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810 - 819

810

The smallest squares containing k 810's :
8100 = 902,
8108102025 = 900452,
681058106468100 = 260970902.

(1 + 2)(3)(4 + 5)(6)(7 + 8)(9)(10) = 8102,
(1)(2 + 3)(4 + 5)(6)(7 + 8 + 9 + 10 + 11)(12 + 13 + 14 + 15) = 8102,
(1)(2 + 3 + 4)(5)(6)(7 + 8 + 9 + 10 + 11)(12 + 13 + 14 + 15) = 8102,
(1)(2 + 3 + ... + 13)(14 + 15 + ... + 121) = 8102.

8102 = (12 + 5)(22 + 5)(52 + 5)(202 + 5) = (52 + 5)(72 + 5)(202 + 5).

154k + 222k + 258k + 810k are squares for k = 1,2,3 (382, 8922, 237322).

Komachi equation: 8102 = 12 * 2342 * 52 * 62 / 782 * 92.

The 4-by-4 magic squares consisting of different squares with constant 810:

02 32152242
42212172 82
132182142112
252 62102 72
     
02 32152242
82212172 42
112182142132
252 62102 72
     
02 82112252
122242 32 92
152 12222102
212132142 22
     
12 62172222
82212162 72
132182112142
242 32122 92
     
12 82132242
142 72222 92
172162112122
182212 62 32

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

811

The smallest squares containing k 811's :
298116 = 5462,
7318118116 = 855462,
281185811811216 = 167685962.

8112 = 373 + 573 + 753.

3-by-3 magic squares consisting of different squares with constant 8112:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(9, 242, 774, 318, 711, 226, 746, 306, 87),(9, 406, 702, 474, 567, 334, 658, 414, 231),
(18, 199, 786, 361, 702, 186, 726, 354, 73),(39, 334, 738, 478, 606, 249, 654, 423, 226),
(42, 174, 791, 359, 714, 138, 726, 343, 114),(46, 438, 681, 537, 494, 354, 606, 471, 262),
(54, 226, 777, 471, 642, 154, 658, 441, 174),(66, 233, 774, 521, 606, 138, 618, 486, 199),
(73, 474, 654, 534, 521, 318, 606, 402, 359),(73, 474, 654, 534, 567, 226, 606, 334, 423),
(114, 262, 759, 297, 726, 206, 746, 249, 198),(114, 375, 710, 550, 486, 345, 585, 530, 186),
(126, 422, 681, 566, 441, 378, 567, 534, 226) 

8112 = 657721, 6 + 57 + 721 = 282,
8112 = 657721, 65 + 7 + 7 + 21 = 102,
8112 = 657721, 6 + 5 + 772 + 1 = 282,
8112 = 657721, 63 + 573 + 73 + 23 + 13 = 4312.

Page of Squares : First Upload November 7, 2005 ; Last Revised September 14, 2009
by Yoshio Mimura, Kobe, Japan

812

The smallest squares containing k 812's :
68121 = 2612,
38126858121 = 1952612,
97812812081296 = 98900362.

8122 = 64 x 65 + 66 x 67 + 68 x 69 + 70 x 71 + ... + 160 x 161.

8122 = 659344, 63 + 53 + 93 + 33 + 43 + 43 = 352,
8122 = 659344, 65 + 9 + 3 + 44 = 112,
8122 = 659344, 65 + 9344 = 972.

Page of Squares : First Upload November 7, 2005 ; Last Revised November 13, 2006
by Yoshio Mimura, Kobe, Japan

813

The smallest squares containing k 813's :
813604 = 9022,
11813081344 = 1086882,
813481381344025 = 285215952.

8132 = 660969, a square with 3 kinds of digits.

Komachi fraction : 36 / 5948721 = (2 / 813)2.

34146k + 149592k + 186177k + 291054k are squares for k = 1,2,3 (8132, 3780452, 1857322892).

The square root of 813 is 28.51315485876650584709233561422081 ...,
and the sum of the squares of its digits is a square 282.

3-by-3 magic squares consisting of different squares with constant 8132:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(18, 306, 753, 558, 543, 234, 591, 522, 198),(19, 88, 808, 392, 709, 68, 712, 388, 59),
(28, 124, 803, 163, 788, 116, 796, 157, 52),(28, 539, 608, 568, 448, 371, 581, 412, 392),
(37, 140, 800, 500, 635, 88, 640, 488, 115),(37, 212, 784, 368, 704, 173, 724, 347, 128),
(44, 248, 773, 448, 653, 184, 677, 416, 172),(52, 352, 731, 389, 632, 332, 712, 371, 128),
(59, 248, 772, 508, 592, 229, 632, 499, 112),(68, 332, 739, 532, 541, 292, 611, 508, 172),
(81, 402, 702, 522, 513, 354, 618, 486, 207),(88, 184, 787, 236, 763, 152, 773, 212, 136),
(88, 275, 760, 325, 712, 220, 740, 280, 187),(88, 397, 704, 443, 616, 292, 676, 352, 283),
(101, 292, 752, 548, 581, 152, 592, 488, 269),(112, 436, 677, 467, 592, 304, 656, 347, 332),
(116, 227, 772, 332, 724, 163, 733, 292, 196),(124, 283, 752, 448, 656, 173, 667, 388, 256),
(126, 462, 657, 513, 558, 294, 618, 369, 378),(136, 443, 668, 493, 488, 424, 632, 476, 187),
(157, 436, 668, 556, 548, 227, 572, 413, 404),(163, 380, 700, 500, 605, 212, 620, 388, 355)

8132 = 660969, 6 + 6 + 0 + 9 + 6 + 9 = 62,
8132 = 660969, 6 + 60 + 9 + 69 = 122,
8132 = 660969, 66 + 0 + 9 + 69 = 122.

