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570

The smallest squares containing k 570's :
257049 = 507²,
257057089 = 16033²,
357057077570116 = 18895954².

570² = 324900, 324 = 18² and 900 = 30².

138^k + 570^k + 1086^k + 1122^k are squares for k = 1,2,3 (54², 1668², 53676²).

(1 + 2 + 3)(4 + 5 + ... + 15)(16 + 17 + ... + 34) = 570².

570² = 324900 appears in the decimal expression of π:
  π = 3.14159•••324900••• (from the 65029th digit).

Page of Squares : First Upload May 23, 2005 ; Last Revised March 15, 2011
by Yoshio Mimura, Kobe, Japan

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571

The smallest squares containing k 571's :
57121 = 239²,
22571757121 = 150239²,
1571571571874704 = 39643052².

571² = 326041, a zigzag square with different digits.

3-by-3 magic squares consisting of different squares with constant 571²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(6, 121, 558, 167, 534, 114, 546, 162, 41),	(6, 194, 537, 238, 489, 174, 519, 222, 86),
(9, 138, 554, 202, 519, 126, 534, 194, 57),	(9, 202, 534, 222, 519, 86, 526, 126, 183),
(9, 222, 526, 306, 446, 183, 482, 279, 126),	(14, 279, 498, 393, 366, 194, 414, 338, 201),
(22, 201, 534, 294, 454, 183, 489, 282, 86),	(30, 135, 554, 345, 446, 90, 454, 330, 105),
(41, 198, 534, 378, 391, 174, 426, 366, 103),	(114, 302, 471, 327, 426, 194, 454, 231, 258)

571² = 326041, 3 + 2 + 6 + 0 + 4 + 1 = 4².

571² + 572² + 573² + 574² + ... + 44972² = 5506295².

Page of Squares : First Upload May 23, 2005 ; Last Revised June 16, 2009
by Yoshio Mimura, Kobe, Japan

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572

The smallest squares containing k 572's :
35721 = 189²,
3572572441 = 59771²,
572354857275721 = 23923939².

572² = 327184, a zigzag square with different digits.

572 is the 3rd integer which is the sum of a square and a prime in 10 ways :
1² + 571, 3² + 563, 5² + 547, 7² + 523, 9² + 491, 15² + 347, 17² + 283, 19² + 211, 21² + 131, 23² + 43.

572² = 327184, 3 + 2 + 7 + 1 + 8 + 4 = 5²,
572² = 327184, 3 + 2 + 7 + 184 = 14².

Page of Squares : First Upload May 23, 2005 ; Last Revised August 9, 2006
by Yoshio Mimura, Kobe, Japan

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573

The smallest squares containing k 573's :
573049 = 757²,
20057357376 = 141624²,
573719573573136 = 23952444².

573² = 328329.

573² = 9³ + 21³ + 26³ + 67³ = 21³ + 32³ + 39³ + 61³.

573² = 25³ + 8⁵ + 6⁷.

573²± 2 are primes.

573328329 = 573².

3-by-3 magic squares consisting of different squares with constant 573²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(4, 292, 493, 332, 403, 236, 467, 284, 172),	(18, 129, 558, 207, 522, 114, 534, 198, 63),
(18, 303, 486, 369, 378, 222, 438, 306, 207),	(28, 299, 488, 328, 392, 259, 469, 292, 152),
(33, 126, 558, 234, 513, 102, 522, 222, 81),	(40, 248, 515, 277, 460, 200, 500, 235, 152),
(40, 355, 448, 380, 352, 245, 427, 280, 260),	(53, 128, 556, 172, 536, 107, 544, 157, 88),
(53, 332, 464, 368, 376, 227, 436, 277, 248),	(64, 203, 532, 392, 404, 107, 413, 352, 184),
(68, 224, 523, 376, 413, 128, 427, 328, 196),	(83, 196, 532, 308, 467, 124, 476, 268, 173),
(116, 277, 488, 352, 424, 157, 437, 268, 256)

573² = 328329, 32 + 8 + 32 + 9 = 9²,
573² = 328329, 32 + 83 + 29 = 12².

573² = 328329 appears in the decimal expression of e:
  e = 2.71828•••328329••• (from the 80208th digit).

Page of Squares : First Upload May 23, 2005 ; Last Revised December 29, 2013
by Yoshio Mimura, Kobe, Japan

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574

The smallest squares containing k 574's :
345744 = 588²,
8574574801 = 92599²,
557457457428369 = 23610537².

