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510

The smallest squares containing k 510's :
51076 = 226²,
51051025 = 7145²,
351095105451024 = 18737532².

510² is the 5th square which is the sum of 10 sixth powers.

510^k + 690^k + 1330^k + 9570^k are squares for k = 1,2,3 (110², 9700², 937700²).

Komachi equation: 510² = - 1³ + 2³ - 34³ - 5³ + 67³ - 8³ - 9³.

510² + 511² + 512² + 513² + ... + 678² = 7748².

(1 + 2 + ... + 5)(6 + 7 + ... + 11)(12 + 13 + ... + 28) = 510².

(1³ + 2³ + ... + 144³)(145³ + 146² + ... + 149³)(150³ + 151² + ... + 510³) = 5402314764000².

Page of Squares : First Upload April 18, 2005 ; Last Revised March 15, 2011
by Yoshio Mimura, Kobe, Japan

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511

The smallest squares containing k 511's :
511225 = 715²,
51151104 = 7152²,
1054511251151121 = 32473239².

511² = 261121, a square consisting of just 3 kinds of digits.

511² = (45 + 46 + 47 + 48 + 49 + 50 + 51)² + (52 + 53 + 54 + 55 + 56 + 57 + 58)².

511² = (1² + 2² + ... + 69²) + (1² + 2² + ... + 76²).

3-by-3 magic squares consisting of different squares with constant 511²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(11, 114, 498, 282, 414, 101, 426, 277, 54),	(34, 234, 453, 261, 398, 186, 438, 219, 146),
(43, 186, 474, 354, 331, 162, 366, 342, 101),	(43, 186, 474, 354, 362, 69, 366, 309, 178),
(66, 246, 443, 282, 389, 174, 421, 222, 186),	(69, 142, 486, 198, 459, 106, 466, 174, 117),
(69, 178, 474, 254, 426, 123, 438, 219, 146),	(69, 318, 394, 354, 254, 267, 362, 309, 186),
(75, 186, 470, 290, 405, 114, 414, 250, 165),	(102, 286, 411, 309, 366, 178, 394, 213, 246)

511² = 261121, 2³ + 6³ + 11³ + 21³ = 104²,
511² = 261121, 2 + 6 + 112 + 1 = 11²,
511² = 261121, 26 + 1 + 1 + 21 = 7².

Page of Squares : First Upload April 18, 2005 ; Last Revised May 21, 2009
by Yoshio Mimura, Kobe, Japan

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512

The smallest squares containing k 512's :
15129 = 123²,
51251281 = 7159²,
933512512512969 = 30553437².

512² = (1² + 7)(181² + 7).

Cubic Polynomial :
(X + 512²)(X + 837²)(X + 1104²) = X³ + 1477²X² + 1164912²X + 473112576².

Komachi equations:
512² = 12³ * 3³ / 4³ + 56³ / 7³ * 8³ - 9³ = 12³ * 3³ / 4³ * 56³ / 7³ * 8³ / 9³
 = - 12³ * 3³ / 4³ + 56³ / 7³ * 8³ + 9³,
512² = 9⁶ + 8⁶ - 7⁶ / 6⁶ * 54⁶ * 3⁶ / 21⁶ = 9⁶ * 8⁶ * 7⁶ * 6⁶ / 54⁶ * 3⁶ / 21⁶
 = 9⁶ * 8⁶ / 7⁶ * 6⁶ / 54⁶ / 3⁶ * 21⁶ = - 9⁶ + 8⁶ + 7⁶ / 6⁶ * 54⁶ * 3⁶ / 21⁶.

175² + 176² + 177² + 178² + ... + 512² = 6565².

512² + 513² + 514² + 515² + ... + 561² = 3795².

512² = 262144, 2³ + 6³ + 2³ + 1³ + 4³ + 4³ = 19²,
512² = 262144, 2² + 6² + 2² + 14² + 4² = 16²,
512² = 262144, 2⁴ + 6⁴ + 2⁴ + 14⁴ + 4⁴ = 200².

512² = 4⁸ + 4⁸ + 4⁸ + 4⁸ = 16⁴ + 16⁴ + 16⁴ + 16⁴.

512² = 262144 appears in the decimal expression of π:
  π = 3.14159•••262144••• (from the 76875th digit).

Page of Squares : First Upload April 18, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

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513

The smallest squares containing k 513's :
151321 = 389²,
151351343521 = 389039²,
1513513513017801 = 38903901².

513² is the 2nd square which is the sum of 4 ninth powers : 1⁹ + 2⁹ + 2⁹ + 4⁹.

