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890

The smallest squares containing k 890's :
28900 = 170²,
2890890289 = 53767²,
289089028900 = 537670².

(343 / 890)² = 0.148527963... (Komachic).

890^k + 12104^k + 22873^k + 35422^k are squares for k = 1,2,3 (267², 43877², 7627923²).

890² + 891² + 892² + ... + 10786² = 646604²,
890² + 891² + 892² + ... + 12401² = 797206².

Page of Squares : First Upload January 9, 2006 ; Last Revised April 1, 2011
by Yoshio Mimura, Kobe, Japan

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891

The smallest squares containing k 891's :
198916 = 446²,
89189136 = 9444²,
148911891078916 = 12202946².

Komachi square sum : 891² = 189² + 432² + 756².

Komachi cubic sum : 891² = 1³ + 3³ + 5³ + 26³ + 49³ + 87³.

(1 + 2)(3 + 4 + 5 + 6 + 7 + 8)(9)(10 + 11 + 12)(13 + 14) = 891².

891² = 793881, 7 + 9 / 3 + 881 = 891.

3-by-3 magic squares consisting of different squares with constant 891²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(3, 384, 804, 624, 573, 276, 636, 564, 267),	(7, 94, 886, 626, 631, 62, 634, 622, 71),
(7, 254, 854, 334, 791, 238, 826, 322, 89),	(10, 455, 766, 505, 634, 370, 734, 430, 265),
(11, 308, 836, 484, 704, 253, 748, 451, 176),	(14, 151, 878, 487, 734, 134, 746, 482, 71),
(18, 279, 846, 441, 738, 234, 774, 414, 153),	(24, 228, 861, 588, 651, 156, 669, 564, 168),
(34, 298, 839, 514, 679, 262, 727, 494, 146),	(34, 521, 722, 554, 578, 391, 697, 434, 346),
(41, 146, 878, 526, 713, 94, 718, 514, 119),	(41, 386, 802, 542, 626, 329, 706, 503, 206),
(46, 182, 871, 377, 794, 146, 806, 361, 118),	(46, 503, 734, 599, 526, 398, 658, 514, 311),
(48, 381, 804, 444, 708, 309, 771, 384, 228),	(50, 370, 809, 409, 710, 350, 790, 391, 130),
(58, 329, 826, 574, 646, 217, 679, 518, 254),	(71, 158, 874, 214, 854, 137, 862, 199, 106),
(71, 346, 818, 398, 746, 281, 794, 343, 214),	(71, 398, 794, 622, 526, 361, 634, 599, 182),
(73, 326, 826, 554, 631, 298, 694, 538, 151),	(84, 501, 732, 579, 588, 336, 672, 444, 381),
(106, 358, 809, 617, 554, 326, 634, 599, 182),	(122, 274, 839, 311, 806, 218, 826, 263, 206),
(126, 423, 774, 522, 666, 279, 711, 414, 342),	(134, 262, 841, 314, 809, 202, 823, 266, 214),
(134, 295, 830, 470, 734, 185, 745, 410, 266),	(137, 434, 766, 526, 662, 281, 706, 409, 358),
(146, 409, 778, 617, 514, 386, 626, 602, 199),	(202, 521, 694, 601, 598, 274, 626, 406, 487),
(204, 501, 708, 552, 636, 291, 669, 372, 456),	(214, 574, 647, 601, 578, 314, 622, 361, 526)

891² = 793881, 7 + 9 + 3 + 8 + 8 + 1 = 6²,
891² = 793881, 7 + 9 + 3 + 881 = 30².

Page of Squares : First Upload January 9, 2006 ; Last Revised October 9, 2009
by Yoshio Mimura, Kobe, Japan

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892

The smallest squares containing k 892's :
168921 = 411²,
8928927049 = 94493²,
108923892828921 = 10436661².

892²± 3 are primes.

1² + 2² + ... + 892² = 236975410, a consecutive square sum with distinct digits.

6135² = 843² + 844² + 845² + ... + 892².