574 = (1² + 2² + 3² + ... + 20²) / (1² + 2²).

574² = 329476, a zigzag square with different digits.

Komachi equation: 574² = - 12² * 34² + 56² + 78² * 9².

574² + 575² + 576² + 577² + ... + 23927² = 2136891².

574² = 329476, 3 + 29 + 4 + 7 + 6 = 7²,
574² = 329476, 32 + 9 + 4 + 76 = 11².

574² = 329476 appears in the decimal expression of π:
  π = 3.14159•••329476••• (from the 17126th digit).

Page of Squares : First Upload May 23, 2005 ; Last Revised June 18, 2010
by Yoshio Mimura, Kobe, Japan

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575

The smallest squares containing k 575's :
1557504 = 1248²,
4575575449 = 67643²,
1257557557595364 = 35462058².

287² + 288² + 289² + 290² + ... + 575² = 7463².

575² = 5⁴ + 10⁴ + 20⁴ + 20⁴.

Komachi equation: 575² = 1² + 2² - 34² + 56² / 7² * 8² * 9².

3-by-3 magic squares consisting of different squares with constant 575²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(10, 75, 570, 213, 530, 66, 534, 210, 37),	(10, 75, 570, 354, 450, 53, 453, 350, 54),
(10, 150, 555, 282, 485, 126, 501, 270, 82),	(10, 150, 555, 402, 395, 114, 411, 390, 98),
(18, 170, 549, 325, 450, 150, 474, 315, 82),	(21, 222, 530, 390, 395, 150, 422, 354, 165),
(26, 270, 507, 357, 390, 226, 450, 325, 150),	(30, 165, 550, 341, 438, 150, 462, 334, 75),
(42, 219, 530, 306, 458, 165, 485, 270, 150),	(42, 330, 469, 405, 350, 210, 406, 315, 258),
(54, 235, 522, 315, 450, 170, 478, 270, 171),	(75, 150, 550, 210, 523, 114, 530, 186, 123),
(75, 298, 486, 350, 411, 198, 450, 270, 235)

575² = 330625, 3 + 30 + 6 + 25 = 8²,
575² = 330625, 3 + 30 + 62 + 5 = 10²,
575² = 330625, 33 + 0 + 6 + 25 = 8²,
575² = 330625, 33 + 0 + 62 + 5 = 10²,
575² = 330625, 330 + 6 + 25 = 19²,
575² = 330625, 330² + 6² + 25² = 331².

575² = 330625 appears in the decimal expression of e:
  e = 2.71828•••330625••• (from the 122184th digit).

Page of Squares : First Upload May 23, 2005 ; Last Revised June 18, 2010
by Yoshio Mimura, Kobe, Japan

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576

The square of 24.

The smallest squares containing k 576's :
576 = 24²,
144576576 = 12024²,
15765767595769 = 3970613².

The squares which begin with 576 and end in 576 are
57611520576 = 240024²,   576044568576 = 758976²,   576117432576 = 759024²,
576803794576 = 759476²,   576876706576 = 759524²,...

1 / 576 = 0.0017361111..., 17² + 3² + 6² + 11² + 11² = 576.

576² = 331776, a square with odd digits except the last digit 6.

576² = (2² - 1)(3² - 1)(7² - 1)(17² - 1) = (5² - 1)(7² - 1)(17² - 1).

576² = 48³ + 48³ + 48³.

Komachi equations:
576² = 12² - 3² * 4² + 56² / 7² * 8² * 9² = 12² / 3² - 4² + 56² / 7² * 8² * 9²
 = 12² / 3² / 4² * 56² / 7² * 8² * 9² = - 12² + 3² * 4² + 56² / 7² * 8² * 9²
 = - 12² / 3² + 4² + 56² / 7² * 8² * 9²,
576² = 1⁴ * 2⁴ / 3⁴ * 4⁴ * 56⁴ / 7⁴ / 8⁴ * 9⁴ = 1⁴ * 2⁴ / 3⁴ * 4⁴ / 56⁴ * 7⁴ * 8⁴ * 9⁴
 = 1⁴ / 2⁴ / 3⁴ / 4⁴ * 56⁴ / 7⁴ * 8⁴ * 9⁴ = 9⁴ * 8⁴ * 7⁴ / 6⁴ * 5⁴ * 4⁴ * 3⁴ / 210⁴
 = 98⁴ / 7⁴ * 6⁴ * 5⁴ * 4⁴ * 3⁴ / 210⁴.

