Squares:860s

860

The smallest squares containing k 860's :
418609 = 647²,
9860688601 = 99301²,
1286079386078601 = 35861949².

The square root of 860 is 29. 3 2 5 7 5 6 5 9 7 23 0 3,
29² = 3² + 2² + 5² + 7² + 5² + 6² + 5² + 9² + 7² + 23² + 0² + 3².

Komachi equations:
860² = 9² * 8² * 7² / 6² * 5² * 43² / 21² = 98² / 7² * 6² * 5² * 43² / 21².

860² = 739600 appears in the decimal expression of e:
e = 2.71828•••739600••• (from the 44657th digit)

Page of Squares : First Upload December 14, 2006 ; Last Revised July 6, 2010
by Yoshio Mimura, Kobe, Japan

861

The smallest squares containing k 861's :
861184 = 928²,
86198611216 = 293596²,
1086178613786176 = 32957224².

A cubic polynomial :
(X + 704²)(X + 861²)(X + 1008²) = X³ + 1501²X^2 + 1274448² + 610993152².

The square root of 861 is 29.3428015022424196611726312851072932061942622...,
and the sum of the squares of its digits is 29² : 3² + 4² + 2² + ... + 6² + 2² + 2².

The square root of 861 is 29.342..., and 29 = 3² + 4² + 2².

3-by-3 magic squares consisting of different squares with constant 861²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(4, 149, 848, 373, 764, 136, 776, 368, 61),	(4, 397, 764, 589, 556, 292, 628, 524, 269),
(11, 92, 856, 548, 661, 64, 664, 544, 67),	(11, 328, 796, 484, 656, 277, 712, 451, 176),
(16, 221, 832, 533, 656, 164, 676, 512, 149),	(16, 419, 752, 524, 592, 341, 683, 464, 244),
(18, 159, 846, 351, 774, 138, 786, 342, 81),	(29, 88, 856, 248, 821, 76, 824, 244, 53),
(29, 296, 808, 584, 587, 236, 632, 556, 181),	(40, 395, 764, 445, 664, 320, 736, 380, 235),
(43, 464, 724, 556, 568, 331, 656, 451, 328),	(49, 238, 826, 434, 721, 182, 742, 406, 161),
(54, 354, 783, 543, 594, 306, 666, 513, 186),	(64, 200, 835, 460, 715, 136, 725, 436, 160),
(64, 304, 803, 341, 748, 256, 788, 299, 176),	(68, 356, 781, 461, 676, 268, 724, 397, 244),
(68, 491, 704, 584, 544, 323, 629, 452, 376),	(76, 508, 691, 556, 499, 428, 653, 484, 284),
(78, 306, 801, 369, 738, 246, 774, 321, 198),	(88, 356, 779, 541, 584, 328, 664, 523, 164),
(90, 486, 705, 570, 495, 414, 639, 510, 270),	(104, 227, 824, 331, 776, 172, 788, 296, 181),
(115, 236, 820, 380, 755, 164, 764, 340, 205),	(116, 307, 796, 499, 676, 188, 692, 436, 269),
(159, 414, 738, 522, 639, 246, 666, 402, 369),	(164, 328, 779, 376, 739, 232, 757, 296, 284),
(164, 491, 688, 533, 604, 304, 656, 368, 419),	(172, 484, 691, 589, 436, 452, 604, 563, 244)

The 4-by-4 magic square consisting of different squares with constant 861:

2²	5²	16²	24²
7²	26²	6²	10²
18²	4²	20²	11²
22²	12²	13²	8²

861² = 741321, 7 + 4 + 1 + 3 + 21 = 6²,
861² = 741321, 7 + 41 + 32 + 1 = 9²,
861² = 741321, 74 + 1 + 3 + 2 + 1 = 9²,
861² = 741321, 7 + 4 + 132 + 1 = 12²,
861² = 741321, 7 + 413 + 21 = 21²,
861² = 741321, 7⁵ + 4⁵ + 13⁵ + 21⁵ = 2115².

