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840 - 849

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).

Page of Squares : First Upload November 28, 2005 ; Last Revised December 7, 2013
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:

A2B2C2
D2E2F2
G2H2K2
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.

Page of Squares : First Upload November 28, 2005 ; Last Revised September 25, 2009
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.

Page of Squares : First Upload November 28, 2005 ; Last Revised September 25, 2006
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:

A2B2C2
D2E2F2
G2H2K2
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).

Page of Squares : First Upload November 28, 2005 ; Last Revised July 6, 2010
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.

Page of Squares : First Upload November 28, 2005 ; Last Revised December 14, 2013
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:

A2B2C2
D2E2F2
G2H2K2
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, 2009
by 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:

02 12192222
62212152122
92202142132
272 22 82 72
      
02 12192222
72282 22 32
112 62202172
262 52 92 82
      
02 62 92272
72212162102
112152222 42
262122 52 12
      
02 62 92272
142242 52 72
173 32222 82
192152162 22
      
02 92182212
112202172 62
142192 82152
232 22132122

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, 2013
by 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:

A2B2C2
D2E2F2
G2H2K2
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).

Page of Squares : First Upload November 28, 2005 ; Last Revised September 25, 2009
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, 2006
by 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:

A2B2C2
D2E2F2
G2H2K2
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:

02 12 82282
132242102 22
142162192 62
222 42182 52

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.

Page of Squares : First Upload November 28, 2005 ; Last Revised July 6, 2010
by Yoshio Mimura, Kobe, Japan