Squares:800s

800

The smallest squares containing k 800's :
80089 = 283²,
3080028004 = 55498²,
15968008008001 = 3995999².

800² = 40³ + 40³ + 80³ = 20⁴ + 20⁴ + 20⁴ + 20⁴.

800² + 801² + 802² + ... + 4223² = 157932².

800² = 640000 appears in the decimal expression of π:
π = 3.14159•••640000••• (from the 37320th digit).

Page of Squares : First Upload October 31, 2005 ; Last Revised September 19, 2006
by Yoshio Mimura, Kobe, Japan

801

The smallest squares containing k 801's :
9801 = 99²,
16801801 = 4099²,
801801462577801 = 28316099².

The squares which begin with 801 and end in 801 are
80145043801 = 283099², 80174488801 = 283151², 801202219801 = 895099²,
801295312801 = 895151², 801649831801 = 895349²,...

Komachi equations: 801² = 9⁶ + 8⁶ - 7⁶ - 6⁶ + 5⁶ - 4⁶ + 3⁶ + 2⁶ - 1⁶.

3-by-3 magic squares consisting of different squares with constant 801²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(1, 448, 664, 512, 511, 344, 616, 424, 287),	(7, 164, 784, 316,721, 148, 736, 308, 71),
(8, 89, 796, 359, 712, 76, 716, 356, 47),	(21, 246, 762, 426, 642, 219, 678, 411, 114),
(28, 239, 764, 424, 644, 217, 679, 412, 104),	(30, 390, 699, 474, 555, 330, 645, 426, 210),
(32, 161, 784, 511, 608, 104, 616, 496, 127),	(44, 209, 772, 313, 716, 176, 736, 292, 121),
(44, 296, 743, 401, 652, 236, 692, 359, 184),	(47, 236, 764, 524, 569, 208, 604, 512, 121),
(47, 356, 716, 524, 593, 124, 604, 404, 337),	(56, 281, 748, 473, 616, 196, 644, 428, 209),
(69, 474, 642, 546, 498, 309, 582, 411, 366),	(71, 352, 716, 404, 604, 337, 688, 391, 124),
(76, 268, 751, 544, 569, 148, 583, 496, 236),	(89, 356, 712, 484, 593, 236, 632, 404, 281),
(103, 344, 716, 436, 628, 239, 664, 359, 268),	(124, 524, 593, 551, 488, 316, 568,359,436),
(138, 309, 726, 366, 678, 219, 699, 294, 258),	(175, 476, 620, 524, 400, 455, 580,505,224)

801² = 641601, 64 + 16 + 0 + 1 = 9²,
801² = 641601, 64 + 160 + 1 = 15².

Page of Squares : First Upload October 31, 2005 ; Last Revised July 2, 2010
by Yoshio Mimura, Kobe, Japan

802

The smallest squares containing k 802's :
38025 = 195²,
3458028025 = 58805²,
426528028028025 = 20652555².

(303 / 802)² = 0.142736985... (Komachic).

802² = 1⁴ + 17⁴ + 23⁴ + 23⁴.

190^k + 338^k + 786^k + 802^k are squares for k = 1,2,3 (46², 1188², 32356²).
173^k + 245^k + 397^k + 629^k are squares for k = 1,2,3 (38², 802², 18202²).

802² + 803² + 804² + ... + 1194² = 19912².

802² = 643204, 6² + 4² + 3² + 2² + 0² + 4² = 9²,
802² = 643204, 64 + 32 + 0 + 4 = 10².

802² = 643204 appears in the decimal expression of e:
e = 2.71828•••643204••• (from the 50126th digit)

Page of Squares : First Upload October 31, 2005 ; Last Revised March 29, 2011
by Yoshio Mimura, Kobe, Japan

803

The smallest squares containing k 803's :
988036 = 994²,
6803280324 = 82482²,
78034803380361 = 8833731².

3-by-3 magic squares consisting of different squares with constant 803²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(6, 178, 783, 438, 657, 146, 673, 426, 102),	(15, 430, 678, 522, 510, 335, 610, 447, 270),
(18, 367, 714, 417, 606, 322, 686, 378, 177),	(30, 322, 735, 497, 570, 270, 630, 465, 178),
(38, 414, 687, 447, 582, 326, 666, 367, 258),	(63, 354, 718, 558, 497, 294, 574, 522, 207),
(78, 241, 762, 538, 582, 129, 591, 498, 218),	(97, 234, 762, 438, 657, 146, 666, 398, 207),
(106, 258, 753, 402, 673, 174, 687, 354, 218),	(106, 417, 678, 543, 538, 246, 582, 426, 353),
(111, 342, 718, 542, 498, 321, 582, 529, 162),	(126, 497, 618, 543, 402, 434, 578, 486, 273)

803² = 644809, 64 + 48 + 0 + 9 = 11².

Page of Squares : First Upload October 31, 2005 ; Last Revised September 14, 2009
by Yoshio Mimura, Kobe, Japan

804

The smallest squares containing k 804's :
40804 = 202²,
823804804 = 28702²,
80484187804804 = 8971298².

The squares which begin with 804 and end in 804 are
8046448804 = 89702², 80486824804 = 283702², 804074476804 = 896702²,
804246652804 = 896798², 804971428804 = 897202²,...

804² = 646416, a square with just three kinds of digits.

804²± 5 are primes.

804²=646416, 64 & 16 are squares.

804²= 31³ + 44³ + 81³.

804² = 646416 is an exchangeable square, 166464 = 408².

556² + 557² + 558² + ... + 804² = 10790².

804² = 646416, 6 + 4 + 6 + 4 + 16 = 6²,
804² = 646416, 6 + 4 + 64 + 1 + 6 = 9²,
804² = 646416, 64 + 6 + 4 + 1 + 6 = 9²,
804² = 646416, 64 + 64 + 16 = 12².

804² = 646416 appears in the decimal expression of e:
e = 2.71828•••646416••• (from the 114647th digit)

Page of Squares : First Upload October 31, 2005 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

805

The smallest squares containing k 805's :
2805625 = 1675²,
8055780516 = 89754²,
4028058058056256 = 63466984².

The square root of 805 is 28. 3 7 2 5 2 19 18 2 2 ..., and
28² = 3² + 7² + 2² + 5² + 2² + 19² + 18² + 2² + 2².

805² = 648025, a square with different digits.

27370^k + 141680^k + 191590^k + 287385^k are squares for k = 1,2,3 (805², 374325², 183391075²).

3-by-3 magic squares consisting of different squares with constant 805²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(4, 300, 747, 540, 555, 220, 597, 500, 204),	(5, 144, 792, 360, 708, 131, 720, 355, 60),
(27, 264, 760, 536, 573, 180, 600, 500, 195),	(36, 248, 765, 285, 720, 220, 752, 261, 120),
(40, 120, 795, 228, 765, 104, 771, 220, 72),	(40, 195, 780, 507, 600, 176, 624, 500, 93),
(40, 381, 708, 480, 580, 285, 645, 408, 256),	(40, 480, 645, 564, 477, 320, 573, 436, 360),
(48, 261, 760, 436, 648, 195, 675, 400, 180),	(60, 355, 720, 555, 540, 220, 580, 480, 285),
(60, 363, 716, 555, 500, 300, 580, 516, 213),	(67, 300, 744, 456, 600, 283, 660, 445, 120),
(85, 180, 780, 324, 725, 132, 732, 300, 149),	(96, 428, 675, 472, 579, 300, 645, 360, 320),
(99, 220, 768, 320, 720, 165, 732, 285, 176),	(120, 445, 660, 480, 492, 419, 635, 456, 192),
(120, 480, 635, 509, 540, 312, 612, 355, 384),	(165, 320, 720, 384, 675, 212, 688, 300, 291),
(171, 428, 660, 540, 555, 220, 572, 396, 405)

805² = 648025, 6 + 4 + 8 + 0 + 2 + 5 = 5²,
805² = 648025, 64 + 80 + 25 = 13².

805² + 806² + 807² + ... + 1883² = 45318².

Page of Squares : First Upload October 31, 2005 ; Last Revised March 29, 2011
by Yoshio Mimura, Kobe, Japan

806

The smallest squares containing k 806's :
80656 = 284²,
36806806201 = 191851²,
1180638806180625 = 34360425².

806²± 3 are primes.

806² = 649636, 64 + 96 + 3 + 6 = 13²,
806² = 649636, 64 + 96 + 36 = 14².

253² + 254² + 255² + ... + 806² = 13019².

The 4-by-4 magic squares consisting of different squares with constant 806:

0²	2²	19²	21²
9²	1²	20²	18²
14²	24²	3²	5²
23²	15²	6²	4²

3²	10²	16²	21²
12²	7²	17²	18²
13²	24²	6²	5²
22²	9²	15²	4²

806² = 649636 appears in the decimal expression of e:
e = 2.71828•••649636••• (from the 80594th digit).

Page of Squares : First Upload October 31, 2005 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

807

The smallest squares containing k 807's :
4280761 = 2069²,
88074807076 = 296774²,
807807980773369 = 28421963².

807² = 651249, a square with different digits.

807²± 2 are primes.

590² + 591² + 592² + ... + 807² = 10355².

807² + 808² + 809² + ... + 2487² = 70397².

807⁵ = 342269084820807,
807 = 3² + 4² + 2² + 2² + 6² + 9² + 0² + 8² + 4² + 8² + 20² + 8² + 0² + 7².

3-by-3 magic squares consisting of different squares with constant 807²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(13, 214, 778, 514, 602, 157, 622, 493, 146),	(14, 178, 787, 563, 566, 118, 578, 547, 134),
(14, 317, 742, 467, 602, 266, 658, 434, 173),	(22, 202, 781, 461, 638, 178, 662, 451, 98),
(22, 253, 766, 307, 706, 242, 746, 298, 77),	(24, 153, 792, 468, 648, 111, 657, 456, 108),
(34, 83, 802, 122, 794, 77, 797, 118, 46),	(35, 382, 710, 430, 610, 307, 682, 365, 230),
(38, 211, 778, 322, 718, 179, 739, 302, 118),	(38, 298, 749, 398, 659, 242, 701, 358, 178),
(50, 157, 790, 365, 710, 118, 718, 350, 115),	(72, 273, 756, 441, 648, 192, 672, 396, 207),
(74, 382, 707, 482, 547, 346, 643, 454, 178),	(77, 518, 614, 562, 406, 413, 574, 467, 322),
(83, 382, 706, 454, 563, 358, 662, 434, 157),	(109, 298, 742, 482, 622, 179, 638, 419, 262),
(118, 349, 718, 382, 662, 259, 701, 302, 262),	(118, 382, 701, 563, 454, 358, 566, 547, 178),
(118, 410, 685, 515, 490, 382, 610, 493, 190),	(130, 470, 643, 518, 445, 430, 605, 482, 230),
(206, 413, 662, 442, 626, 253, 643, 298, 386)

807² = 651249, 6 + 5 + 12 + 4 + 9 = 6²,
807² = 651249, 65 + 1 + 2 + 4 + 9 = 9²,
807² = 651249, 6 + 5 + 124 + 9 = 12²,
807² = 651249, 651 + 249 = 30²,
807² = 651249, 6512 + 49 = 81².

Page of Squares : First Upload October 31, 2005 ; Last Revised December 29, 2013
by Yoshio Mimura, Kobe, Japan

808

The smallest squares containing k 808's :
58081 = 241²,
80838081 = 8991²,
80898080080896 = 8994336².

808² is the eight square which is the sum of 10 sixth powers.

808² = 31² + 33² + 35² + 37² + 39² + 41² + 43² + ... + 157².

808² = 86³ + 7⁵ + 1⁷.

808² = 652864, 6 + 5 + 28 + 6 + 4 = 7²,
808² = 652864, 6² + 5² + 2² + 8² + 64² = 65²,
808² = 652864, 65³ + 2³ + 8³ + 64³ = 733².

808² + 809² + 810² + ... + 1671² = 37164².
808² + 809² + 810² + ... + 13106² = 866201².

Page of Squares : First Upload October 31, 2005 ; Last Revised January 6, 2011
by Yoshio Mimura, Kobe, Japan

809

The smallest squares containing k 809's :
2809 = 53²,
809231809 = 28447²,
58091880970809 = 7621803².

The squares which begin with 809 and end in 809 are
809231809 = 28447², 8090462809 = 89947², 80910095809 = 284447²,
80970409809 = 284553², 809004905809 = 899447²,...

809² is the ninth square which is the sum of 10 sixth powers.

Komachi equation: 809² = 9² + 8² - 7² * 6² + 54² * 3² / 2² * 10².

3-by-3 magic squares consisting of different squares with constant 809²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(1, 288, 756, 432, 639, 244, 684, 404, 153),	(9, 384, 712, 568, 504, 279, 576, 503, 264),
(12, 89, 804, 496, 636, 63, 639, 492, 64),	(24, 287, 756, 532, 564, 231, 609, 504, 172),
(33, 264, 764, 316, 708, 231, 744, 289, 132),	(36, 449, 672, 537, 516, 316, 604, 432, 321),
(44, 348, 729, 513, 576, 244, 624, 449, 252),	(48, 471, 656, 504, 496, 393, 631, 432, 264),
(100, 384, 705, 516, 575, 240, 615, 420, 316),	(120, 316, 735, 415, 660, 216, 684, 345, 260),
(153, 476, 636, 516, 552, 289, 604, 351, 408),	(156, 359, 708, 447, 636, 224, 656, 348, 321)

809² = 654481, 6 + 5 + 4 + 48 + 1 = 8²,
809² = 654481, 6 + 5 + 44 + 8 + 1 = 8²,
809² = 654481, 6 + 5 + 4 + 4 + 81 = 10².

Page of Squares : First Upload October 31, 2005 ; Last Revised July 2, 2010
by Yoshio Mimura, Kobe, Japan