Cubic polynomials :
(X + 576²)(X + 688²)(X + 903²) = X³ + 1273²X² + 901968²X + 357848064²,
(X + 576²)(X + 744²)(X + 1393²) = X³ + 1681²X² + 1378968²X + 596961792².

576² = 331776, 3 + 3 + 17 + 7 + 6 = 6².

Page of Squares : First Upload May 23, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

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577

The smallest squares containing k 577's :
5776 = 76²,
2857757764 = 53458²,
4577577359577616 = 67657796².

577² = 332929, a square consisting of just 3 kinds of digits.

Komachi equation: 577² = 1² - 2² + 34² + 56² / 7² * 8² * 9².

577² = 15³ + 43³ + 63³.

3-by-3 magic squares consisting of different squares with constant 577²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(8, 252, 519, 372, 399, 188, 441, 332, 168),	(15, 180, 548, 348, 440, 135, 460, 327, 120),
(36, 177, 548, 208, 516, 153, 537, 188, 96),	(57, 296, 492, 404, 372, 177, 408, 327, 244),
(89, 288, 492, 372, 348, 271, 432, 359, 132)

577² = 332929, 3 + 3 + 29 + 29 = 8²,
577² = 332929, 3 + 329 + 29 = 19².

Page of Squares : First Upload May 23, 2005 ; Last Revised June 18, 2010
by Yoshio Mimura, Kobe, Japan

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578

The smallest squares containing k 578's :
5257849 = 2293²,
25786578724 = 160582²,
9557865784578244 = 97764338².

578^k + 2074^k + 2890^k + 4862^k are squares for k = 1,2,3 (102², 6052², 384948²).
578^k + 2754^k + 6358^k + 8806^k are squares for k = 1,2,3 (136², 11220², 980288²).

25² + 26² + 27² + 28² + ... + 578² = 8033².

(1² + 2² + ... + 24²)(25² + 26² + ... + 578²) = 562310².

578² = 17⁴ + 17⁴ + 17⁴ + 17⁴.

578² = 334084, 3 + 34 + 0 + 8 + 4 = 7²,
578² = 334084, 33 + 4 + 0 + 8 + 4 = 7²,
578² = 334084, 3 + 34 + 0 + 84 = 11²,
578² = 334084, 33 + 4 + 0 + 84 = 11².

Page of Squares : First Upload May 23, 2005 ; Last Revised March 15, 2011
by Yoshio Mimura, Kobe, Japan

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579

The smallest squares containing k 579's :
45796 = 214²,
5792579881 = 76109²,
4894579579857984 = 69961272².

4825^k + 106922^k + 110396^k + 113098^k are squares for k = 1,2,3 (579², 190877², 63360549²).

Komachi equation: 579² = 9³ + 8³ * 7³ + 6³ + 54³ + 3³ * 2³ + 10³.

3-by-3 magic squares consisting of different squares with constant 579²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(1, 122, 566, 214, 526, 113, 538, 209, 46),	(1, 242, 526, 302, 449, 206, 494, 274, 127),
(17, 74, 574, 266, 511, 58, 514, 262, 49),	(31, 118, 566, 358, 449, 74, 454, 346, 97),
(34, 319, 482, 346, 398, 239, 463, 274, 214),	(36, 288, 501, 339, 396, 252, 468, 309, 144),
(38, 146, 559, 239, 514, 118, 526, 223, 94),	(45, 204, 540, 396, 405, 120, 420, 360, 171),
(46, 190, 545, 335, 454, 130, 470, 305, 146),	(46, 305, 490, 385, 350, 254, 430, 346, 175),
(49, 254, 518, 298, 434, 241, 494, 287, 94),	(98, 214, 529, 254, 497, 154, 511, 206, 178),
(98, 374, 431, 401, 266, 322, 406, 353, 214)

579² = 335241, 3² + 3² + 5² + 2² + 4² + 1² = 8²,
579² = 335241, 3 + 3 + 5 + 24 + 1 = 6²,
579² = 335241, 3 + 35 + 2 + 41 = 9²,
579² = 335241, 33 + 5 + 2 + 41 = 9²,
579² = 335241, 335 + 241 = 24².

Page of Squares : First Upload May 23, 2005 ; Last Revised March 15, 2011
by Yoshio Mimura, Kobe, Japan

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