880
The smallest squares containing k 880's :
38809 = 1972,
3988048801 = 631512,
188018806880025 = 137119952.
8802 + 8812 + 8822 + ... + 80332 = 4154432.
Page of Squares : First Upload December 26, 2005 ; Last Revised October 2, 2006by Yoshio Mimura, Kobe, Japan
881
The smallest squares containing k 881's :
11881 = 1092,
88190881 = 93912,
1418816881881 = 11911412.
The squares which begin with 881 and end in 881 are
88190881 = 93912, 8815519881 = 938912, 88125265881 = 2968592,
88144265881 = 2968912, 881046926881 = 9386412,...
8812 = 776161, a square with 3 kinds of digits.
8812 = 503 + 643 + 733.
8817 = 4 1 1 9 3 7 5 2 8 3 6 0 8 6 6 18 8 5 6 1, and
42+12+12+92+32+72+52+22+82+32+62+02+82+62+62+182+82+52+62+12 = 881.
8812 = 1! + 7! + 7! + 8! + 9! + 9!
(12 + 8)(32 + 8)(72 + 8)(92 + 8) = 8812 + 8.
8812 + 8822 + 8832 + ... + 69292 = 3326952,
8812 + 8822 + 8832 + ... + 9032 = 42782.
3-by-3 magic squares consisting of different squares with constant 8812:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(9, 172, 864, 332, 801, 156, 816, 324, 73), | (9, 224, 852, 588, 636, 161, 656, 567, 156), |
(24, 207, 856, 431, 744, 192, 768, 424, 81), | (28, 471, 744, 609, 548, 324, 636, 504, 343), |
(33, 276, 836, 564, 649, 192, 676, 528, 201), | (48, 399, 784, 504, 656, 303, 721, 432, 264), |
(84, 289, 828, 393, 756, 224, 784, 348, 201), | (96, 404, 777, 588, 609, 244, 649, 492, 336), |
(105, 244, 840, 460, 735, 156, 744, 420, 215), | (116, 432, 759, 543, 564, 404, 684, 521, 192), |
(136, 471, 732, 564, 612, 289, 663, 424, 396) |
8812 = 776161, 74 + 74 + 64 + 14 + 64 + 14 = 862,
8812 = 776161, 7 + 76 + 16 + 1 = 102,
8812 = 776161, 77 + 6 + 16 + 1 = 102,
8812 = 776161, 776 + 1 + 6 + 1 = 282.
by Yoshio Mimura, Kobe, Japan
882
The smallest squares containing k 882's :
88209 = 2972,
18882882225 = 1374152,
2488298823888201 = 498828512.
8822 = 33 + 573 + 843.
8822 = (42 + 4)(1992 + 4) = (12 + 5)(112 + 5)(322 + 5) = (12 + 5)(22 + 5)(32 + 5)(322 + 5)
= (22 + 5)(32 + 5)(42 + 5)(172 + 5) = (32 + 5)(72 + 5)(322 + 5) = (42 + 5)(112 + 5)(172 + 5).
882k + 2478k + 16338k + 24402k are squares for k = 1,2,3 (2102, 294842, 43482602).
The integral triangle of sides 689, 16810, 17493 has square area 8822.
Komachi equations:
8822 = 982 * 72 / 62 * 542 * 32 / 212 = 982 / 72 / 62 * 542 / 32 * 212.
(1 + 2 + ... + 6)(7 + 8 + ... + 20)(21 + 22 + ... + 28) = 8822,
(1 + 2 + 3)(4 + 5 + ... + 24)(25 + 26 + ... + 38) = 8822,
(1 + 2 + 3)(4 + 5 + ... + 17)(18 + 19 + ... + 45) = 8822,
(1 + 2 + 3)(4 + 5 + ... + 10)(11 + 12 + ... + 73) = 8822.
(12 + 22 + ... + 762)(772 + 782 + ... + 4402)(4412 + ... + 8822) = 291281690702.
The 4-by-4 magic squares consisting of different squares with constant 882:
|
|
|
|
8822 = 777924, 7 + 7 + 7 + 9 + 2 + 4 = 62.
Page of Squares : First Upload December 26, 2005 ; Last Revised December 7, 2013by Yoshio Mimura, Kobe, Japan
883
The smallest squares containing k 883's :
8836 = 942,
288388324 = 169822,
388368883883929 = 197070772.
8832 = 779689, a square every digit of which is greater than 5.
8832 = 13 + 103 + 923,
The square root of 883 is 29. 7 1 5 3 1 5 9 1 6 2 0 7 2 5 13 8 8 11 8 7 ..., and
292 = 72 + 12 + 52 + 32 + 12 + 52 + 92 + 12 + 62 + 22 + 02 + 72 + 22 + 52 + 132 + 82 + 82 + 112 + 82 + 72.
