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540 - 549

540

The smallest squares containing k 540's :
254016 = 5042,
5409455401 = 735492,
235402954065409 = 153428472.

5402 = 23 + 163 + 663 = 93 + 153 + 663.

5402 + 5412 + 5422 + 5432 + ... + 5632 = 27022.

Komachi equations:
5402 = 92 * 82 * 72 * 62 * 52 / 42 * 32 / 212 = 92 * 82 / 72 * 62 * 52 / 42 / 32 * 212
 = - 92 - 82 * 72 - 62 + 5432 + 22 * 12 = - 92 - 82 * 72 - 62 + 5432 + 22 / 12.

(1 + 2)(3 + 4 + 5)(6)(7 + 8)(9)(10) = 5402,
(1 + 2)(3 + 4 + 5 + 6)(7 + 8)(9 + 10 + 11)(12) = 5402,
(1)(2)(3 + 4 + 5 + 6)(7 + 8)(9)(10 + 11 + 12 + 13 + 14) = 5402,
(1)(2 + 3 + 4 + 5 + 6)(7 + 8 + 9 + 10 + 11)(12)(13 + 14) = 5402,
(1 + 2 + 3)(4 + 5 + 6 + 7 + 8)(9 + 10 + 11)(12 + 13 + 14 + 15) = 5402,
(1)(2)(3)(4 + 5 + 6 + 7 + 8)(9 + 10 + 11)(12 + 13 + 14 + 15) = 5402,
(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8)(9)(10 + 11 + 12 + 13 + 14)(15) = 5402,
(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8)(9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18)(19 + 20 + 21) = 5402,
(1 + 2 + 3 + 4)(5 + 6 + 7 + 8 + 9 + 10)(11 + 12 + ... + 37) = 5402,
(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9)(10)(11 + 12 + ... + 37) = 5402,
(1)(2 + 3 + ... + 25)(26 + 27 + ... + 49) = 5402,
(1 + 2)(3 + 4 + 5 + 6 + 7)(8 + 9 + ... + 88) = 5402.

(12 + 22 + ... + 42)(52 + 62 + ... + 1802)(1812 + 1822 + ... + 5402) = 545886002.

(13 + 23 + ... + 3693)(3703 + 3713 + ... + 4493)(4503 + 4513 + ... + 5403) = 5363409636585002.

5402 = 291600 appears in the decimal expression of e:
  e = 2.71828•••291600••• (from the 55639th digit)

Page of Squares : First Upload May 9, 2005 ; Last Revised June 15, 2010
by Yoshio Mimura, Kobe, Japan

541

The smallest squares containing k 541's :
541696 = 7362,
26654154121 = 1632612,
5413541966554176 = 735767762.

5412 = 292681, 29 + 26 + 8 + 1 = 82,
5412 = 292681, 2+9 + 2+6 + 81 = 102,
5412 = 292681, 29 + 2 + 68 + 1 = 102,
5412 = 292681, 292 + 68 + 1 = 192.

3-by-3 magic squares consisting of different squares with constant 5412:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(3, 216, 496, 336, 388, 171, 424, 309, 132),(8, 204, 501, 291, 424, 168, 456, 267, 116),
(8, 309, 444, 381, 312, 224, 384, 316, 213),(21, 116, 528, 368, 384, 99, 396, 363, 64),
(36, 171, 512, 213, 476, 144, 496, 192, 99),(36, 269, 468, 307, 396, 204, 444, 252, 179),
(60, 280, 459, 309, 360, 260, 440, 291, 120),(84, 235, 480, 360, 384, 125, 395, 300, 216)

5412 + 5422 + 5432 + 5442 + ... + 35652 = 1227052.

Page of Squares : First Upload May 9, 2005 ; Last Revised June 1, 2009
by Yoshio Mimura, Kobe, Japan

542

The smallest squares containing k 542's :
54289 = 2332,
3542154256 = 595162,
515420215425424 = 227028682.

5422 = 293764, a square with different digits.

10k + 218k + 542k + 674k are squares for k = 1,2,3 (382, 8922, 218122).

5422 = 293764, 29 + 3 + 7 + 6 + 4 = 72.

Page of Squares : First Upload May 9, 2005 ; Last Revised March 15, 2011
by Yoshio Mimura, Kobe, Japan

543

The smallest squares containing k 543's :
543169 = 7372,
83543543521 = 2890392,
205435433654361 = 143330192.