Page of Squares : First Upload December 14, 2006 ; Last Revised October 2, 2009
by Yoshio Mimura, Kobe, Japan

862

The smallest squares containing k 862's :
286225 = 535²,
5862658624 = 76568²,
93786286286224 = 9684332².

(1³ + 2³ + ... + 62³)(63³ + 64² + ... + 122³)(123³ + 124² + ... + 862³) = 5261410123200².

862² = 743044, 7 + 4 + 30 + 4 + 4 = 7²,
862² = 743044, 74 + 3 + 0 + 44 = 11².

862² = 743044 appears in the decimal expression of π:
π = 3.14159•••743044••• (from the 52294th digit).

Page of Squares : First Upload December 14, 2006 ; Last Revised September 28, 2006
by Yoshio Mimura, Kobe, Japan

863

The smallest squares containing k 863's :
863041 = 929²,
78639863184 = 280428²,
2863401863863696 = 53510764².

863² = 33³ + 54³ + 82³.

3-by-3 magic squares consisting of different squares with constant 863²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(10, 210, 837, 405, 738, 190, 762, 395, 90),	(18, 91, 858, 442, 738, 69, 741, 438, 62),
(18, 414, 757, 514, 603, 342, 693, 458, 234),	(21, 262, 822, 318, 762, 251, 802, 309, 78),
(21, 558, 658, 602, 462, 411, 618, 469, 378),	(42, 206, 837, 606, 603, 118, 613, 582, 174),
(53, 162, 846, 378, 766, 123, 774, 363, 118),	(62, 174, 843, 213, 822, 154, 834, 197, 102),
(62, 330, 795, 570, 613, 210, 645, 510, 262),	(66, 267, 818, 378, 746, 213, 773, 342, 174),
(102, 549, 658, 582, 442, 459, 629, 498, 318),	(123, 294, 802, 498, 683, 174, 694, 438, 267),
(123, 386, 762, 602, 507, 354, 606, 582, 197),	(162, 395, 750, 550, 630, 213, 645, 438, 370),
(165, 370, 762, 462, 690, 235, 710, 363, 330)

863² = 744769, 7 + 4 + 4 + 76 + 9 = 10²,
863² = 744769, 74 + 4 + 7 + 6 + 9 = 10²,
863² = 744769, 7 + 4 + 4 + 769 = 28².

Page of Squares : First Upload December 14, 2006 ; Last Revised October 2, 2009
by Yoshio Mimura, Kobe, Japan

864

The smallest squares containing k 864's :
8649 = 93²,
148644864 = 12192²,
86486428836864 = 9299808².

The squares which begin with 864 and end in 864 are
864327214864 = 929692², 864542916864 = 929808², 8641789054864 = 2939692²,
8642471076864 = 2939808², 8644728996864 = 2940192²,...

864² = 746496, a zigzag square.

864² is the second square which is the sum of 3 fifth powers : 12⁵ + 12⁵ + 12⁵.

864² = 72³ + 72³.

30² + 864 = 42², 30² - 864 = 6².

A cubic polynomial :
(X + 264²)(X + 864²)(X + 1273²) = X³ + 1561²X² + 1172472² + 290366208².

Komachi equations:
864² = 1² * 2² * 3² / 4² * 56² / 7² * 8² * 9² = 9² * 8² * 7² * 6² * 5² * 4² * 3² / 210²
= 9² * 8² / 7² / 6² / 5² * 4² * 3² * 210²,
864² = 12³ * 3³ / 4³ * 56³ / 7³ + 8³ * 9³.

854² + 855² + 856² + ... + 864² = 2849².

(1³ + 2³ + ... + 519³)(520³ + 521³ + ... + 584³)(585³ + 586³ + ... + 864³) = 4697366113440000²,
(1³ + 2³ + ... + 504³)(505³ + 506³ + ... + 584³)(585³ + 586³ + ... + 864³) = 4819471604640000².

The square root of 864 is 29. 3 9 3 8 7 6 9 1 3 3 9 8 1 3 7 17 ...,
and 29² = 3² + 9² + 3² + 8² + 7² + 6² + 9² + 1² + 3² + 3² + 9² + 8² + 1² + 3² + 7² + 17².