513² is the 5th square which is the sum of 9 eighth powers.

513²± 2 are primes.

(X + 76²)(X + 192²)(X + 513²) = X³ + 553²X² + 106932²X + 7485696².

A + B, A + C, A + D, B + C, B + D, C + D are squares for A = 513, B = 1008, C = 1696, D = 6048.

513² is the sum of consecutive primes 3, 5, 7, 11,..., 1949.

513² = 4⁰ + 4⁵ + 4⁹ = 64³ + 4⁵ + 1⁷.

(1)(2 + 3 + ... + 55)(56 + 57 + 58) = 513².

513^k + 1292^k + 3382^k + 3838^k are squares for k = 1,2,3 (95², 5301², 312265²).

3-by-3 magic squares consisting of different squares with constant 513²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(6, 123, 498, 267, 426, 102, 438, 258, 69),	(8, 71, 508, 332, 388, 49, 391, 328, 52),
(20, 160, 487, 305, 388, 140, 412, 295, 80),	(20, 263, 440, 335, 340, 188, 388, 280, 185),
(21, 222, 462, 318, 357, 186, 402, 294, 123),	(28, 199, 472, 241, 412, 188, 452, 232, 71),
(32, 127, 496, 287, 416, 88, 424, 272, 97),	(32, 167, 484, 196, 452, 143, 473, 176, 92),
(32, 281, 428, 308, 332, 241, 409, 272, 148),	(36, 207, 468, 288, 396, 153, 423, 252, 144),
(56, 193, 472, 353, 328, 176, 368, 344, 97),	(64, 148, 487, 188, 463, 116, 473, 164, 112),
(78, 309, 402, 354, 258, 267, 363, 318, 174),	(104, 263, 428, 332, 364, 143, 377, 248, 244),
(113, 284, 412, 316, 368, 167, 388, 217, 256)

513² = 263169, 2 + 6 + 3 + 16 + 9 = 6²,
513² = 263169, 2 + 6 + 3 + 1 + 69 = 9²,
513² = 263169, 2 + 63 + 1 + 6 + 9 = 9²,
513² = 263169, 2³ + 63³ + 1³ + 6³ + 9³ = 501².

Page of Squares : First Upload April 18, 2005 ; Last Revised December 29, 2013
by Yoshio Mimura, Kobe, Japan

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514

The smallest squares containing k 514's :
514089 = 717²,
51445951489 = 226817²,
51465142514241 = 7173921².

514² is the second square which is the sum of 9 ninth powers.

442² + 443² + 444² + 445² + ... + 514² = 4088².

Komachi equation: 514² = 9³ - 8³ + 7³ + 65³ - 4³ * 3³ - 21³.

514² = 264196, 2 + 6 + 41 + 9 + 6 = 8²,
514² = 264196, 26² + 4² + 19² + 6² = 33²,
514² = 264196, 264 + 1 + 96 = 19²,
514² = 264196, 264 + 19 + 6 = 17².

Page of Squares : First Upload April 18, 2005 ; Last Revised June 11, 2010
by Yoshio Mimura, Kobe, Japan

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515

The smallest squares containing k 515's :
51529 = 227²,
51513565156 = 226966²,
38515515851569 = 6206087².

515² = 265225, a square consisting of just 3 kinds of digits.

515² = 265225, 2 * 65 * 2 * 2 - 5 = 26 * 5 * 2 * 2 - 5 = 515.

3-by-3 magic squares consisting of different squares with constant 515²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(6, 142, 495, 342, 369, 110, 385, 330, 90),	(15, 70, 510, 114, 498, 65, 502, 111, 30),
(15, 70, 510, 250, 447, 54, 450, 246, 47),	(15, 210, 470, 250, 414, 177, 450, 223, 114),
(15, 210, 470, 362, 330, 159, 366, 335, 138),	(30, 254, 447, 335, 330, 210, 390, 303, 146),
(30, 335, 390, 358, 294, 225, 369, 258, 250),	(34, 138, 495, 225, 450, 110, 462, 209, 90),
(65, 282, 426, 330, 351, 182, 390, 250, 225),	(70, 210, 465, 318, 385, 126, 399, 270, 182)

515² = 265225, 26 + 5 + 225 = 16².

Page of Squares : First Upload April 18, 2005 ; Last Revised May 21, 2009
by Yoshio Mimura, Kobe, Japan

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516

The smallest squares containing k 516's :
60516 = 246²,
516516529 = 22727²,
251658151605169 = 15863737².

The squares which begin with 516 and end in 516 are
51640744516 = 227246²,   51644380516 = 227254²,   516595812516 = 718746²,
516607312516 = 718754²,   5160829888516 = 2271746²,...

