272 52 82
172 42182152
232102122 92

8542 = 729316, 72 + 9 + 3 + 16 = 102.

8542 = 729316 appears in the decimal expression of π:
  π = 3.14159•••729316••• (from the 47149th digit).

Page of Squares : First Upload December 5, 2006 ; Last Revised March 29, 2011
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:

A2B2C2
D2E2F2
G2H2K2
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.

Page of Squares : First Upload December 5, 2006 ; Last Revised September 25, 2009
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).

Page of Squares : First Upload December 5, 2006 ; Last Revised September 25, 2006
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:

A2B2C2
D2E2F2
G2H2K2
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, 2009
by 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:

12 22182232
42272 72 82
202 52172122
212102142112

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.

Page of Squares : First Upload December 5, 2006 ; Last Revised September 30, 2011
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:

A2B2C2
D2E2F2
G2H2K2
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.

Page of Squares : First Upload December 5, 2006 ; Last Revised September 25, 2009
by Yoshio Mimura, Kobe, Japan