840
The smallest squares containing k 840's :
48400 = 2202,
684084025 = 261552,
84079840284036 = 91695062.
372 + 840 = 472, 372 - 840 = 232,
292 + 840 = 412, 292 - 840 = 12.
The integral triangle of sides 1131, 1285, 1904 (or 801, 1825, 2176) has square area 8402.
Komachi equations:
8402 = 92 / 82 * 72 / 62 * 52 * 42 * 322 */ 12 = 92 * 82 * 72 * 62 / 542 * 32 / 22 * 102
= 982 / 72 / 62 * 542 / 32 * 22 * 102.
8402 = 705600, 70 / 5 * 60 + 0 = 840.
8402 = 31 x 32 + 32 x 33 + 33 x 34 + 34 x 35 + ... + 128 x 129.
8402 = 52 x 53 x 54 + 54 x 55 x 56 + 56 x 57 x 58 + 58 x 59 x 60.
8402 = (22 - 1)(32 - 1)(62 - 1)(292 - 1) = (22 - 1)(52 - 1)(992 - 1) = (32 - 1)(42 - 1)(62 - 1)(132 - 1)
= (52 - 1)(62 - 1)(292 - 1) = (62 - 1)(112 - 1)(132 - 1).
(1)(2)(3 + 4)(5)(6)(7)(8)(9 + 10 + 11) = 8402,
(1 + 2 + 3 + 4)(5)(6)(7)(8)(9 + 10 + 11 + 12) = 8402,
(1 + 2 + 3)(4)(5)(6)(7 + 8 + 9 + 10 + 11 + 12 + 13)(14) = 8402,
(1)(2)(3)(4)(5)(6)(7 + 8 + 9 + 10 + 11 + 12 + 13)(14) = 8402,
(1 + 2 )(3)(4)(5 + 6 + 7 + 8 + 9 + 10 + 11)(12 + 13)(14) = 8402,
(1)(2)(3 + 4 + 5 + 6 + 7)(8)(9 + 10 + 11 + 12)(13 + 14 + 15) = 8402,
(1 + 2 + 3 + 4)(5 + 6 + 7 + 8 + 9 + 10 + 11)(12 + 13 + ... + 51) = 8402,
(1 + 2 + ... + 20)(21 + 22 + ... + 84) = 8402.
(13 + 23 + ... + 93)(103 + 113 + ... + 203)(213 + 223 + ... + 8403) = 32603917502,
(13 + 23 + ... + 6713)(6723 + 6733 + ... + 8403) = 613032900482.
8402 = 705600 appears in the decimal expressions of e:
e = 2.71828•••705600••• (from the 18244th digit).
by Yoshio Mimura, Kobe, Japan
841
the square of 29.
The smallest squares containing k 841's :
841 = 292,
218418841 = 147792,
484108416841 = 6957792.
The squares which begin with 841 and end in 841 are
8412741841 = 917212, 84116820841 = 2900292, 841294362841 = 9172212,
841400763841 = 9172792, 841753035841 = 9174712,...
292 = 707281, a zigzag square.
841 = 292, the second mosaic square (81 = 92, 4 = 22).
8412 = 707281, 7 + 0 + 7 + 2 + 8 + 1 = 52,
8412 = 707281, 703 + 723 + 813 = 11172.
13 - 23 + 33 - 43 + ... - 8403 + 8413 = 172612.
8412 = 293 + 293 + 873.
3-by-3 magic squares consisting of different squares with constant 8412:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(9, 168, 824, 472, 681, 144, 696, 464, 87) | (9, 392, 744, 536, 576, 297, 648, 471, 256) |
(12, 256, 801, 441, 684, 212, 716, 417, 144) | (31, 144, 828, 576, 607, 84, 612, 564, 121) |
(32, 201, 816, 471, 672, 184, 696, 464, 87) | (63, 156, 824, 264, 788, 129, 796, 249, 108) |
(68, 444, 711, 576, 543, 284, 609, 464, 348) | (84, 348, 761, 383, 696, 276, 744, 319, 228) |
(96, 417, 724, 572, 564, 249, 609, 464, 348) | (108, 436, 711, 536, 513, 396, 639, 504, 212) |
(121, 348, 756, 396, 696, 257, 732, 319, 264) | (129, 408, 724, 472, 639, 276, 684, 364, 327) |
(135, 284, 780, 420, 705, 184, 716, 360, 255) |
8412 + 8422 + 8432 + ... + 22482 = 599282,
8412 + 8422 + 8432 + ... + 24572 = 689152,
8412 + 8422 + 8432 + ... + 7274642 = 3582259802,
8412 + 8422 + 8432 + ... + 8351132 = 4406133272.
