700
The smallest squares containing k 700's :
247009 = 4972,
770007001 = 277492,
70047007009281 = 83694092.
(251 / 700)2 = 0.128573469... (Komachic),
(631 / 700)2 = 0.812573469... (Komachic).
Komachi equations:
7002 = 92 * 82 * 72 / 62 * 52 / 42 / 32 * 22 * 102 = 982 / 72 * 62 * 52 / 42 / 32 * 22 * 102
= 982 / 72 / 62 * 52 * 42 * 32 / 22 * 102.
7002 = 104 + 204 + 204 + 204.
Page of Squares : First Upload February 13, 2006 ; Last Revised June 25, 2010by Yoshio Mimura, Kobe, Japan
701
The smallest squares containing k 701's :
1270129 = 11272,
11370170161 = 1066312,
567017011270161 = 238121192.
7012 = 491401, a zigzag square.
(580 / 701)2 = 0.684573291... (Komachic).
3-by-3 magic squares consisting of different squares with constant 7012:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(24, 235, 660, 285, 600, 224, 640, 276, 75), | (24, 341, 612, 387, 504, 296, 584, 348, 171), |
(44, 147, 684, 339, 604, 108, 612, 324, 109), | (60, 395, 576, 424, 480, 285, 555, 324, 280), |
(64, 252, 651, 483, 456, 224, 504, 469, 132), | (72, 171, 676, 336, 604, 117, 611, 312, 144), |
(72, 381, 584, 469, 408, 324, 516, 424, 213), | (84, 429, 548, 467, 444, 276, 516, 332, 339), |
(144, 368, 579, 444, 501, 208, 523, 324, 336) |
7012 = 491401, 49 + 14 + 0 + 1 = 82,
7012 = 491401, 4 + 91 + 4 + 0 + 1 = 102.
by Yoshio Mimura, Kobe, Japan
702
The smallest squares containing k 702's :
70225 = 2652,
3470270281 = 589092,
170268702177025 = 130487052.
702 = (12 + 22 + 32 + ... + 31592) / (12 + 22 + 32 + ... + 3552).
7022 = 492804, a zigzag square.
Komachi equations:
7022 = 12 * 22 - 32 + 42 + 52 - 62 + 782 * 92 = - 12 * 22 + 32 - 42 - 52 + 62 + 782 * 92.
1 / 702 = 0.0014245014245014245...,
12 + 422 + 42 + 5012 + 4242 + 52 + 0142 + 2452 = 7022,
1 / 702 = 0.0014245014245014245...,
12 + 4242 + 5012 + 422 + 42 + 52 + 0142 + 2452 = 7022,
1 / 702 = 0.0014245014245014245...,
142 + 2452 + 012 + 422 + 42 + 5012 + 4242 + 52 = 7022,
1 / 702 = 0.0014245014245014245...,
142 + 2452 + 012 + 4242 + 5012 + 422 + 42 + 52 = 7022.
The 4-by-4 magic squares consisting of different squares with constant 702:
|
|
|
|
|
|
|
7022 = 492804, 49 + 28 + 0 + 4 = 92,
7022 = 492804, 492 + 80 + 4 = 242,
7022 = 492804, 4 + 92 + 804 = 302,
7022 = 492804, 492 + 804 = 362,
7022 = 492804, 49280 + 4 = 2222.
by Yoshio Mimura, Kobe, Japan
703
The smallest squares containing k 703's :
703921 = 8392,
17037036676 = 1305262,
703770703370329 = 265286772.
Kaprekar : 7032 = 494209, and 494 + 209 = 703.
1 / 703 = 0.0014224751066856330014224751066856330014224...,
and the sum of the squares of its digits is 703.
The square root of 703 is 26.51 ..., and 26 = 52 + 12.
7032 = 494209, 494 + 209 = 703.
22k + 24k + 58k + 65k are squares for k = 1,2,3 (132, 932, 7032).