3-by-3 magic squares consisting of different squares with constant 8832:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(17, 402, 786, 606, 577, 282, 642, 534, 287), | (18, 186, 863, 593, 642, 126, 654, 577, 138), |
(18, 242, 849, 298, 801, 222, 831, 282, 98), | (18, 486, 737, 534, 593, 378, 703, 438, 306), |
(33, 222, 854, 438, 746, 177, 766, 417, 138), | (33, 350, 810, 550, 642, 255, 690, 495, 242), |
(42, 495, 730, 530, 570, 417, 705, 458, 270), | (54, 273, 838, 458, 726, 207, 753, 422, 186), |
(81, 402, 782, 478, 639, 378, 738, 458, 159), | (82, 417, 774, 513, 654, 298, 714, 422, 303), |
(177, 334, 798, 402, 753, 226, 766, 318, 303), | (222, 431, 738, 462, 702, 271, 719, 318, 402) |
8832 = 779689, 7 + 7 + 9 + 68 + 9 = 102.
Page of Squares : First Upload December 26, 2005 ; Last Revised September 7, 2013by Yoshio Mimura, Kobe, Japan
884
The smallest squares containing k 884's :
14884 = 1222,
884884009 = 297472,
2588488418884 = 16088782.
The squares which begin with 884 and end in 884 are
88433674884 = 2973782, 884310782884 = 9403782, 884769746884 = 9406222,
8840976730884 = 29733782, 8842427798884 = 29736222,...
8842 = 781456, a square with different digits.
8842 = 781456, 781 + 4 + 56 = 292.
Page of Squares : First Upload December 26, 2005 ; Last Revised October 2, 2006by Yoshio Mimura, Kobe, Japan
885
The smallest squares containing k 885's :
788544 = 8882,
50885885241 = 2255792,
2885068858588569 = 537128372.
Komachi equation: 8852 = 13 + 233 - 43 * 53 + 63 * 73 + 893.
1/ 885 = 0.00 1 1 2 9 9 4 3 5 0 2 8 2 4 8 5 8 7 5 7 0 6 2 1 4 6 8 9 2 6 5 ...,
the sum of the squares of its digits is 885 : 12 + 12 + 22 + 92 + ... +62 + 52 = 885.
3-by-3 magic squares consisting of different squares with constant 8852:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(9, 162, 870, 450, 750, 135, 762, 441, 90), | (19, 92, 880, 208, 856, 85, 860, 205, 40), |
(20, 212, 859, 485, 716, 188, 740, 475, 100), | (20, 301, 832, 560, 640, 245, 685, 532, 176), |
(20, 485, 740, 560, 580, 365, 685, 460, 320), | (40, 205, 860, 460, 740, 155, 755, 440, 140), |
(40, 260, 845, 299, 800, 232, 832, 275, 124), | (40, 260, 845, 520, 691, 188, 715, 488, 184), |
(40, 352, 811, 460, 685, 320, 755, 436, 152), | (40, 460, 755, 565, 568, 376, 680, 499, 268), |
(40, 565, 680, 596, 520, 397, 653, 440, 404), | (43, 320, 824, 376, 740, 307, 800, 365, 100), |
(54, 153, 870, 222, 846, 135, 855, 210, 90), | (64, 245, 848, 595, 640, 140, 652, 560, 211), |
(77, 536, 700, 580, 560, 365, 664, 427, 400), | (85, 496, 728, 560, 595, 340, 680, 428, 371), |
(90, 345, 810, 615, 558, 306, 630, 594, 183), | (91, 188, 860, 340, 805, 140, 812, 316, 155), |
(100, 365, 800, 475, 700, 260, 740, 400, 275), | (100, 421, 772, 475, 628, 404, 740, 460, 155), |
(128, 365, 796, 400, 740, 275, 779, 320, 272), | (135, 330, 810, 450, 729, 222, 750, 378, 279), |
(140, 392, 781, 440, 715, 280, 755, 344, 308), | (140, 440, 755, 595, 608, 244, 640, 469, 392), |
(152, 461, 740, 565, 520, 440, 664, 548, 205), | (155, 460, 740, 604, 485, 428, 628, 580, 229), |
(205, 440, 740, 484, 688, 275, 712, 341, 400) |
Page of Squares : First Upload December 26, 2005 ; Last Revised July 9, 2010
by Yoshio Mimura, Kobe, Japan
886
The smallest squares containing k 886's :
478864 = 6922,
4688688676 = 684742,
34788668868864 = 58981922.
Komachi square sum : 8862 = 52 + 732 + 2692 + 8412.
5902 + 5912 + 5922 + ... + 8862 = 128042.
The square root of 886 is 29. 7 6 5 7 5 2 13 22 ...,
and 292 = 72 + 62 + 52 + 72 + 52 + 22 + 132 + 222 = 292.
8862 = 784996, 7 + 84 + 9 + 96 = 142,
8862 = 784996, 7 + 84 + 99 + 6 = 142.
by Yoshio Mimura, Kobe, Japan
887
The smallest squares containing k 887's :
887364 = 9422,
29488788729 = 1717232,
22887887988879225 = 1512874352.
8872 = 786769, a zigzag square.
8872 = 786769, a square every digit of which is greater than 5.
8872 = 144 + 224 + 224 + 234.
