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580 - 589

580

The smallest squares containing k 580's :
58081 = 2412,
160558085809 = 4006972,
58055804958096 = 76194362.

580 = (12 + 22 + 32 + ... + 872) / (12 + 22 + 32 + ... + 102).

5802± 3 are primes.

Komachi equations:
5802 = 12 * 22 * 342 + 562 / 72 * 82 * 92 = 92 * 872 * 62 / 542 / 32 * 22 * 102.

5802 = 383 + 513 + 533.

5802 = (42 + 4)(52 + 4)(242 + 4).

32 + 42 + 52 + 62 + ... + 5802 = 80752.

(12)(22)(32 + 42 + ... + 5802) = 161502.

Page of Squares : First Upload May 30, 2005 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

581

The smallest squares containing k 581's :
5812921 = 24112,
581581456 = 241162,
1658158142958144 = 407204882.

1 / 581 = 0.001721170395869191049..., the sum of the squares of its digits is 581.

5812 + 5822 + 5832 + 5842 + ... + 6762 = 61642.

3-by-3 magic squares consisting of different squares with constant 5812:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(3, 76, 576, 384, 432, 59, 436, 381, 48),(4, 267, 516, 309, 436, 228, 492, 276, 139),
(12, 139, 564, 356, 444, 117, 459, 348, 76),(36, 292, 501, 373, 396, 204, 444, 309, 212),
(48, 131, 564, 219, 528, 104, 536, 204, 93),(60, 195, 544, 355, 444, 120, 456, 320, 165),
(96, 347, 456, 381, 384, 212, 428, 264, 291),(99, 248, 516, 324, 456, 157, 472, 261, 216),
(104, 264, 507, 291, 468, 184, 492, 221, 216) 

5812 = 337561, 3 + 3 + 7 + 5 + 6 + 1 = 52,
5812 = 337561, 33 + 75 + 61 = 132.

Page of Squares : First Upload May 30, 2005 ; Last Revised June 22, 2009
by Yoshio Mimura, Kobe, Japan

582

The smallest squares containing k 582's :
535824 = 7322,
5822758249 = 763072,
2958249582582849 = 543897932.

(385 / 582)2 = 0.437598162... (Komachic).

The square root of 582 is 24.1246761636296374141...,
242 = 122 + 42 + 62 + 72 + 62 + 12 + 62 + 32 + 62 + 22 + 92 + 62 + 32 + 72 + 42 + 12 + 42 + 12.

5822 = (32 + 3)(1682 + 3).

138k + 417k + 582k + 888k are squares for k = 1,2,3 (452, 11492, 311852).
51410k + 61498k + 66154k + 159662k are squares for k = 1,2,3 (5822, 1905082, 687609722).

The 4-by-4 magic square consisting of different squares with constant 582:

02 22 72232
102202 92 12
112132162 62
192 32142 42

5822 = 338724, 333 + 83 + 73 + 23 + 43 = 1922,
5822 = 338724, 33 + 87 + 24 = 122.

Page of Squares : First Upload May 30, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

583

The smallest squares containing k 583's :
458329 = 6772,
53058358336 = 2303442,
458358340583025 = 214093052.

5832 = 339889, a square consisting of just 3 kinds of digits.

(510 / 583)2 = 0.765249831... (Komachic).

46057k + 46640k + 114268k + 132924k are squares for k = 1,2,3 (5832, 1871432,635592432).

(12 + 22 + ... + 2332)(2342 + 2352 + ... + 4662)(4672 + 4682 + ... + 5832) = 637732566122.

5832 = 44 + 64 + 94 + 244.

3-by-3 magic squares consisting of different squares with constant 5832:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(3, 322, 486, 398, 354, 237, 426, 333, 218),(18, 106, 573, 326, 477, 78, 483, 318, 74),
(42, 115, 570, 270, 510, 83, 515, 258, 90),(45, 210, 542, 342, 430, 195, 470, 333, 90),
(83, 246, 522, 402, 403, 126, 414, 342, 227),(102, 189, 542, 218, 522, 141, 531, 178, 162),
(106, 237, 522, 318, 466, 147, 477, 258, 214),(141, 358, 438, 398, 258, 339, 402, 381, 182)

5832 = 339889, 3 + 3 + 98 + 8 + 9 = 112.