Page of Squares : First Upload November 7, 2005 ; Last Revised March 29, 2011
by Yoshio Mimura, Kobe, Japan

814

The smallest squares containing k 814's :
181476 = 4262,
68148146704 = 2610522,
57881481408144 = 76079882.

8142 = 662596, 6 + 62 + 5 + 96 = 132,
8142 = 662596, 66 + 2 + 5 + 96 = 132.

8142 = 333 + 383 + 833.

Kaprekar : 8142 = 662596 : 62 + 62 + 252 + 92 + 62 = 814.

1 / 814 = 0.0012285..., and 12 + 22 + 282 + 52 = 814.

154k + 814k + 1298k + 2090k are squares for k = 1,2,3 (662, 25962, 1089002).
106k + 266k + 814k + 1314k are squares for k = 1,2,3 (502, 15722, 531802).

The 4-by-4 magic square consisting of different squares with constant 814:

 12 42112262
142132202 72
162232 22 52
192102172 82

Page of Squares : First Upload November 7, 2005 ; Last Revised March 29, 2011
by Yoshio Mimura, Kobe, Japan

815

The smallest squares containing k 815's :
815409 = 9032,
81518815225 = 2855152,
18158157815121 = 42612392.

3-by-3 magic squares consisting of different squares with constant 8152:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(5, 90, 810, 414, 698, 75, 702, 411, 50),(5, 90, 810, 558, 590, 69, 594, 555, 58),
(6, 267, 770, 517, 594, 210, 630, 490, 165),(14, 165, 798, 498, 630, 139, 645, 490, 90),
(30, 85, 810, 202, 786, 75, 789, 198, 50),(30, 338, 741, 390,645, 310, 715, 366, 138),
(30, 418, 699, 510, 555, 310, 635, 426, 282),(30, 510, 635, 554, 453, 390, 597, 446, 330),
(42, 405, 706, 570, 490, 315, 581, 510, 258),(50, 315, 750, 426, 630, 293, 693, 410, 126),
(50, 315, 750, 546, 570, 203, 603, 490, 246),(69, 258, 770, 330, 715, 210, 742, 294, 165),
(75, 310, 750, 446, 645, 222, 678, 390, 229),(75, 310, 750, 526, 555, 282, 618, 510, 149),
(85, 462, 666, 510, 490, 405, 630, 459, 238),(90, 490, 645, 555, 510, 310, 590, 405, 390),
(114, 405, 698, 498, 590, 261, 635, 390, 330),(222, 446, 645, 554, 453, 390, 555, 510, 310)

8152 = 664225, 6 + 6 + 4 + 2 + 2 + 5 = 52,
8152 = 664225, 62 + 62 + 42 + 22 + 22 + 52 = 112,
8152 = 664225, 62 + 642 + 22 + 252 = 692,
8152 = 664225, 63 + 63 + 43 + 23 + 253 = 1272.

Page of Squares : First Upload November 7, 2005 ; Last Revised September 14, 2009
by Yoshio Mimura, Kobe, Japan

816

The smallest squares containing k 816's :
10816 = 1042,
8168164 = 28582,
816081602048164 = 285671422.

The squares which begin with 816 and end in 816 are
816124332816 = 9033962,   816500188816 = 9036042,   8160186412816 = 28566042,
8161854754816 = 28568962,   8163043266816 = 28571042,...

8168164 = 28582.

8162 = 665856, a square with 3 kinds of digits.

8162 = (22 + 8)(42 + 8)(482 + 8).

Komachi equation: 8162 = 93 + 873 - 63 + 543 / 33 + 23 + 103.

8162 = 665856, 6 + 6 + 5 + 8 + 5 + 6 = 62,
8162 = 665856, 6 + 6 + 5 + 8 + 56 = 92,
8162 = 665856, 6 + 6 + 58 + 5 + 6 = 92,
8162 = 665856, 665 + 8 + 56 = 272,
8162 = 665856, 665 + 856 = 392.

1 / 816 = 0.001225..., and 1225 = 352.

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

817

The smallest squares containing k 817's :
81796 = 2862,
81781700625 = 2859752,
381725081781796 = 195377862.

817 = (12 + 22 + 32 + ... + 3222) / (12 + 22 + 32 + ... + 342).

8172 = 2732 + 3762 + 6722 = 2762 + 6732 + 3722.

Komachi square sum : 8172 = 52 + 462 + 3982 + 7122.