5432 = 294849, a zigzag square.

Komachi equations:
5432 = 92 - 82 - 72 + 62 + 5432 - 22 * 12 = 92 - 82 - 72 + 62 + 5432 - 22 / 12
 = - 92 + 82 + 72 - 62 + 5432 + 22 * 12 = - 92 + 82 + 72 - 62 + 5432 + 22 / 12,
5432 = 13 * 233 - 453 + 63 + 73 + 83 * 93,
5432 = - 14 + 234 + 44 + 564 / 74 + 84 + 94.

5432 = 294849, 2 + 9 + 4 + 8 + 4 + 9 = 62.

3-by-3 magic squares consisting of different squares with constant 5432:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(2, 146, 523, 314, 427, 118, 443, 302, 86),(2, 166, 517, 197, 482, 154, 506, 187, 62),
(5, 118, 530, 250, 470, 107, 482, 245, 50),(22, 133, 526, 373, 386, 82, 394, 358, 107),
(34, 98, 533, 142, 517, 86, 523, 134, 58),(43, 110, 530, 230, 485, 82, 490, 218, 85),
(43, 254, 478, 358, 373, 166, 406, 302, 197),(53, 154, 518, 322, 427, 94, 434, 298, 133),
(58, 251, 478, 338, 358, 229, 421, 322, 118),(63, 192, 504, 288, 441, 132, 456, 252, 153),
(91, 302, 442, 358, 299, 278, 398, 338, 149),(107, 226, 482, 278, 443, 146, 454, 218, 203),
(146, 203, 482, 302, 406, 197, 427, 298, 154) 

Page of Squares : First Upload May 9, 2005 ; Last Revised June 15, 2010
by Yoshio Mimura, Kobe, Japan

544

The smallest squares containing k 544's :
12544 = 1122,
4954470544 = 703882,
54406954475449 = 73761072.

The squares which begin with 544 and end in 544 are
54469958544 = 2333882,   544071462544 = 7376122,   544478700544 = 7378882,
544809324544 = 7381122,   5440033782544 = 23323882,...

5442 = 295936, a zigzag square.

5442 = 295936, 2 + 95 + 93 + 6 = 142.

Page of Squares : First Upload May 9, 2005 ; Last Revised August 9, 2006
by Yoshio Mimura, Kobe, Japan

545

The smallest squares containing k 545's :
954529 = 9772,
5458845456 = 738842,
545545945455625 = 233569252.

13 - 23 + 33 - 43 + 53 - 63 + ... + 5433 - 5443 + 5453 = 90092.

Komachi Fractions : 243 / 8019675 = (3 / 545)2, 8019675 / 432 = (545 / 4)2.

5452 = 297025, 2 + 9 + 7 + 0 + 2 + 5 = 52.

3-by-3 magic squares consisting of different squares with constant 5452:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(0, 300, 455, 327, 364, 240, 436, 273, 180),(4, 135, 528, 228, 480, 121, 495, 220, 60),
(12, 175, 516, 320, 420, 135, 441, 300, 112),(12, 284, 465, 360, 345, 220, 409, 312, 180),
(40, 129, 528, 255, 472, 96, 480, 240, 95),(40, 231, 492, 345, 392, 156, 420, 300, 175),
(40, 255, 480, 345, 360, 220, 420, 320, 135),(68, 249, 480, 276, 432, 185, 465, 220, 180),
(81, 180, 508, 220, 480, 135, 492, 185, 144),(84, 212, 495, 288, 441, 140, 455, 240, 180)

Page of Squares : First Upload May 9, 2005 ; Last Revised June 1, 2009
by Yoshio Mimura, Kobe, Japan

546

The smallest squares containing k 546's :
546121 = 7392,
5468454601 = 739492,
854654645463529 = 292344772.

28938k + 64974k + 70434k + 133770k are squares for k = 1,2,3 (5462, 1670762, 551514602).

The integral triangle of sides 696, 865, 1183 (or 2809, 4395, 7202) has square area 5462.

Komachi fraction : 675 / 8049132 = (5 / 546)2.

Komachi equations:
5462 = 92 * 82 * 72 * 652 / 42 / 32 * 22 / 102 = 982 / 72 * 652 * 42 * 32 / 22 / 102.