Page of Squares : First Upload May 30, 2005 ; Last Revised March 15, 2011
by Yoshio Mimura, Kobe, Japan

584

The smallest squares containing k 584's :
29584 = 1722,
358458489 = 189332,
25842958458409 = 50835972.

The squares which begin with 584 and end in 584 are
584285584 = 241722,   58405355584 = 2416722,   58480781584 = 2418282,
584197291584 = 7643282,   584723267584 = 7646722,...

1 / 584 = 0.00171232..., 12 + 72 + 12 + 232 + 22 = 584.

(457 / 584)2 = 0.612359847... (Komachic).

5842 = 341056, a square with different digits.

5842 = 341056, 3 + 4 + 1 + 0 + 56 = 82,
5842 = 341056, 3 + 41 + 0 + 56 = 102.

Page of Squares : First Upload May 30, 2005 ; Last Revised August 21, 2006
by Yoshio Mimura, Kobe, Japan

585

The smallest squares containing k 585's :
58564 = 2422,
225854958564 = 4752422,
585358539425856 = 241941842.

5392 + 5402 + 5412 + 5422 + ... + 5852 = 38542,
2102 + 2112 + 2122 + 2132 + ... + 5852 = 79902.

5852 = (22 + 9)(62 + 9)(242 + 9).

19110k + 86970k + 105105k + 131040k are squares for k = 1,2,3 (5852, 1901252,638439752).

Komachi equations:
5852 = 122 - 32 * 42 + 52 / 62 * 782 * 92 = 122 / 32 - 42 + 52 / 62 * 782 * 92
  = 122 / 32 / 42 * 52 / 62 * 782 * 92 = - 122 + 32 * 42 + 52 / 62 * 782 * 92
 = - 122 / 32 + 42 + 52 / 62 * 782 * 92.

(1 + 2 + 3 + 4 + 5)(6 + 7)(8 + 9 + 10)(11 + 12 + 13 + 14 + 15) = 5852,
(1 + 2 + ... + 26)(27 + 28 + ... + 51) = 5852,
(1 + 2 + ... + 25)(26 + 27 + ... + 52) = 5852.

3-by-3 magic squares consisting of different squares with constant 5852:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(0, 225, 540, 351, 432, 180, 468, 324, 135),(1, 140, 568, 268, 505, 124, 520, 260, 65),
(6, 258, 525, 330, 435, 210, 483, 294, 150),(7, 76, 580, 260, 520, 65, 524, 257, 40),
(8, 169, 560, 215, 520, 160, 544, 208, 55),(16, 212, 545, 412, 391, 140, 415, 380, 160),
(20, 121, 572, 385, 428, 104, 440, 380, 65),(28, 71, 580, 104, 572, 65, 575, 100, 40),
(40, 265, 520, 296, 440, 247, 503, 280, 104),(40, 265, 520, 400, 392, 169, 425, 344, 208),
(41, 280, 512, 320, 440, 215, 488, 265, 184),(44, 208, 545, 292, 481, 160, 505, 260, 140),
(52, 161, 560, 364, 448, 95, 455, 340, 140),(52, 280, 511, 364, 385, 248, 455, 340, 140),
(65, 260, 520, 380, 377, 236, 440, 364, 127),(65, 260, 520, 380, 415, 160, 440, 320, 215),
(78, 195, 546, 246, 510, 147, 525, 210, 150),(100, 313, 484, 400, 316, 287, 415, 380, 160),
(100, 377, 436, 400, 364, 223, 415, 260, 320) 

5852 = 342225, 3 + 4 + 2 + 2 + 25 = 62,
5852 = 342225, 3 + 4 + 2 + 22 + 5 = 62,
5852 = 342225, 3 + 4 + 22 + 2 + 5 = 62,
5852 = 342225, 34 + 22 + 25 = 92.

Page of Squares : First Upload May 30, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

586

The smallest squares containing k 586's :
586756 = 7662,
5861586721 = 765612,
658635865865604 = 256639022.

5862 = 343396, 3 + 43 + 3 + 9 + 6 = 82,
5862 = 343396, 343 + 3 + 9 + 6 = 192.