10^k + 218^k + 542^k + 674^k are squares for k = 1,2,3 (38², 892², 21812²).
154^k + 222^k + 258^k + 810^k are squares for k = 1,2,3 (38², 892², 23732²).

892² = 795664, 79 + 5 + 6 + 6 + 4 = 10².

1² + 2² + 3² + ... + 892² = 236975410, the first 9-digit sum with different digits,
(there are 3 such sums in all.)

Page of Squares : First Upload January 9, 2006 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

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893

The smallest squares containing k 893's :
389376 = 624²,
52089389361 = 228231²,
2388938938893529 = 48876773².

893² = 797449, a square with 3 kinds of digits.

(828 / 893)² = 0.859721436... (Komachic).

A cubic polynomial :
(X + 196)(X + 528)(X + 693) = X³ + 893²X² + 403788²X + 71717184².

517^k + 893^k + 3525^k + 3901^k are squares for k = 1,2,3 (94², 5358², 322514²).

3-by-3 magic squares consisting of different squares with constant 893²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(12, 173, 876, 264, 852, 43, 853, 204, 168),	(12, 173, 876, 524, 708, 147, 723, 516, 92),
(12, 264, 853, 597, 632, 204, 664, 573, 168),	(16, 93, 888, 267, 848, 84, 852, 264, 43),
(21, 92, 888, 412, 789, 72, 792, 408, 61),	(36, 488, 747, 587, 576, 348, 672, 477, 344),
(43, 168, 876, 492, 736, 117, 744, 477, 128),	(43, 300, 840, 600, 632, 195, 660, 555, 232),
(48, 331, 828, 448, 708, 309, 771, 432, 128),	(72, 384, 803, 484, 693, 288, 747, 412, 264),
(83, 324, 828, 468, 723, 236, 756, 412, 237),	(93, 432, 776, 624, 587, 252, 632, 516, 363),
(124, 237, 852, 348, 804, 173, 813, 308, 204),	(132, 588, 659, 621, 412, 492, 628, 531, 348),
(216, 547, 672, 597, 408, 524, 628, 576, 267)

893² = 797449, 7 + 97 + 4 + 4 + 9 = 11².

Page of Squares : First Upload January 9, 2006 ; Last Revised April 1, 2011
by Yoshio Mimura, Kobe, Japan

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894

The smallest squares containing k 894's :
89401 = 299²,
894189409 = 29903²,
8949894489469924 = 94603882².

Komachi square sum : 894² = 5² + 69² + 347² + 821².

894² = 5³ + 52³ + 87³.

59898^k + 167178^k + 206514^k + 365646^k are squares for k = 1,2,3 (894², 455940², 250160868²).

(3² + 4)(5² + 4)(6² + 4)(7² + 4) = 894² + 4,
(1² + 4)(2² + 4)(3² + 4)(5² + 4)(7² + 4) = 894² + 4.

The 4-by-4 magic squares consisting of different squares with constant 894:

894² = 799236, 7 + 9 + 9 + 2 + 3 + 6 = 6²,
894² = 799236, 7 + 9 + 92 + 36 = 12²,
894² = 799236, 7 + 99 + 2 + 36 = 12²,
894² = 799236, 79 + 9 + 236 = 18².

894² = 799236 appears in the decimal expression of π
  π = 3.14159•••799236••• (from the 65187th digit).

Page of Squares : First Upload January 9, 2006 ; Last Revised April 1, 2011
by Yoshio Mimura, Kobe, Japan

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895

The smallest squares containing k 895's :
5089536 = 2256²,
198950789521 = 446039²,
8958958895104 = 2993152².