5462 = 298116, 2 * 9 + 8 * 11 * 6 = 546.

5462 = (122 + 3)(452 + 3) = (22 + 3)(62 + 3)(332 + 3) = (62 + 3)(72 + 3)(122 + 3).

5462 = (1 + 2)(3 + 4)(5 + 6 + 7 + 8)(9 + 10 + 11 + 12)(13).

The 4-by-4 magic squares consisting of different squares with constant 546:

02 12162172
82212 42 52
112 22152142
192102 72 62
      
12 52 62222
82122172 72
92192102 22
202 42112 32

5462 = 298116, 2 + 9 + 8 + 1 + 16 = 62,
5462 = 298116, 2 + 9 + 8 + 11 + 6 = 62.

5462 = 298116 appears in the decimal expression of π:
  π = 3.14159•••298116••• (from the 71474th digit).

Page of Squares : First Upload May 9, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

547

The smallest squares containing k 547's :
5476 = 742,
18754754704 = 1369482,
1354707547547904 = 368063522.

5472 = 299209, a square with just 3 kinds of digits.

12 + 22 + ... + 5472 = 54705470.

Komachi equation: 5472 = 92 + 872 * 62 + 542 * 32 + 22 * 102.

5472 = 299209, 2 + 9 + 9 + 20 + 9 = 72,
5472 = 299209, 29 + 9 + 2 + 0 + 9 = 72,
5472 = 299209, 29920 + 9 = 1732.

3-by-3 magic squares consisting of different squares with constant 5472:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(2, 273, 474, 354, 362, 207, 417, 306, 178),(33, 326, 438, 378, 303, 254, 394, 318, 207),
(38, 159, 522, 369, 378, 142, 402, 362, 81),(54, 222, 497, 367, 354, 198, 402, 353, 114),
(66, 178, 513, 207, 486, 142, 502, 177, 126),(70, 222, 495, 303, 430, 150, 450, 255, 178),
(79, 282, 462, 318, 402, 191, 438, 241, 222),(138, 303, 434, 326, 402, 177, 417, 214, 282)

5472 = 299209 appears in the decimal expression of e:
  e = 2.71828•••299209••• (from the 31102nd digit)

Page of Squares : First Upload May 9, 2005 ; Last Revised June 15, 2010
by Yoshio Mimura, Kobe, Japan

548

The smallest squares containing k 548's :
95481 = 3092,
5484735481 = 740592,
632255485485481 = 251446912.

5482 = 300304, a square with just 3 kinds of digits.

5482 = 44 + 44 + 164 + 224.

548, 549 and 550 are three consecutive integers having square factors (the 8th case).

5482 = 300304, 30 + 0 + 30 + 4 = 82.

Page of Squares : First Upload May 9, 2005 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

549

The smallest squares containing k 549's :
549081 = 7412,
10549549521 = 1027112,
1854954954979344 = 430691882.

5492 = 301401, 3 + 0 + 1 + 4 + 0 + 1 = 32,
5492 = 301401, 30 + 1 + 4 + 0 + 1 = 62,
5492 = 301401, 3 + 0 + 140 + 1 = 122.

16104k + 58194k + 106689k + 120414k are squares for k = 1,2,3 (5492, 1718372,562280312).

Komachi equation: 5492 = 13 * 23 + 33 * 43 - 563 + 783 + 93.

3-by-3 magic squares consisting of different squares with constant 5492:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(5, 176, 520, 376, 380, 125, 400, 355, 124),(6, 231, 498, 282, 426, 201, 471, 258, 114),
(8, 104, 539, 259, 476, 88, 484, 253, 56),(16, 163, 524, 251, 464, 152, 488, 244, 61),
(29, 196, 512, 316, 413, 176, 448, 304, 91),(30, 201, 510, 345, 390, 174, 426, 330, 105),
(42, 174, 519, 246, 471, 138, 489, 222, 114),(61, 244, 488, 328, 376, 229, 436, 317, 104),
(64, 356, 413, 379, 328, 224, 392, 259, 284),(68, 149, 524, 236, 484, 107, 491, 212, 124),
(107, 316, 436, 356, 292, 299, 404, 341, 148),(128, 349, 404, 376, 236, 323, 379, 352, 184)
Page of Squares : First Upload May 9, 2005 ; Last Revised March 15, 2011
by Yoshio Mimura, Kobe, Japan