887 is the first integer which can be written as the sum of a prime and a square in 11 ways:
22 + 883, 82 + 823, 102 + 787, 122 + 743, 142 + 691, 162 + 631, 182 + 563, 202 + 487, 242 + 311, 262 + 211, 282 + 103.
3-by-3 magic squares consisting of different squares with constant 8872:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(3, 94, 882, 486, 738, 77, 742, 483, 54), | (3, 302, 834, 426, 762, 157, 778, 339, 258), |
(3, 302, 834, 454, 717, 258, 762, 426, 157), | (3, 426, 778, 454, 717, 258, 762, 302, 339), |
(3, 426, 778, 498, 643, 354, 734, 438, 237), | (14, 93, 882, 522, 714, 67, 717, 518, 66), |
(14, 333, 822, 387, 742, 294, 798, 354, 157), | (22, 186, 867, 354, 797, 162, 813, 342, 94), |
(22, 354, 813, 582, 643, 186, 669, 498, 302), | (22, 438, 771, 582, 589, 318, 669, 498, 302), |
(42, 221, 858, 338, 798, 189, 819, 318, 122), | (42, 534, 707, 626, 483, 402, 627, 518, 354), |
(45, 438, 770, 490, 630, 387, 738, 445, 210), | (67, 258, 846, 582, 626, 237, 666, 573, 122), |
(78, 346, 813, 606, 573, 302, 643, 582, 186), | (90, 237, 850, 445, 750, 162, 762, 410, 195), |
(99, 482, 738, 582, 522, 419, 662, 531, 258), | (130, 330, 813, 435, 738, 230, 762, 365, 270), |
(141, 302, 822, 402, 762, 211, 778, 339, 258) |
8872 = 786769, 7 + 86 + 7 + 69 = 132,
8872 = 786769, 78 + 6 + 76 + 9 = 132.
by Yoshio Mimura, Kobe, Japan
888
The smallest squares containing k 888's :
88804 = 2982,
97888888384 = 3128722,
988858588838884 = 314461222.
8882 + 8892 + 8902 + ... + 578262 = 80283992.
138k + 417k + 582k + 888k are squares for k = 1,2,3 (452, 11492, 311852).
8882 = 788544, 7 + 8 + 8 + 5 + 4 + 4 = 62,
8882 = 788544, 7 + 8 + 8 + 54 + 4 = 92,
8882 = 788544, 7 + 8 + 85 + 44 = 122,
8882 = 788544, 7 + 88 + 5 + 44 = 122,
8882 = 788544, 78 + 8 + 54 + 4 = 122,
8882 = 788544, 7 + 885 + 4 + 4 = 302.
8882 = 788544 appears in the decimal expression of π:
π = 3.14159•••788544••• (from the 31609th digit).
by Yoshio Mimura, Kobe, Japan
889
The smallest squares containing k 889's :
6889 = 832,
5537889889 = 744172,
9488968893889 = 30804172.
The squares which begin with 889 and end in 889 are
8898714889 = 943332, 88903559889 = 2981672, 889092468889 = 9429172,
889405544889 = 9430832, 889563989889 = 9431672,...
8892 = 790321, a square with different digits.
8892 = 30 + 31 + 34 + 36 + 37 + 39 + 310 + 311 + 312.
A cubic polynomial :
(X + 2092)(X + 5282)(X + 6842) = X3 + 8892X2 + 4037882X + 754807682.
8892 + 8902 + 8912 + ... + 90972 = 5007492.
3-by-3 magic squares consisting of different squares with constant 8892:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(12, 151, 876, 209, 852, 144, 864, 204, 47), | (12, 279, 844, 591, 628, 216, 664, 564, 177), |
(16, 132, 879, 564, 681, 92, 687, 556, 96), | (36, 87, 884, 128, 876, 81, 879, 124, 48), |
(36, 425, 780, 600, 564, 335, 655, 540, 264), | (39, 264, 848, 432, 736, 249, 776, 423, 96), |
(60, 236, 855, 495, 720, 164, 736, 465, 180), | (81, 492, 736, 624, 556, 303, 628, 489, 396), |
(88, 369, 804, 561, 648, 236, 684, 484, 297), | (96, 367, 804, 529, 624, 348, 708, 516, 151), |
(111, 304, 828, 576, 657, 164, 668, 516, 279), | (124, 456, 753, 489, 668, 324 732, 369, 344), |
(126, 371, 798, 462, 714, 259, 749, 378, 294), | (144, 344, 807, 488, 711, 216, 729, 408, 304), |
(177, 396, 776, 524, 681, 228, 696, 412, 369), | (204, 556, 663, 591, 408, 524, 632, 561, 276) |
8892 = 790321, 72 + 92 + 02 + 32 + 22 + 12 = 122,
8892 = 790321, 7 + 9 + 0 + 32 + 1 = 72,
8892 = 790321, 7 + 90 + 3 + 21 = 112,
8892 = 790321, 79 + 0 + 321 = 202.
by Yoshio Mimura, Kobe, Japan