3-by-3 magic squares consisting of different squares with constant 895²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(5, 210, 870, 570, 670, 165, 690, 555, 130),	(5, 354, 822, 570, 635, 270, 690, 522, 229),
(14, 405, 798, 555, 630, 310, 702, 490, 261),	(30, 130, 885, 171, 870, 122, 878, 165, 54),
(30, 130, 885, 410, 789, 102, 795, 402, 86),	(30, 390, 805, 427, 714, 330, 786, 373, 210),
(30, 390, 805, 630, 581, 258, 635, 558, 294),	(30, 410, 795, 630, 555, 310, 635, 570, 270),
(54, 453, 770, 490, 630, 405, 747, 446, 210),	(75, 270, 850, 346, 795, 222, 822, 310, 171),
(75, 294, 842, 450, 742, 219, 770, 405, 210),	(75, 450, 770, 518, 651, 330, 726, 418, 315),
(130, 390, 795, 555, 662, 234, 690, 459, 338),	(130, 555, 690, 597, 570, 346, 654, 410, 453),
(165, 570, 670, 618, 410, 501, 626, 555, 318),	(198, 486, 725, 570, 635, 270, 661, 402, 450),
(210, 526, 693, 555, 630, 310, 670, 357, 474)

895² = 801025, 8 + 0 + 1 + 0 + 2 + 5 = 4².

Page of Squares : First Upload January 9, 2006 ; Last Revised October 9, 2009
by Yoshio Mimura, Kobe, Japan

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896

The smallest squares containing k 896's :
26896 = 164²,
498896896 = 22336²,
88968962416896 = 9432336².

The squares which begin with 896 and end in 896 are
8961272896 = 94664²,   89602040896 = 299336²,   896172728896 = 946664²,
896498410896 = 946836²,   8960060408896 = 2993336²,...

1² + 2² + ... + 896² = 240175936, a consecutive square sum with distinct digits.

(345 / 896)² = 0.148259376... (Komachic).

896² = (5² + 7)(7² + 7)(21² + 7).

896² is the 6th square which is the sum of 7 sixth powers : 4⁶ + 4⁶ + 4⁶ + 4⁶ + 8⁶ + 8⁶ + 8⁶.

A cuic polynomial :
(X + 896)(X + 1269)(X + 2268) = X³ + 2749²X² + 3702132²X + 2578770432².

Komachi equation: 896² = - 9² * 8² + 7² * 6² * 5² * 4² + 32² * 10².

1² + 2² + 3² + ... + 896² = 240175936, the second 9-digit sum with different digits (see 892).

896² + 897² + 898² + ... + 6513² = 303107².

896² = 802816, 8 + 0 + 2 + 8 + 1 + 6 = 5²,
896² = 802816, 8² + 0² + 2² + 8² + 1² + 6² = 13²,
896² = 802816, 80 + 2 + 81 + 6 = 13²,
896² = 802816, 80² + 2² + 8² + 16² = 82².

Page of Squares : First Upload January 9, 2006 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

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897

The smallest squares containing k 897's :
678976 = 824²,
95589798976 = 309176²,
128974089768976 = 11356676².

897²± 2 are primes.

897² = 25³ + 68³ + 78³.

897^k + 2806^k + 9890^k + 12328^k are squares for k = 1,2,3 (161², 16077², 1692271²).

897² + 898² + 899² + ... + 1425² = 26933²,
897² + 898² + 899² + ... + 1760² = 39732².

3-by-3 magic squares consisting of different squares with constant 897²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(7, 152, 884, 268, 844, 143, 856, 263, 52),	(7, 524, 728, 616, 532, 377, 652, 497, 364),
(18, 261, 858, 453, 738, 234, 774, 438, 117),	(21, 162, 882, 558, 693, 114, 702, 546, 117),
(25, 172, 880, 628, 625, 140, 640, 620, 103),	(31, 208, 872, 488, 728, 191, 752, 481, 88),
(32, 89, 892, 212, 868, 79, 871, 208, 52),	(32, 353, 824, 464, 712, 287, 767, 416, 208),
(40, 455, 772, 620, 572, 305, 647, 520, 340),	(44, 292, 847, 487, 704, 268, 752, 473, 124),
(52, 143, 884, 416, 788, 103, 793, 404, 112),	(52, 208, 871, 572, 679, 128, 689, 548, 172),
(52, 416, 793, 527, 628, 364, 724, 487, 208),	(52, 472, 761, 572, 569, 392, 689, 508, 268),
(54, 318, 837, 507, 702, 234, 738, 459, 222),	(65, 472, 760, 520, 640, 353, 728, 415, 320),
(68, 199, 872, 532, 712, 121, 719, 508, 172),	(68, 256, 857, 364, 793, 208, 817, 332, 164),
(88, 313, 836, 583, 616, 292, 676, 572, 143),	(89, 448, 772, 592, 551, 388, 668, 548, 241),
(102, 486, 747, 549, 558, 438, 702, 507, 234),	(128, 359, 812, 409, 752, 268, 788, 332, 271),
(136, 548, 697, 583, 476, 488, 668, 527, 284),	(143, 388, 796, 572, 656, 217, 676, 473, 352),
(208, 383, 784, 416, 752, 257, 767, 304, 352),	(212, 497, 716, 584, 628, 263, 647, 404, 472)