Page of Squares : First Upload January 16, 2006 ; Last Revised August 21, 2006
by Yoshio Mimura, Kobe, Japan

587

The smallest squares containing k 587's :
15876 = 1262,
13835875876 = 1176262,
35878458758769 = 59898632.

1 / 587 = 0.00170357751277683134582623..., the sum of the squares of its digits is 587.

5872 = 344569, a square whose digits are nondecreasing.

3-by-3 magic squares consisting of different squares with constant 5872:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(9, 182, 558, 362, 441, 138, 462, 342, 119),(25, 138, 570, 390, 430, 87, 438, 375, 110),
(38, 234, 537, 327, 438, 214, 486, 313, 102),(38, 234, 537, 375, 438, 110, 450, 313, 210),
(39, 178, 558, 222, 522, 151, 542, 201, 102),(42, 366, 457, 393, 322, 294, 434, 327, 222),
(102, 354, 457, 394, 297, 318, 423, 362, 186),(137, 318, 474, 366, 423, 178, 438, 254, 297)

5872 = 344569, 33 + 43 + 43 + 53 + 63 + 93 = 352,
5872 = 344569, 3 + 4 + 45 + 69 = 112,
5872 = 344569, 3 + 44 + 5 + 69 = 112.

Page of Squares : First Upload May 30, 2005 ; Last Revised June 22, 2009
by Yoshio Mimura, Kobe, Japan

588

The smallest squares containing k 588's :
358801 = 5992,
15889358809 = 1260532,
1195885882785889 = 345815832.

5882± 5 are primes.

5882 = 143 + 703, the tenth square which is the sum of 2 cubes.

5882 = (12 + 3)(22 + 3)(92 + 3)(122 + 3) = (22 + 3)(32 + 3)(52 + 3)(122 + 3)
= (52 + 3)(92 + 3)(122 + 3).

Komachi cube sum : 5882 = 53 + 83 + 93 + 243 + 313 + 673.

5882 = 143 + 153 + 163 + 173 + 183 + ... + 343.

5882 + 5892 + 5902 + 5912 + ... + 70112 = 3388662,
5882 + 5892 + 5902 + 5912 + ... + 83242 = 4384302.

(1 + 2 + 3)(4 + 5 + ... + 24)(25 + 26 + ... + 31) = 5882.

(12 + 22 + ... + 592)(602 + 612 + ... + 2202)(2212 + 2222 + ... + 5882) = 39789411202.

5882 = 345744, 345 + 744 = 332.

Page of Squares : First Upload May 30, 2005 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

589

The smallest squares containing k 589's :
589824 = 7682,
25895890084 = 1609222,
158958965899449 = 126078932.

1 / 589 = 0.00169..., 169 = 132.

5892 = 346921, a square with different digits.

Komachi equations:
5892 = 1232 + 42 + 562 / 72 * 82 * 92,
5892 = 93 * 83 - 73 - 63 + 53 * 43 - 323 - 103.

5892 = 153 + 413 + 653.

1832 + 1842 + 1852 + ... + 5892 = 81402.

(13 + 23 + ... + 893)(903 + 913 + ... + 1853)(1863 + 1873 + ... + 5893) = 115866412200002.

3-by-3 magic squares consisting of different squares with constant 5892:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(3, 284, 516, 396, 381, 212, 436, 348, 189),(4, 108, 579, 387, 436, 84, 444, 381, 68),
(4, 123, 576, 261, 516, 112, 528, 256, 51),(24, 212, 549, 243, 504, 184, 536, 219, 108),
(24, 283, 516, 312, 444, 229, 499, 264, 168),(40, 339, 480, 411, 360, 220, 420, 320, 261),
(45, 240, 536, 360, 436, 165, 464, 315, 180),(67, 156, 564, 276, 509, 108, 516, 252, 131),
(96, 284, 507, 392, 411, 156, 429, 312, 256),(96, 339, 472, 364, 408, 219, 453, 256, 276)

5892 = 346921, 3 + 4 + 6 + 9 + 2 + 1 = 52,
5892 = 346921, 34 + 6 + 921 = 312.

Page of Squares : First Upload May 30, 2005 ; Last Revised June 18, 2010
by Yoshio Mimura, Kobe, Japan