3-by-3 magic squares consisting of different squares with constant 8172:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(16, 252, 777, 567, 556, 192, 588, 543, 164),(24, 372, 727, 407, 636, 312, 708, 353, 204),
(24, 497, 648, 528, 504, 367, 623, 408, 336),(48, 137, 804, 396, 708, 97, 713, 384, 108),
(84, 353, 732, 457, 588, 336, 672, 444, 137),(87, 468, 664, 552, 524, 297, 596, 417, 372),
(92, 225, 780, 375, 708, 160, 720, 340, 183),(92, 351, 732, 489, 612, 232, 648, 412, 279),
(108, 340, 735, 560, 567, 180, 585, 480, 308),(132, 393, 704, 543, 484, 372, 596, 528, 183)

8172 = 667489, 667 + 489 = 342.

Page of Squares : First Upload November 7, 2005 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan

818

The smallest squares containing k 818's :
3818116 = 19542,
818818225 = 286152,
6818281881856 = 26111842.

8182± 3 are primes.

8182 = 13 + 463 + 833.

8182 = 669124, 6 + 69 + 1 + 24 = 102,
8182 = 669124, 66 + 9 + 1 + 24 = 102,
8182 = 669124, 669 + 1 + 2 + 4 = 262.

8182 + 8192 + 8202 + ... + 29882 = 933532.

The square root of 818 is 28. 6 0 0 6 9 9 2 9 2 1 5 0 1 8 3 3 6 8 7 3 9 8 7 ...,
and 282 = 62 + 02 + 02 + 62 + 92 + 92 + 22 + 92 + ... + 32 + 92 + 82 + 72.

Page of Squares : First Upload November 7, 2005 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

819

The smallest squares containing k 819's :
281961 = 5312,
9819819025 = 990952,
2566819819819264 = 506637922.

819 = 12 + 22 + 32 + ... + 132.

8192 = (5 + 6 + 7 + ... + 25)2 + (26 + 27 + 28 + ... + 46)2.

A cubic polynomial :
(X + 842)(X + 8192)(X + 11682) = X3 + 14292X2 + 9640682X + 803537282.

3-by-3 magic squares consisting of different squares with constant 8192:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(1, 166, 802, 394, 703, 146, 718, 386, 79),(1, 338, 746, 394, 689, 202, 718, 286, 271),
(1, 338, 746, 542, 559, 254, 614, 494, 223),(3, 156, 804, 516, 624, 123, 636, 507, 96),
(10, 145, 806, 415, 694, 130, 706, 410, 65),(22, 289, 766, 326, 706, 257, 751, 298, 134),
(24, 348, 741, 411, 636, 312, 708, 381, 156),(26, 143, 806, 559, 586, 122, 598, 554, 79),
(26, 178, 799, 286, 751, 158, 767, 274, 86),(26, 286, 767, 442, 641, 254, 689, 422, 134),
(26, 337, 746, 442, 634, 271, 689, 394, 202),(26, 442, 689, 559, 494, 338, 598, 481, 286),
(26, 466, 673, 559, 502, 326, 598, 449, 334),(31, 254, 778, 362, 694, 241, 734, 353, 86),
(31, 370, 730, 470, 590, 319, 670, 431, 190),(34, 481, 662, 527, 494, 386, 626, 442, 289),
(46, 278, 769, 463, 626, 254, 674, 449, 122),(47, 286, 766, 526, 598, 191, 626, 481, 218),
(48, 339, 744, 429, 624, 312, 696, 408, 141),(58, 431, 694, 466, 554, 383, 671, 422, 206),
(62, 191, 794, 506, 634, 113, 641, 482, 166),(65, 130, 806, 206, 785, 110, 790, 194, 95),
(65, 430, 694, 494, 575, 310, 650, 394, 305),(69, 240, 780, 480, 645, 156, 660, 444, 195),
(81, 414, 702, 558, 486, 351, 594, 513, 234),(91, 364, 728, 532, 581, 224, 616, 448, 301),
(118, 374, 719, 529, 586, 218, 614, 433, 326),(132, 411, 696, 564, 552, 219, 579, 444, 372),
(143, 286, 754, 334, 718, 209, 734, 271, 242),(146, 326, 737, 418, 671, 214, 689, 338, 286),
(156, 507, 624, 564, 384, 453, 573, 516, 276),(194, 367, 706, 398, 674, 241, 689, 286, 338),
(206, 526, 593, 559, 338, 494, 562, 529, 274) 

The 4-by-4 magic square consisting of different squares with constant 819:

 02 12172232
 32212152122
 92192162112
272 42 72 52

8192 = 670761, 6 + 7 + 0 + 7 + 61 = 92,
8192 = 670761, 67 + 0 + 7 + 6 + 1 = 92,
8192 = 670761, 6 + 70 + 7 + 61 = 122,
8192 = 670761, 67 + 0 + 76 + 1 = 122.

7242 + 7252 + 7262 + ... + 8192 = 75642.

Page of Squares : First Upload November 7, 2005 ; Last Revised September 18, 2009
by Yoshio Mimura, Kobe, Japan