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530

The smallest squares containing k 530's :
253009 = 503²,
1530530884 = 39122²,
1453053053058489 = 38118933².

530 = (1² + 2² + 3² + ... + 52²) / (1² + 2² + 3² + ... + 6²).

190^k + 314^k + 410^k + 530^k are squares for k = 1,2,3 (38², 764², 15988²).

The square root of 530 is 23.0217288664426764419..., and
23² = 0² + 2² + 1² + 7² + 2² + 8² + 8² + 6² + 6² + 4² + 4² + 2² + 6² + 7² + 6² + 4² + 4² + 1² + 9².

The square root of 530 is 23.02172886644..., and
23² = 0² + 2² + 17² + 2² + 8² + 8² + 6² + 6² + 4² + 4².

Page of Squares : First Upload April 25, 2005 ; Last Revised March 15, 2011
by Yoshio Mimura, Kobe, Japan

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531

The smallest squares containing k 531's :
125316 = 354²,
135316565316 = 367854²,
531655315753104 = 23057652².

531² = 38² + 39² + 40² + 41² + ... + 96².

531² is the 6th square which is the sum of 10 sixth powers.

531² = 32³ + 40³ + 57³.

Cubic Polynomial :
(X + 531²)(X + 832²)(X + 3024²) = X³ + 3181²X² + 3017232²X + 1335979008².

45489^k + 48498^k + 62304^k + 125670^k are squares for k = 1,2,3 (531², 155229², 49343175²).

Komachi equation: 531² = - 1³ * 2³ - 3³ - 45³ + 6³ - 7³ + 8³ * 9³.

3-by-3 magic squares consisting of different squares with constant 531²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(12, 291, 444, 336, 348, 219, 411, 276, 192),	(17, 194, 494, 254, 431, 178, 466, 242, 79),
(24, 123, 516, 276, 444, 93, 453, 264, 84),	(26, 142, 511, 287, 434, 106, 446, 271, 98),
(31, 158, 506, 326, 394, 143, 418, 319, 74),	(38, 226, 479, 346, 353, 194, 401, 326, 122),
(46, 319, 422, 353, 334, 214, 394, 262, 241),	(49, 106, 518, 166, 497, 86, 502, 154, 79),
(49, 226, 478, 302, 406, 161, 434, 257, 166),	(60, 219, 480, 285, 420, 156, 444, 240, 165),
(65, 194, 490, 350, 385, 106, 394, 310, 175),	(82, 214, 479, 241, 446, 158, 466, 193, 166),
(86, 298, 431, 367, 346, 166, 374, 271, 262),	(98, 266, 449, 334, 383, 154, 401, 254, 238)

531² = 281961, 2 + 8 + 19 + 6 + 1 = 6²,
531² = 281961, 2 + 8 + 1 + 9 + 61 = 9²,
531² = 281961, 28 + 196 + 1 = 15².

531² = 281961 appears in the decimal expression of π:
  π = 3.14159•••281961••• (from the 49109th digit).

Page of Squares : First Upload April 25, 2005 ; Last Revised March 15, 2011
by Yoshio Mimura, Kobe, Japan

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532

The smallest squares containing k 532's :
5329 = 73²,
95325327504 = 308748²,
53240532085321 = 7296611².

532² = (1² + 3)(2² + 3)(4² + 3)(23² + 3) = (1² + 3)(4² + 3)(61² + 3) = (4² + 3)(5² + 3)(23² + 3).

Komachi equation: 532² = 12² + 34² * 5² - 6² + 7² * 8² * 9².

532² = 24³ + 42³ + 58³.

532² + 533² + 534² + 535² + ... + 6315² = 289682²,
532² + 533² + 534² + 535² + ... + 7564² = 379782².

532² = 283024, 2 + 8 + 30 + 24 = 8²,
532² = 283024, 28² + 3² + 0² + 24² = 37²,
532² = 283024, 28 + 30 + 2 + 4 = 8²,
532² = 283024, 283 + 0 + 2 + 4 = 17².

Page of Squares : First Upload April 25, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

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533

The smallest squares containing k 533's :
53361 = 231²,
43533153316 = 208646²,
753374533533561 = 27447669².

60^k + 241^k + 282^k + 378^k are squares for k = 1,2,3 (31², 533², 9521²).