864² = 746496, 7 + 4 + 6 + 4 + 9 + 6 = 6²,
864² = 746496, 74 + 6 + 496 = 24²,
864² = 746496, 74 + 649 + 6 = 27².

Page of Squares : First Upload December 14, 2006 ; Last Revised July 22, 2011
by Yoshio Mimura, Kobe, Japan

865

The smallest squares containing k 865's :
1478656 = 1216²,
86586593536 = 294256²,
7286586535865025 = 85361505².

865² = 1³ + 70³ + 74³.

1 / 865 = 0.001156..., and 1156 = 34².

Komachi equation: 865² = 9³ * 8³ + 76³ - 5³ / 4³ * 32³ + 1³.

3-by-3 magic squares consisting of different squares with constant 865²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(0, 260, 825, 519, 660, 208, 692, 495, 156),	(8, 240, 831, 456, 705, 208, 735, 440, 120),
(12, 384, 775, 559, 588, 300, 660, 505, 240),	(36, 145, 852, 552, 660, 89, 665, 540, 120),
(36, 273, 820, 327, 764, 240, 800, 300, 135),	(44, 345, 792, 495, 660, 260, 708, 440, 231),
(57, 324, 800, 476, 657, 300, 720, 460, 135),	(64, 135, 852, 252, 820, 111, 825, 240, 100),
(71, 240, 828, 540, 660, 145, 672, 505, 204),	(72, 321, 800, 496, 672, 225, 705, 440, 240),
(72, 460, 729, 604, 495, 372, 615, 540, 280),	(100, 303, 804, 600, 604, 153, 615, 540, 280),
(100, 516, 687, 600, 537, 316, 615, 440, 420),	(111, 420, 748, 548, 615, 264, 660, 440, 345),
(120, 440, 735, 540, 615, 280, 665, 420, 360),	(135, 460, 720, 512, 540, 441, 684, 495, 188),
(168, 449, 720, 575, 600, 240, 624, 432, 415)

865² = 748225, 7 + 48 + 2 + 2 + 5 = 8²,
865² = 748225, 7 + 4 + 82 + 2 + 5 = 10².

Page of Squares : First Upload December 14, 2006 ; Last Revised July 6, 2010
by Yoshio Mimura, Kobe, Japan

866

The smallest squares containing k 866's :
186624 = 432²,
17866866889 = 133667²,
186686668668889 = 13663333².

114745^k + 161509^k + 213469^k + 260233^k are squares for k = 1,2,3 (866², 390566², 181864330²).
130333^k + 135529^k + 239449^k + 244645^k are squares for k = 1,2,3 (866², 390566², 181864330²).

866² = 749956, 7 + 4 + 9 + 95 + 6 = 11²,
866² = 749956, 7 + 4 + 99 + 5 + 6 = 11²,
866² = 749956, 7 + 49 + 9 + 56 = 11².

Page of Squares : First Upload December 14, 2006 ; Last Revised March 29, 2011
by Yoshio Mimura, Kobe, Japan

867

The smallest squares containing k 867's :
586756 = 766²,
7486748676 = 86526²,
378678678670224 = 19459668².

867² = 751689, a square with different digits.

867² = 4³ + 36³ + 89³ = 17³ + 51³ + 85³.

(1² + 2² + ... + 433²)(434²)(435² + 436² + ... + 867²) = 31195991610².

867² = 751689, 7² + 5² + 1² + 6² + 8² + 9² = 16².