897² = 804609, 8³ + 0³ + 4³ + 6³ + 0³ + 9³ = 39²,
897² = 804609, 8 + 0 + 4 + 60 + 9 = 9².

897² = 804609 appears in the decimal expression of π
  π = 3.14159•••804609••• (from the 25022th digit).

Page of Squares : First Upload January 9, 2006 ; Last Revised December 29, 2013
by Yoshio Mimura, Kobe, Japan

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898

The smallest squares containing k 898's :
589824 = 768²,
117089889856 = 342184²,
2465898689889856 = 49657816².

898² = 806404, a square with even digits.

(3² - 4)(4² - 4)(10² - 4)(12² - 4) = 898² - 4.

2! + 2! + 8! + 8! + 9! + 9! = 898².

Page of Squares : First Upload January 9, 2006 ; Last Revised October 2, 2006
by Yoshio Mimura, Kobe, Japan

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899

The smallest squares containing k 899's :
148996 = 386²,
18999589921 = 137839²,
69899899899456 = 8360616².

899² = 808201, 80 + 820 - 1 = 899.

899² = 49³ + 60³ + 78³.

899² + 900² + 901² + ... + 1137² = 15774²,
899² + 900² + 901² + ... + 1210² = 18694²,
899² + 900² + 901² + ... + 2457² = 68596²,
899² + 900² + 901² + ... + 2810² = 84606²,
899² + 900² + 901² + ... + 8698² = 468130².

(1³ + ... + 836³)(837³ + ... + 899³) = 71061982992²,
(1³ + ... + 125³)(126³ + ... + 476³)(477³ + ... + 899³) = 346305700377000².

3-by-3 magic squares consisting of different squares with constant 899²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(1, 174, 882, 414, 783, 154, 798, 406, 81),	(15, 226, 870, 530, 705, 174, 726, 510, 145),
(26, 225, 870, 495, 730, 174, 750, 474, 145),	(26, 303, 846, 342, 786, 271, 831, 314, 138),
(33, 334, 834, 594, 618, 271, 674, 561, 198),	(54, 321, 838, 433, 726, 306, 786, 422, 111),
(66, 161, 882, 273, 846, 134, 854, 258, 111),	(66, 294, 847, 609, 638, 174, 658, 561, 246),
(78, 334, 831, 534, 687, 226, 719, 474, 258),	(98, 321, 834, 609, 638, 174, 654, 546, 287),
(114, 593, 666, 622, 534, 369, 639, 414, 478),	(126, 522, 721, 566, 609, 342, 687, 406, 414),
(134, 513, 726, 618, 474, 449, 639, 566, 282),	(161, 522, 714, 582, 609, 314, 666, 406, 447)

899² = 808201, 8² + 0² + 8² + 20² + 1² = 23²,
899² = 808201, 8³ + 0³ + 8³ + 20³ + 1³ = 95²,
899² = 808201, 80 + 8 + 201 = 17²,
899² = 808201, 80 + 8201 = 91².

Page of Squares : First Upload January 9, 2006 ; Last Revised October 9, 2009
by Yoshio Mimura, Kobe, Japan

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