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:
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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:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
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.
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.
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:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
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.
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:
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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:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
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.
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, 2013by 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:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
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, 2013by 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.
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:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
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:
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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, 2009by Yoshio Mimura, Kobe, Japan