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700 - 709

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, 2010
by 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:

A2B2C2
D2E2F2
G2H2K2
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.

Page of Squares : First Upload August 22, 2005 ; Last Revised August 4, 2009
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:

02 22132232
62242 92 32
152112162102
212 12142 82
      
12 22162212
42252 52 62
182 32152122
192 82142 92
      
12 42182192
52222122 72
102112152162
242 92 32 62
      
12 42182192
52232122 22
102112152162
242 62 32 92
12 42182192
82252 32 22
142 52152162
212 62122 92
      
12 52102242
112222 42 92
162 72192 62
182122152 32
      
12 52102242
112232 42 62
162 22192 92
182122152 32

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.

Page of Squares : First Upload August 22, 2005 ; Last Revised June 25, 2010
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:

A2B2C2
D2E2F2
G2H2K2
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.

Page of Squares : First Upload August 22, 2005 ; Last Revised August 4, 2009
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).

Page of Squares : First Upload August 22, 2005 ; Last Revised January 16, 2014
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:

A2B2C2
D2E2F2
G2H2K2
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:

02 52142222
72242 42 82
162 22182112
202102132 62

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).

Page of Squares : First Upload August 22, 2005 ; Last Revised September 14, 2009
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.

Page of Squares : First Upload February 13, 2006 ; Last Revised August 31, 2006
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:

A2B2C2
D2E2F2
G2H2K2
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).

Page of Squares : First Upload August 22, 2005 ; Last Revised August 4, 2009
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.

Page of Squares : First Upload August 22, 2005 ; Last Revised August 31, 2006
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:

A2B2C2
D2E2F2
G2H2K2
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, 2009
by Yoshio Mimura, Kobe, Japan