(13 + 23 + ... + 8403)(8413) = 86146825802,
(13 + 23 + ... + 1193)(1203 + 1213 + ... + 8413) = 25274814602,
(13 + 23 + ... + 6713)(6723 + 6733 + ... + 8403)(8413) = 14951259409806722.
by Yoshio Mimura, Kobe, Japan
842
The smallest squares containing k 842's :
842724 = 9182,
258842842756 = 5087662,
384284234842561 = 196031692.
8422 = 708964, a square with different digits.
8422 = 708964, 70 + 8 * 96 + 4 = 842.
8422 = 708964, 70 + 89 + 6 + 4 = 132.
(13 + 23 + ... + 6023)(6033 + 6043 + ... + 8423) = 553547849402,
(13 + 23 + ... + 7443)(7453 + 7463 + ... + 8423) = 614411065802.
by Yoshio Mimura, Kobe, Japan
843
The smallest squares containing k 843's :
498436 = 7062,
18438436 = 42942,
148430508438436 = 121832062.
8432 = 710649, a square with different digits.
8432 + 8442 + 8452 + ... + 8922 = 61352.
The square root of 843 is 29. 0 3 4 4 6 2 2 8 19 15 9 5, and
292 = 02 + 32 + 42 + 42 + 62 + 22 + 22 + 82 + 192 + 152 + 92 + 52.
Komachi equation: 8432 = - 122 + 32 + 42 * 52 * 62 * 72 + 82 * 92.
3-by-3 magic squares consisting of different squares with constant 8432:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(2, 247, 806, 286, 758, 233, 793, 274, 82) | (2, 538, 649, 569, 478, 398, 622, 439, 362) |
(10, 343, 770, 518, 610, 265, 665, 470, 218) | (14, 182, 823, 398, 727, 154, 743, 386, 98) |
(22, 89, 838, 362, 758, 71, 761, 358, 58) | (22, 167, 826, 574, 602, 137, 617, 566, 98) |
(23, 218, 814, 478, 674, 167, 694, 457, 142) | (27, 156, 828, 252, 792, 141, 804, 243, 72) |
(27, 372, 756, 492, 621, 288, 684, 432, 237) | (36, 432, 723, 588, 531, 288, 603, 492, 324) |
(41, 322, 778, 398, 679, 302, 742, 382, 119) | (55, 310, 782, 418, 670, 295, 730, 407, 110) |
(58, 394, 743, 446, 617, 362, 713, 418, 166) | (58, 457, 706, 503, 586, 338, 674, 398, 313) |
(82, 470, 695, 505, 530, 418, 670, 457, 230) | (86, 313, 778, 418, 694, 233, 727, 362, 226) |
(119, 538, 638, 578, 518, 329, 602, 391, 442) | (134, 377, 742, 553, 602, 206, 622, 454, 343) |
(217, 526, 622, 574, 358, 503, 578, 553, 266) |
8432 = 710649, 7 + 1 + 0 + 64 + 9 = 92,
8432 = 710649, 71 + 0 + 64 + 9 = 122,
8432 = 710649, 710 + 6 + 4 + 9 = 272.
8432 = 710649 appears in the decimal expression of e:
e = 2.71828•••710649••• (from the 49151st digit).
by Yoshio Mimura, Kobe, Japan
844
The smallest squares containing k 844's :
3844 = 622,
844599844 = 290622,
58448440844964 = 76451582.
The squares which begin with 844 and end in 844 are
844599844 = 290622, 84426275844 = 2905622, 844447047844 = 9189382,
844674959844 = 9190622, 8441569971844 = 29054382,...