Komachi equation: 533² = - 1 * 2 + 3 + 456 * 7 * 89.

533² + 534² + 535² + 536² + ... + 1940² = 48840²,
533² + 534² + 535² + 536² + ... + 21157² = 1776775².

3-by-3 magic squares consisting of different squares with constant 533²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(0, 117, 520, 205, 480, 108, 492, 200, 45),	(11, 72, 528, 312, 429, 52, 432, 308, 51),
(12, 133, 516, 164, 492, 123, 507, 156, 52),	(12, 173, 504, 333, 396, 128, 416, 312, 117),
(24, 213, 488, 348, 376, 147, 403, 312, 156),	(52, 156, 507, 276, 443, 108, 453, 252, 124),
(84, 187, 492, 268, 444, 123, 453, 228, 164),	(117, 264, 448, 312, 403, 156, 416, 228, 243)

533² = 284089, 28 + 4 + 0 + 8 + 9 = 7²,
533² = 284089, 28 + 4 + 0 + 89 = 11².

Page of Squares : First Upload April 25, 2005 ; Last Revised March 15, 2011
by Yoshio Mimura, Kobe, Japan

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534

The smallest squares containing k 534's :
35344 = 188²,
5345344 = 2312²,
53453455348969 = 7311187².

534²± 5 are primes.

534 = (1² + 2² + 3² + ... + 44²) / (1² + 2² + 3² + 4² + 5²).

534² = (2² + 2)(218² + 2).

29370^k + 42186^k + 65682^k + 147918^k are squares for k = 1,2,3 (534², 169812², 60167916²).

Komachi equations:
534² = - 98² - 7² - 6² + 543² - 2² * 1² = - 98² - 7² - 6² + 543² - 2² / 1².

The 4-by-4 magic square consisting of different squares with constant 534:

534² = 285156, 2 + 8 + 5 + 15 + 6 = 6²,
534² = 285156, 2 + 85 + 1 + 56 = 12²,
534² = 285156, 28 + 5156 = 72²,
534² = 285156, 285 + 156 = 21².

Page of Squares : First Upload April 25, 2005 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

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535

The smallest squares containing k 535's :
535824 = 732²,
45351535681 = 212959²,
535575352535001 = 23142501².

1 / 535 = 0.0018691588..., 1² + 8² + 6² + 9² + 15² + 8² + 8² = 535.

Komachi equation: 535² = 98³ - 76³ - 5³ * 4³ * 3³ + 2³ + 1³.

3-by-3 magic squares consisting of different squares with constant 535²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(14, 102, 525, 195, 490, 90, 498, 189, 50),	(30, 162, 509, 365, 366, 138, 390, 355, 90),
(30, 310, 435, 365, 330, 210, 390, 285, 230),	(45, 126, 518, 210, 482, 99, 490, 195, 90),
(45, 210, 490, 266, 435, 162, 462, 230, 141),	(50, 243, 474, 285, 390, 230, 450, 274, 93),
(50, 285, 450, 366, 310, 237, 387, 330, 166),	(78, 285, 446, 310, 390, 195, 429, 230, 222),
(90, 195, 490, 355, 366, 162, 390, 338, 141),	(90, 275, 450, 355, 306, 258, 390, 342, 131)

The square root of 535 is 23.130067012440755140165..., and 23² = 1² + 3² + 0² + 0² + 6² + 7² + 0² + 1² + 2² + 4² + 4² + 0² + 7² + 5² + 5² + 1² + 4² + 0² + 16² + 5²,
The square root of 535 is 23.130067012440755..., and 23² = 13² + 0² + 0² + 6² + 7² + 0² + 12² + 4² + 4² + 0² + 7² + 5² + 5².

Page of Squares : First Upload April 25, 2005 ; Last Revised June 15, 2010
by Yoshio Mimura, Kobe, Japan

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536

The smallest squares containing k 536's :
45369 = 213²,
1536953616 = 39204²,
5369675369536 = 2317256².

The squares which begin with 536 and end in 536 are
5364683536 = 73244²,   5366441536 = 73256²,   536181275536 = 732244²,
536198849536 = 732256²,   536913769536 = 732744²,...

The square root of 536 is 23.15167380558045094711162..., and 23² = 1² + 5² + 1² + 6² + 7² + 3² + 8² + 0² + 5² + 5² + 8² + 0² + 4² + 5² + 0² + 9² + 4² + 7² + 1² + 1² + 1² + 6² + 2².

536² = 287296, 28 + 72 + 96 = 14².

536² = 287296 appears in the decimal expression of e:
  e = 2.71828•••287296••• (from the 138442nd digit).

Page of Squares : First Upload April 25, 2005 ; Last Revised August 3, 2006
by Yoshio Mimura, Kobe, Japan

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537

The smallest squares containing k 537's :
15376 = 124²,
22537215376 = 150124²,
2320537537553764 = 48171958².

3-by-3 magic squares consisting of different squares with constant 537²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(3, 126, 522, 342, 402, 99, 414, 333, 78),	(4, 73, 532, 367, 388, 56, 392, 364, 47),
(4, 188, 503, 272, 433, 164, 463, 256, 92),	(7, 256, 472, 296, 392, 217, 448, 263, 136),
(17, 232, 484, 344, 367, 188, 412, 316, 137),	(18, 246, 477, 378, 333, 186, 381, 342, 162),
(32, 113, 524, 316, 428, 73, 433, 304, 92),	(49, 292, 448, 368, 308, 241, 388, 329, 172),
(55, 188, 500, 340, 400, 113, 412, 305, 160),	(76, 143, 512, 172, 496, 113, 503, 148, 116),
(80, 260, 463, 335, 388, 160, 412, 265, 220),	(92, 239, 472, 311, 412, 148, 428, 248, 209)

537² = 288369, 2 + 8 + 8 + 3 + 6 + 9 = 6²,
537² = 288369, 28 + 8 + 36 + 9 = 9²,
537² = 288369, 2 + 883 + 6 + 9 = 30².

Page of Squares : First Upload December 26, 2005 ; Last Revised May 21, 2009
by Yoshio Mimura, Kobe, Japan

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538

The smallest squares containing k 538's :
53824 = 232²,
39538538649 = 198843²,
538075385388304 = 23196452².

538² = 289444, ending in 444.

126^k + 538^k + 1050^k + 1422^k are squares for k = 1,2,3 (56², 1852², 64736²).
210^k + 241^k + 538^k + 692^k are squares for k = 1,2,3 (41², 933², 22591²).

538² = 289444, 289 = 17², and 4 = 2².

538⁷ = 13045956607324926592, and 1² + 3² + 0² + 4² + 5² + 9² + 5² + 6² + 6² + 0² + 7² + 3² + 2² + 4² + 9² + 2² + 6² + 5² + 9² + 2² = 538.

538² = 289444, 28 + 9 + 4 + 4 + 4 = 7².

Page of Squares : First Upload April 25, 2005 ; Last Revised March 15, 2011
by Yoshio Mimura, Kobe, Japan

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539

The smallest squares containing k 539's :
835396 = 914²,
539539984 = 23228²,
353925394539025 = 18812905².

(353 / 539)² = 0.428915637... (Komachic).

539² = 290521, 2 * 9 + 0 + 521 = 539.

539² = 132² + 264² + 451² = 154² + 462² + 231².

Cubic Polynomial :
(X + 348²)(X + 539²)(X + 1584²) = X³ + 1709²X² + 1033428²X + 297114048².

3-by-3 magic squares consisting of different squares with constant 539²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(6, 73, 534, 249, 474, 62, 478, 246, 39),	(6, 233, 486, 262, 426, 201, 471, 234, 118),
(6, 334, 423, 359, 318, 246, 402, 279, 226),	(9, 134, 522, 314, 423, 114, 438, 306, 71),
(9, 162, 514, 206, 474, 153, 498, 199, 54),	(18, 71, 534, 226, 486, 57, 489, 222, 46),
(30, 186, 505, 375, 370, 114, 386, 345, 150),	(46, 201, 498, 327, 386, 186, 426, 318, 89),
(46, 258, 471, 366, 361, 162, 393, 306, 206),	(87, 334, 414, 366, 342, 199, 386, 249, 282)

539² + 540² + 541² + 542² + ... + 585² = 3854².

539² = 290521, 2 + 9 + 0 + 52 + 1 = 8²,
539² = 290521, 2 + 90 + 5 + 2 + 1 = 10².

Page of Squares : First Upload April 25, 2005 ; Last Revised August 17, 2013
by Yoshio Mimura, Kobe, Japan

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