3-by-3 magic squares consisting of different squares with constant 7032:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(6, 118, 693, 363, 594, 98, 602, 357, 66), | (27, 198, 674, 334, 597, 162, 618, 314, 117), |
(27, 246, 658, 294, 602, 213, 638, 267, 126), | (30, 278, 645, 370, 555, 222, 597, 330, 170), |
(37, 222, 666, 426, 523, 198, 558, 414, 107), | (42, 181, 678, 469, 498, 162, 522, 462, 91), |
(62, 366, 597, 453, 478, 246, 534, 363, 278), | (69, 278, 642, 422, 498, 261, 558, 411, 118), |
(90, 422, 555, 453, 390, 370, 530, 405, 222), | (138, 334, 603, 363, 558, 226, 586, 267, 282) |
7032 = 494209, 4 + 9 + 42 + 0 + 9 = 82,
7032 = 494209, 49 + 4 + 2 + 0 + 9 = 82,
7032 = 494209, 49 + 42 + 0 + 9 = 102.
by Yoshio Mimura, Kobe, Japan
704
The smallest squares containing k 704's :
2704 = 522,
597704704 = 244482,
1704570470464 = 13055922.
The squares which begin with 704 and end in 704 are
7047266704 = 839482, 70462640704 = 2654482, 704008258704 = 8390522,
704672944704 = 8394482, 704847560704 = 8395522,...
7042 = 495616, a zigzag square.
7042± 3 are primes.
7042 = (12 + 7)(22 + 7)(52 + 7)(132 + 7) = (12 + 7)(22 + 7)(752 + 7) = (132 + 7)(532 + 7)
= (22 + 7)(32 + 7)(52 + 7)(92 + 7) = (22 + 7)(32 + 7)(532 + 7) = (32 + 7)(52 + 7)(312 + 7)
= (52 + 7)(92 + 7)(132 + 7) = (92 + 7)(752 + 7).
Cubic polunomial :
(X + 7042)(X + 8612)(X + 10082) = X3 + 15012X2 + 12744482X + 6109931522.
704k + 1034k + 1870k + 6193k are squares for k = 1,2,3 (992, 65892, 4954952).
81k + 136k + 304k + 704k are squares for k = 1,2,3 (352, 7832, 194952).
(13 + 23 + ... + 1493)(1503 + 1513 + ... + 3753)(3763 + 3773 + ... + 7043) = 1850843003625002.
7042 = 495616, 4 + 95 + 6 + 16 = 112,
7042 = 495616, 49 + 5 + 61 + 6 = 112,
7042 = 495616, 49 + 56 + 16 = 112.
7042 = 495616 appears in the decimal expression of e:
e = 2.71828•••495616••• (from the 58306th digit).
by Yoshio Mimura, Kobe, Japan
705
The smallest squares containing k 705's :
7056 = 842,
77052877056 = 2775842,
77705705927056 = 88150842.
7052 = 497025, a square with different digits.
Komachi Square Sum : 7052 = 82 + 322 + 4762 + 5192.
7052 = 413 + 523 + 663.
3-by-3 magic squares consisting of different squares with constant 7052:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(4, 280, 647, 497, 460, 196, 500, 455, 200), | (7, 200, 676, 476, 500, 143, 520, 455, 140), |
(20, 185, 680, 295, 620, 160, 640, 280, 95), | (20, 185, 680, 335, 596, 172, 620, 328, 71), |
(20, 260, 655, 295, 592, 244, 640, 281, 92), | (20, 260, 655, 335, 580, 220, 620, 305, 140), |
(20, 295, 640, 433, 500, 244, 556, 400, 167), | (20, 335, 620, 368, 524, 295, 601, 332, 160), |
(25, 80, 700, 224, 665, 68, 668, 220, 49), | (25, 356, 608, 400, 508, 281, 580, 335, 220), |
(25, 400, 580, 484, 412, 305, 512, 409, 260), | (27, 414, 570, 486, 402, 315, 510, 405, 270), |
(49, 332, 620, 460, 455, 280, 532, 424, 185), | (80, 172, 679, 220, 655, 140, 665, 196, 128), |
(80, 223, 664, 440, 536, 127, 545, 400, 200), | (80, 400, 575, 440, 479, 272, 545, 328, 304), |
(88, 391, 580, 484, 388, 335, 505, 440, 220), | (90, 213, 666, 270, 630, 165, 645, 234, 162), |
(90, 354, 603, 405, 522, 246, 570, 315, 270), | (95, 280, 640, 448, 520, 161, 536, 385, 248), |
(113, 316, 620, 484, 487, 160, 500, 400, 295), | (140, 305, 620, 392, 556, 185, 569, 308, 280) |
The 4-by-4 magic square consisting of different squares with constant 705:
|
7052 = 497025, 49 + 7 + 0 + 25 = 92,
7052 = 497025, 49 + 70 + 25 = 122.
7052 = 497025 appears in the decimal expression of e:
e = 2.71828•••497025••• (from the 27101st digit).
by Yoshio Mimura, Kobe, Japan
706
The smallest squares containing k 706's :
2706025 = 16452,
7064570601 = 840512,
7697062970670649 = 877329072.
7062 = 94 + 154 + 154 + 254.
7062 = 498436, 49 + 84 + 36 = 132,
7062 = 498436, 4 + 984 + 36 = 322.
by Yoshio Mimura, Kobe, Japan
707
The smallest squares containing k 707's :
70756 = 2662,
7079707881 = 841412,
67074707707225 = 81899152.
7072 = 499849, a square with just 3 kinds of digits.
(296 / 707)2 = 0.175284936... (Komachic).
7072 + 7082 + 7092 + ... + 96842 = 5501372.
3-by-3 magic squares consisting of different squares with constant 7072:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(18, 166, 687, 463, 522, 114, 534, 447, 122), | (18, 225, 670, 330, 590, 207, 625, 318, 90), |
(18, 302, 639, 401, 522, 258, 582, 369, 158), | (31, 198, 678, 342, 598, 159, 618, 321, 122), |
(33, 246, 662, 438, 527, 174, 554, 402, 177), | (54, 158, 687, 193, 666, 138, 678, 177, 94), |
(66, 337, 618, 482, 474, 207, 513, 402, 274), | (78, 257, 654, 303, 606, 202, 634, 258, 177), |
(78, 354, 607, 446, 447, 318, 543, 418, 174), | (95, 330, 618, 390, 543, 230, 582, 310, 255), |
(138, 337, 606,417,534, 202, 554, 318, 303), | (177, 438, 526, 482, 306, 417, 486, 463, 222) |
7072 = 499849, 4 + 99 + 84 + 9 = 142,
7072 = 499849, 49 + 98 + 49 = 142.
7072 = 499849 appears in the decimal expression of e:
e = 2.71828•••499849••• (from the 97279th digit).
by Yoshio Mimura, Kobe, Japan
708
The smallest squares containing k 708's :
47089 = 2172,
35708527089 = 1889672,
708770870808001 = 266227512.
7082 = 501264, a square with different digits.
(12 + 6)(52 + 6)(62 + 6)(72 + 6) = 7082 + 6.
(13 + 23 + ... + 1363)(1373 + 1383 + ... + 2923)(2933 + 2943 + ... + 7083) = 961938424487042.
The square root of 708 is 26. 6 0 8 2 6 9 3 9 13 0 0 14 ...,
and 262 = 62 + 02 + 82 + 22 + 62 + 92 + 32 + 92 + 132 + 02 + 02 + 142.
7082 = 283 + 423 + 743 = 144 + 164 + 164 + 244.
7082 = 501264, 5 + 0 + 1 + 26 + 4 = 62,
7082 = 501264, 5 + 0 + 12 + 64 = 92,
7082 = 501264, 50 + 1 + 26 + 4 = 92.
by Yoshio Mimura, Kobe, Japan
709
The smallest squares containing k 709's :
2709316 = 16462,
70905570961 = 2662812,
709170960709696 = 266302642.
7092 = 502681, a square with different digits.
The square root of 709 is 26. 6 2 7 0 5 3 9 11 3 8 8 6 9 4 9 ...,
and 262 = 62 + 22 + 72 + 02 + 52 + 32 + 92 + 112 + 32 + 82 + 82 + 62 + 92 + 42 + 92.
3-by-3 magic squares consisting of different squares with constant 7092:
A2 | B2 | C2 |
D2 | E2 | F2 |
G2 | H2 | K2 |
where (A, B, C, D, E, F, G, H, K) = | |
(3, 84, 704, 384, 592, 69, 596, 381, 48), | (24, 448, 549, 477, 396, 344, 524, 381, 288), |
(29, 312, 636, 444, 504, 227, 552, 389, 216), | (36, 232, 669, 416, 549, 168, 573, 384, 164), |
(52, 276, 651, 309, 596, 228, 636, 267, 164), | (56, 123, 696, 228, 664, 99, 669, 216, 92), |
(56, 219, 672, 483, 504, 124, 516, 448, 189), | (84, 360, 605, 480, 475, 216, 515, 384, 300), |
(99, 216, 668, 344, 603, 144, 612, 304, 189), | (132, 459, 524, 484, 444, 267, 501, 308, 396) |
7092 = 502681, 50 + 2 + 68 + 1 = 112.
Page of Squares : First Upload August 22, 2005 ; Last Revised August 4, 2009by Yoshio Mimura, Kobe, Japan