3-by-3 magic squares consisting of different squares with constant 867²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(17, 170, 850, 250, 815, 158, 830, 242, 65),	(17, 170, 850, 550, 655, 142, 670, 542, 95),
(17, 206, 842, 374, 758, 193, 782, 367, 74),	(17, 298, 814, 374, 737, 262, 782, 346, 143),
(17, 298, 814, 578, 641, 82, 646, 502, 287),	(17, 514, 698, 578, 527, 374, 646, 458, 353),
(34, 182, 847, 238, 817, 166, 833, 226, 82),	(34, 238, 833, 602, 593, 194, 623, 586, 142),
(47, 142, 854, 446, 737, 98, 742, 434, 113),	(47, 326, 802, 598, 593, 206, 626, 542, 257),
(49, 418, 758, 602, 562, 271, 622, 511, 322),	(58, 257, 826, 527, 646, 238, 686, 518, 113),
(58, 478, 721, 511, 602, 358, 698, 401, 322),	(72, 459, 732, 516, 612, 333, 693, 408, 324),
(82, 479, 718, 529, 542, 422, 682, 478, 241),	(84, 243, 828, 432, 732, 171, 747, 396, 192),
(95, 410, 758, 458, 670, 305, 730, 367, 290),	(117, 408, 756, 564, 612, 243, 648, 459, 348),
(158, 458, 719, 502, 641, 298, 689, 362, 382),	(193, 382, 754, 446, 703, 242, 718, 334, 353),
(238, 527, 646, 554, 602, 287, 623, 334, 502)

867² = 751689, 7 + 5 + 1 + 6 + 8 + 9 = 6²,
867² = 751689, 7 + 51 + 6 + 8 + 9 = 9²,
867² = 751689, 7² + 5² + 1² + 6² + 8² + 9² = 16²,
867² = 751689, 7 + 5168 + 9 = 72²,
867² = 751689, 75 + 1689 = 42².

Page of Squares : First Upload December 14, 2006 ; Last Revised October 2, 2009
by Yoshio Mimura, Kobe, Japan

868

The smallest squares containing k 868's :
386884 = 622²,
6038688681 = 77709²,
1868598687198681 = 43227291².

1 / 868 = 0.0011520737327188940092165898617 ...,
and the sum of the squares of its digits is 868 : 1² + 1² + 5² + ... + 6² + 1² + 7².

The square root of 868 is 29. 4 6 1 8 3 9 7 2 5 3 12 4 7 0 7 17 ...,
and 29² = 4² + 6² + 1² + 8² + 3² + 9² + 7² + 2² + 5² + 3² + 12² + 4² + 7² + 0² + 7² + 17².

868²± 3 are primes.

868² = (1² + 3)(5² + 3)(82² + 3).

868² = 753424, 7 + 5 + 3 + 4 + 2 + 4 = 5²,
868² = 753424, 7 + 53 + 424 = 22².

868² = 753424 appears in the decimal expression of e:
e = 2.71828•••753424••• (from the 32012nd digit)

Page of Squares : First Upload December 14, 2006 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

869

The smallest squares containing k 869's :
2486929 = 1577²,
128698692516 = 358746²,
869869078692849 = 29493543².

869² = (2² + 7)(262² + 7).

(1² + 2² + 3² + ... + 375²) + (1² + 2² + 3² + ... + 845²) = (1² + 2² + 3² + ... + 869²).

517^k + 869^k + 3245^k + 7469^k are squares for k = 1,2,3 (110², 8206², 672034²).

3-by-3 magic squares consisting of different squares with constant 869²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(3, 276, 824, 536, 648, 219, 684, 509, 168),	(4, 93, 864, 579, 644, 72, 648, 576, 59),
(4, 171, 852, 444, 732, 149, 747, 436, 84),	(12, 364, 789, 564, 597, 284, 661, 516, 228),
(21, 348, 796, 492, 661, 276, 716, 444, 213),	(40, 219, 840, 381, 760, 180, 780, 360, 131),
(60, 445, 744, 544, 600, 315, 675, 444, 320),	(66, 297, 814, 418, 726, 231, 759, 374, 198),
(67, 444, 744, 516, 579, 392, 696, 472, 219),	(93, 284, 816, 336, 768, 229, 796, 291, 192),
(96, 483, 716, 571, 576, 312, 648, 436, 381),	(108, 509, 696, 544, 504, 453, 669, 492, 256),
(131, 408, 756, 564, 536, 387, 648, 549, 184),	(144, 320, 795, 355, 756, 240, 780, 285, 256),
(144, 411, 752, 444, 688, 291, 733, 336, 324)

869² = 755161, 7 + 5 + 5 + 1 + 6 + 1 = 5²,
869² = 755161, 7 + 5 + 516 + 1 = 23².

Page of Squares : First Upload December 14, 2006 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan