Squares:820s

820

The smallest squares containing k 820's :
88209 = 297²,
388208209 = 19703²,
58203182068201 = 7629101².

1 / 820 = 0.00121..., and 121 = 11².

820² + 821² + 822² + ... + 840² = 841² + 842² + 843² + ... + 860².

Komachi equation: 820² = - 1³ * 2³ - 3³ * 4³ + 5³ + 67³ + 8³ * 9³.

Page of Squares : First Upload November 14, 2005 ; Last Revised September 7, 2011
by Yoshio Mimura, Kobe, Japan

821

The smallest squares containing k 821's :
582169 = 763²,
6482182144 = 80512²,
2298219821821504 = 47939752².

821² = 674041, 6 + 74 + 0 + 41 = 11².

1 / 821 = 0.00121..., and 121 = 11².

3-by-3 magic squares consisting of different squares with constant 821²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(11, 216, 792, 504, 627, 164, 648, 484, 141),	(20, 165, 804, 204, 780, 155, 795, 196, 60),
(21, 88, 816, 144, 804, 83, 808, 141, 36),	(24, 252, 781, 363, 704, 216, 736, 339, 132),
(24, 267, 776, 371, 696, 228, 732, 344, 141),	(27, 164, 804, 564, 588, 101, 596, 549, 132),
(36, 372, 731, 556, 549, 252, 603, 484, 276),	(45, 160, 804, 420, 696, 115, 704, 405, 120),
(56, 396, 717, 501, 552, 344, 648, 461, 204),	(141, 412, 696, 456, 624, 277, 668, 339, 336),
(192, 504, 619, 531, 556, 288, 596, 333, 456),	(204, 493, 624, 531, 564, 272, 592, 336, 459)

821² = 674041 appears in the decimal expression of e:
e = 2.71828•••674041••• (from the 43654th digit).

Page of Squares : First Upload November 14, 2005 ; Last Revised September 18, 2009
by Yoshio Mimura, Kobe, Japan

822

The smallest squares containing k 822's :
18225 = 135²,
1882258225 = 43385²,
822690682282225 = 28682585².

Komachi cubic sum : 822² = 4³ + 5³ + 9³ + 16³ + 23³ + 87³.

822² = 675684, 6 + 7 + 5 + 6 + 8 + 4 = 6²,
822² = 675684, 6 + 7 + 56 + 8 + 4 = 9²,
822² = 675684, 67 + 5 + 68 + 4 = 12².

1 / 822 = 0.00121..., and 121 = 11².

226^k + 610^k + 822^k + 1478^k are squares for k = 1,2,3 (56², 1812², 63424²).

The 4-by-4 magic squares consisting of different squares with constant 822:

0²	1²	14²	25²
5²	22²	13²	12²
11²	16²	21²	2²
26²	9²	4²	7²

0²	2²	17²	23²
5²	21²	16²	10²
11²	19²	14²	12²
26²	4²	9²	7²

1²	7²	14²	24²
9²	25²	4²	10²
16²	2²	21²	11²
22²	12²	13²	5²

Page of Squares : First Upload November 14, 2005 ; Last Revised March 29, 2011
by Yoshio Mimura, Kobe, Japan

823

The smallest squares containing k 823's :
82369 = 287²,
18238232401 = 135049²,
1823823618237504 = 42706248².

(797 / 823)² = 0.937814562... (Komachic).

Komachi equations:
823² = 9876² * 5² / 4² / 3² * 2² / 10² = 9876² / 5² / 4² / 3² / 2² * 10².

3-by-3 magic squares consisting of different squares with constant 823²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(22, 126, 813, 243, 778, 114, 786, 237, 58),	(22, 387, 726, 483, 594, 302, 666, 418, 243),
(30, 302, 765, 477, 630, 230, 670, 435, 198),	(30, 435, 698, 573, 490, 330, 590, 498, 285),
(54, 198, 797, 498, 643, 126, 653, 474, 162),	(58, 282, 771, 579, 562, 162, 582, 531, 238),
(77, 186, 798, 402, 707, 126, 714, 378, 157),	(78, 301, 762, 509, 582, 282, 642, 498, 131),
(78, 366, 733, 414, 653, 282, 707, 342, 246),	(93, 426, 698, 562, 477, 366, 594, 518, 237),
(147, 274, 762, 342, 723, 194, 734, 282, 243),	(194, 477, 642, 558, 554, 243,573,378, 454)

823² = 677329, 67 + 73 + 29 = 13²,
823² = 677329, 67 + 7329 = 86².

Page of Squares : First Upload November 14, 2005 ; Last Revised July 6, 2010
by Yoshio Mimura, Kobe, Japan

824

The smallest squares containing k 824's :
53824 = 232²,
82428241 = 9079²,
13824282482404 = 3718102².

The squares which begin with 824 and end in 824 are
824042741824 = 907768², 824885365824 = 908232², 824950759824 = 908268²,
8241102215824 = 2870732², 8241308909824 = 2870768²,...

824² = 678976, 6 * 7 + 8 * 97 + 6 = 824.

824² = 678976, a square every digit of which is greater than 5.

824² = 678976, 6 + 78 + 9 + 76 = 13²,
824² = 678976, 67 + 89 + 7 + 6 = 13².

1 / 824 = 0.00121..., and 121 = 11².

824² = 678976 appears in the decimal expression of π:
π = 3.14159•••678976••• (from the 9771st digit),
(678976 is the fourth 6-digit square in the expression of π.)

Page of Squares : First Upload November 14, 2005 ; Last Revised September 7, 2013
by Yoshio Mimura, Kobe, Japan

825

The smallest squares containing k 825's :
968256 = 984²,
108251528256 = 329016²,
482527825178256 = 21966516².

825² = 680625, a zigzag square.

825² = (3² + 6)(213² + 6).

Komachi equation: 825² = 9² * 87² + 65² * 4² + 3² * 2² - 10².

825² = 40³ + 44³ + 81³.

3-by-3 magic squares consisting of different squares with constant 825²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(4, 128, 815, 535, 620, 100, 628, 529, 80),	(16, 455, 688, 488, 560, 359, 665, 400, 280),
(18, 285, 774, 474, 630, 243, 675, 450, 150),	(23, 364, 740, 436, 623, 320, 700, 400, 175),
(25, 200, 800, 320, 740, 175, 760, 305, 100),	(25, 200, 800, 416, 688, 185, 712, 409, 80),
(25, 200, 800, 520, 625, 140, 640, 500, 145),	(25, 320, 760, 520, 584, 263, 640, 487, 184),
(28, 215, 796, 400, 700, 175, 721, 380, 128),	(32, 124, 815, 340, 745, 100, 751, 332, 80),
(40, 241, 788, 500, 620, 215, 655, 488, 116),	(40, 452, 689, 500, 535, 380, 655, 436, 248),
(45, 150, 810, 282, 765, 126, 774, 270, 93),	(45, 366, 738, 450, 630, 285, 690, 387, 234),
(47, 196, 800, 560, 580, 175, 604, 553, 100),	(52, 361, 740, 536, 548, 305, 625, 500, 200),
(79, 280, 772, 380, 700, 215, 728, 335, 196),	(100, 175, 800, 212, 784, 145, 791, 188, 140),
(100, 305, 760, 340, 712, 241, 745, 284, 212),	(100, 305, 760, 535, 604, 172, 620, 472, 271),
(100, 392, 719, 425, 644, 292, 700, 335, 280),	(124, 343, 740, 560, 580, 175, 593, 476, 320),
(140, 520, 625, 553, 404, 460, 596, 497, 280),	(145, 368, 724, 500, 620, 215, 640, 401, 332),
(145, 500, 640, 556, 535, 292, 592, 380, 431),	(150, 450, 675, 522, 579, 270, 621, 378, 390),
(151, 332, 740, 460, 655, 200, 668, 376, 305),	(200, 460, 655, 500, 599, 268, 625, 332, 424)

825² = 680625, 6 + 8 + 0 + 62 + 5 = 9²,
825² = 680625, 6² + 8² + 0² + 62² + 5² = 63²,
825² = 680625, 68 + 0 + 6 + 2 + 5 = 9².

(1³ + 2³ + ... + 87³)(88³ + 89³ + ... + 374³)(375³ + 376³ + ... + 825³) = 89372170918800².

1 / 825 = 0.00121..., and 121 = 11².

Page of Squares : First Upload November 14, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

826

The smallest squares containing k 826's :
826281 = 909²,
82682649 = 9093²,
826582682623201 = 28750351².

Komachi square sum : 826² = 7² + 9² + 41² + 56² + 823².

17346^k + 80122^k + 185850^k + 398958^k are squares for k = 1,2,3 (826², 447692², 265405364²).

826² = 682276, 6 + 8 + 2 + 27 + 6 = 7²,
826² = 682276, 6 + 8 + 22 + 7 + 6 = 7²,
826² = 682276, 6 + 82 + 27 + 6 = 11²,
826² = 682276, 6 + 822 + 7 + 6 = 29²,
826² = 682276, 68³ + 2³ + 2³ + 76³ = 868².

1 / 826 = 0.00121..., and 121 = 11².

Page of Squares : First Upload November 14, 2005 ; Last Revised March 29, 2011
by Yoshio Mimura, Kobe, Japan

827

The smallest squares containing k 827's :
1378276 = 1174²,
128278276 = 11326²,
2382782774827536 = 48813756².

827² = 683929, a zigzag square.

827² = 683929, 6 + 839 - 2 * 9 = 827.

827² = 9³ + 12³ + 88³ = 32³ + 64³ + 73³.

The square root of 827 is 28. 7 5 7 6 0 7 6 8 9 0 9 6 8 10 5 4 3 8, and
28² = 7² + 5² + 7² + 6² + 0² + 7² + 6² + 8² + 9² + 0² + 9² + 6² + 8² + 10² + 5² + 4² + 3² + 8².

3-by-3 magic squares consisting of different squares with constant 827²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(6, 217, 798, 567, 582, 154, 602, 546, 153),	(18, 89, 822, 201, 798, 82, 802, 198, 39),
(18, 234, 793, 494, 633, 198, 663, 478, 126),	(18, 422, 711, 521, 558, 318, 642, 441, 278),
(26, 87, 822, 282, 774, 73, 777, 278, 54),	(30, 215, 798, 423, 690, 170, 710, 402, 135),
(73, 366, 738, 558, 567, 226, 606, 478, 297),	(82, 318, 759, 462, 649, 222, 681, 402, 242),
(87, 242, 786, 514, 633, 138, 642, 474, 217),	(102, 270, 775, 375, 710, 198, 730, 327, 210),
(102, 359, 738, 458, 642, 249, 681, 378, 278),	(138, 487, 654, 567, 534, 278, 586, 402, 423),
(186, 458, 663, 487, 606, 282, 642, 327, 406)

827² = 683929, 6 + 8 + 39 + 2 + 9 = 8²,
827² = 683929, 683 + 92 + 9 = 28².

Page of Squares : First Upload November 14, 2005 ; Last Revised September 18, 2009
by Yoshio Mimura, Kobe, Japan

828

The smallest squares containing k 828's :
8281 = 91²,
82882816 = 9104²,
598287148288281 = 24459909².

828² = 48³ + 66³ + 66³ = 6⁴ + 8⁴ + 16⁴ + 28⁴ = 6⁴ + 12⁴ + 24⁴ + 24⁴.

828² = (20 + 21 + 22)² + (23 + 24 + 25)² + (26 + 27 + 28)² + ... + (86 + 87 + 88)².

13³ + 828 = 55², 13³ - 828 = 37².

A cubic polynomial:
(X + 828²)(X + 2688²)(X + 4531²) = X³ + 5333²X² + 12936948²X + 10084483584².

Kaprekar : 828⁵ = 389181048634368 :
3² + 8² + 9² + 18² + 10² + 4² + 8² + 6² + 3² + 4² + 3² + 6² + 8² = 828.

Komachi equations:
828² = 1² * 23² * 4² * 56² / 7² / 8² * 9² = 1² * 23² * 4² / 56² * 7² * 8² * 9².

828² = 685584, 6 + 8 + 5 + 5 + 8 + 4 = 6²,
828² = 685584, 6 + 8 + 5 + 58 + 4 = 9²,
828² = 685584, 6 + 8 + 55 + 8 + 4 = 9²,
828² = 685584, 6 + 8 + 558 + 4 = 24².

A, B, C, A+B, B+C, and C+A are squares for A = 828², B = 2035², C = 3120².

828² = 685584 appears in the decimal expression of e:
e = 2.71828•••685584••• (from the 81709th digit).

Page of Squares : First Upload November 14, 2005 ; Last Revised July 22, 2011
by Yoshio Mimura, Kobe, Japan

829

The smallest squares containing k 829's :
82944 = 288²,
8298299025 = 91095²,
4460829829829136 = 66789444².

829² = 687241, a square with different digits.

3-by-3 magic squares consisting of different squares with constant 829²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(21, 244, 792, 396, 693, 224, 728, 384, 99),	(21, 456, 692, 584, 483, 336, 588, 496, 309),
(28, 339, 756, 531, 588, 244, 636, 476, 237),	(36, 267, 784, 512, 624, 189, 651, 476, 192),
(36, 404, 723, 548, 531, 324, 621, 492, 244),	(44, 309, 768, 363, 684, 296, 744, 352, 99),
(72, 244, 789, 296, 747, 204, 771, 264, 152),	(75, 404, 720, 496, 600, 285, 660, 405, 296),
(96, 523, 636, 552, 516, 341, 611, 384, 408),	(188, 429, 684, 516, 604, 237, 621, 372, 404)

829² = 687241, 6 + 8 + 7 + 2 + 41 = 8²,
829² = 687241, 6 + 87 + 2 + 4 + 1 = 10²,
829² = 687241, 68 + 7 + 24 + 1 = 10².

(1³ + 2³ + ... + 155³)(156³ + 157³ + ... + 169³)(170³ + 171³ + ... + 829³) = 32239967760000².

Page of Squares : First Upload November 14, 2005 ; Last Revised September 18, 2009
by Yoshio Mimura, Kobe, Japan