844, 845, 846, 847 and 848 are five consecutive integers having square factors (the first case).
8442 = 712336, 7 + 1 + 2 + 3 + 36 = 72,
8442 = 712336, 7 + 1 + 2 + 33 + 6 = 72.
by Yoshio Mimura, Kobe, Japan
845
The smallest squares containing k 845's :
2845969 = 16872,
8459584576 = 919762,
138458452984569 = 117668372.
8452 = 714025, a square with different digits.
8452 = (33 + 34 + 35 + ... + 45)2 + (46 + 47 + 48 + ... + 58)2.
8452 = 133 + 193 + 893.
3-by-3 magic squares consisting of different squares with constant 8452:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(0, 208, 819, 325, 756, 192, 780, 315, 80) | (0, 325, 780, 429, 672, 280, 728, 396, 165) |
(0, 325, 780, 507, 624, 260, 676, 468, 195) | (0, 507, 676, 595, 480, 360, 600, 476, 357) |
(3, 204, 820, 260, 780, 195, 804, 253, 60) | (12, 91, 840, 315, 780, 80, 784, 312, 45) |
(36, 323, 780, 573, 564, 260, 620, 540, 195) | (45, 440, 720, 568, 549, 300, 624, 468, 325) |
(60, 195, 820, 280, 780, 165, 795, 260, 120) | (60, 280, 795, 539, 600, 252, 648, 525, 136) |
(69, 192, 820, 480, 685, 120, 692, 456, 165) | (80, 315, 780, 540, 620, 195, 645, 480, 260) |
(80, 435, 720, 468, 624, 325, 699, 368, 300) | (80, 435, 720, 540, 528, 379, 645, 496, 228) |
(91, 312, 780, 588, 584, 165, 600, 525, 280) | (120, 260, 795, 360, 741, 188, 755, 312, 216) |
(120, 388, 741, 480, 645, 260, 685, 384, 312) | (125, 228, 804, 300, 771, 172, 780, 260, 195) |
(125, 300, 780, 444, 692, 195, 708, 381, 260) | (168, 549, 620, 576, 532, 315, 595, 360, 480) |
8452 = 714025, 71 + 4 + 0 + 25 = 102,
8452 = 714025, 71 + 4025 = 642.
(13 + 23 + ... + 2293)(2303 + 2313 + ... + 7053)(7063 + 7073 + ... + 8453) = 16720548675600002.
Page of Squares : First Upload November 28, 2005 ; Last Revised September 25, 2009by Yoshio Mimura, Kobe, Japan
846
The smallest squares containing k 846's :
8464 = 922,
846984609 = 291032,
846998462084644 = 291032382.
8462 = 715716.
8462 = 715716, a square with odd digits except the last digit 6.
Komachi Square Sum : 8462 = 1982 + 3242 + 7562.
Komachi Cube Sum : 8462 = 93 + 323 + 483 + 513 + 763.
The 4-by-4 magic squares consisting of different squares with constant 846:
|
|
|
|
|
8462 = 715716, 7 + 1 + 5 + 7 + 16 = 62,
8462 = 715716, 7 + 15 + 7 + 1 + 6 = 62,
8462 = 715716, 7 + 1 + 57 + 16 = 92,
8462 = 715716, 71 + 57 + 16 = 122,
8462 = 715716, 7 + 1 + 5 + 716 = 272,
8462 = 715716, 715 + 7 + 1 + 6 = 272.
(13 + 23 + ... + 1023)(1033 + 1043 + ... + 3623)(3633 + 3643 + ... + 8463) = 1211701605285602.
Page of Squares : First Upload November 28, 2005 ; Last Revised August 17, 2013by Yoshio Mimura, Kobe, Japan
847
The smallest squares containing k 847's :
5184729 = 22772,
8847847969 = 940632,
750884784784729 = 274022772.
8472 = 333 + 883.
8472 = 114 + 224 + 224 + 224.
3-by-3 magic squares consisting of different squares with constant 8472:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(18, 219, 818, 478, 678, 171, 699, 458, 138) | (28, 336, 777, 588, 567, 224, 609, 532, 252) |
(37, 342, 774, 414, 683, 282, 738, 366, 197) | (53, 138, 834, 366, 757, 102, 762, 354, 107) |
(54, 197, 822, 582, 606, 107, 613, 558, 174) | (54, 213, 818, 267, 782, 186, 802, 246, 117) |
(54, 342, 773, 498, 613, 306, 683, 474, 162) | (62, 411, 738, 549, 582, 278, 642, 458, 309) |
(78, 235, 810, 485, 678, 150, 690, 450, 197) | (78, 261, 802, 298, 762, 219, 789, 262, 162) |
(82, 402, 741, 501, 622, 282, 678, 411, 298) | (90, 278, 795, 453, 690, 190, 710, 405, 222) |
(102, 386, 747, 586, 507, 342, 603, 558, 206) | (123, 498, 674, 566, 453, 438, 618, 514, 267) |
(170, 522, 645, 570, 405, 478, 603, 530, 270) | (181, 438, 702, 522, 618, 251, 642, 379, 402) |
(213, 478, 666, 534, 603, 262, 622, 354, 453) |
8472 = 717409, 72 + 12 + 72 + 42 + 02 + 92 = 142,
8472 = 717409, 7 + 1 + 7 + 40 + 9 = 82.
8472 = 717409 appears in the decimal expression of e:
e = 2.71828•••717409••• (from the 95465th digit).
by Yoshio Mimura, Kobe, Japan
848
The smallest squares containing k 848's :
228484 = 4782,
848848225 = 291352,
166784878488484 = 129145222.
8482 is the fifth square which is the sum of 8 sixth powers : 4, 4, 6, 6, 6, 6, 8, 8.
8482 = 719104, 7 + 1 + 9 + 104 = 112.
Page of Squares : First Upload November 28, 2005 ; Last Revised September 25, 2006by Yoshio Mimura, Kobe, Japan
849
The smallest squares containing k 849's :
1849 = 432,
660849849 = 257072,
28493784933849 = 53379572.
The squares which begin with 849 and end in 849 are
84947182849 = 2914572, 84997320849 = 2915432, 849083002849 = 9214572,
849241500849 = 9215432, 849543793849 = 9217072,...
Komachi equation: 8492 = 92 + 8762 - 52 * 4322 / 102.
3-by-3 magic squares consisting of different squares with constant 8492:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(1, 92, 844, 524, 664, 73, 668, 521, 56) | (1, 148, 836, 316, 776, 137, 788, 311, 56), |
(1, 220, 820, 580, 599, 160, 620, 560, 151) | (4, 169, 832, 364, 752, 151, 767, 356, 76), |
(4, 476, 703, 568, 521, 356, 631, 472, 316) | (8, 496, 689, 529, 536, 392, 664, 433, 304), |
(16, 308, 791, 553, 604, 224, 644, 511, 212) | (17, 344, 776, 584, 568, 239, 616, 529, 248), |
(28, 256, 809, 556, 617, 176, 641, 524, 188) | (41, 368, 764, 512, 599, 316, 676, 476, 193), |
(49, 256, 808, 472, 664, 239, 704, 463, 104) | (52, 304, 791, 356, 727, 256, 769, 316, 172), |
(55, 340, 776, 440, 676, 265, 724, 385, 220) | (69, 342, 774, 522, 594, 309, 666, 501, 162), |
(76, 400, 745, 575, 524, 340, 620, 535, 224) | (80, 224, 815, 335, 760, 176, 776, 305, 160), |
(116, 407, 736, 511, 556, 388, 668, 496, 169) | (137, 344, 764, 536, 631, 188, 644, 452, 319), |
(164, 503, 664, 532, 584, 311, 641, 356, 428) | (171, 498, 666, 582, 414, 459, 594, 549, 258), |
(176, 361, 748, 407, 704, 244, 724, 308, 319) |
The 4-by-4 magic square consisting of different squares with constant 849:
|
8492 = 720801, 7 + 20 + 8 + 0 + 1 = 62,
8492 = 720801, 72 + 0 + 8 + 0 + 1 = 92,
8492 = 720801, 720 + 8 + 0 + 1 = 272,
8492 = 720801, 720 + 801 = 392.
by Yoshio Mimura, Kobe, Japan