516² = 266256, a square consisting of just 3 kinds of digits.

516² + 517² + 518² + 519² + ... + 1573² = 35397².

516² = 266256, 2 + 6 + 62 + 5 + 6 = 9²,
516² = 266256, 2 + 66 + 2 + 5 + 6 = 9²,
516² = 266256, 2 + 66 + 256 = 18²,
516² = 266256, 26 + 62 + 56 = 12²,
516² = 266256, 266 + 2 + 56 = 18².

Page of Squares : First Upload April 18, 2005 ; Last Revised August 1, 2006
by Yoshio Mimura, Kobe, Japan

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517

The smallest squares containing k 517's :
1517824 = 1232²,
517517001 = 22749²,
517706517517225 = 22753165².

517 = (1² + 2² + 3² + ... + 164²) / (1² + 2² + 3² + ... + 20²).

517² = 267289, 267 * 2 - 8 - 9 = 517.

517² is the 2nd square which is the sum of 8 sixth powers :
  2⁶ + 2⁶ + 2⁶ + 2⁶ + 2⁶ + 3⁶ + 4⁶ + 8⁶.

517^k + 893^k + 3525^k + 3901^k are squares for k = 1,2,3 (94², 5358², 322514²).
517^k + 869^k + 3245^k + 7469^k are squares for k = 1,2,3 (110², 8206², 672034²).
98^k + 212^k + 305^k + 346^k are squares for k = 1,2,3 (31², 517², 8959²).

Komachi equation: 517² = 1³ * 2³ + 3³ + 45³ - 6³ + 7³ * 8³ + 9³.

3-by-3 magic squares consisting of different squares with constant 517²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(3, 228, 464, 284, 387, 192, 432, 256, 123),	(24, 157, 492, 283, 408, 144, 432, 276, 67),
(27, 112, 504, 336, 387, 68, 392, 324, 93),	(32, 312, 411, 348, 291, 248, 381, 292, 192),
(40, 180, 483, 315, 392, 120, 408, 285, 140)

517² = 267289, 2 + 6 + 72 + 89 = 13².

Page of Squares : First Upload April 18, 2005 ; Last Revised March 15, 2011
by Yoshio Mimura, Kobe, Japan

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518

The smallest squares containing k 518's :
5184 = 72²,
518518441 = 22771²,
51851851877329 = 7200823².

(1³ + 2³ + ... + 399³)(400³ + 401³ + ... + 518³) = 8632045800².

217^k + 218^k + 272^k + 518^k are squares for k = 1,2,3 (35², 661², 13405²).

The 4-by-4 magic square consisting of different squares with constant 518:

518² = 268324, 2 + 6 + 8 + 3 + 2 + 4 = 5².

Page of Squares : First Upload April 18, 2005 ; Last Revised March 15, 2011
by Yoshio Mimura, Kobe, Japan

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519

The smallest squares containing k 519's :
51984 = 228²,
51925192641 = 227871²,
6519235199051961 = 80741781².

Komachi equation: 519² = 12⁶ / 3⁶ * 4⁶ / 56⁶ * 7⁶ - 8⁶ + 9⁶.

3-by-3 magic squares consisting of different squares with constant 519²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(14, 131, 502, 334, 382, 109, 397, 326, 74),	(14, 142, 499, 226, 451, 122, 467, 214, 74),
(14, 238, 461, 314, 371, 182, 413, 274, 154),	(19, 170, 490, 290, 410, 131, 430, 269, 110),
(26, 67, 514, 98, 506, 61, 509, 94, 38), (26,	227, 466, 362, 326, 179, 371, 334, 142),
(26, 301, 422, 326, 338, 221, 403, 254, 206),	(38, 109, 506, 254, 446, 77, 451, 242, 86),
(58, 221, 466, 259, 418, 166, 446, 214, 157),	(72, 264, 441, 324, 369, 168, 399, 252, 216),
(77, 274, 434, 314, 322, 259, 406, 301, 118),	(94, 179, 478, 206, 458, 131, 467, 166, 154),
(94, 250, 445, 275, 406, 170, 430, 205, 206)

519² = 269361, 2 + 6 + 9 + 3 + 61 = 9²,
519² = 269361, 2 + 69 + 3 + 6 + 1 = 9².

519² = 269361 appears in the decimal expression of π:
  π = 3.14159•••269361••• (from the 65818th digit).

Page of Squares : First Upload April 18, 2005 ; Last Revised June 11, 2010
by Yoshio Mimura, Kobe, Japan

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