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470

The smallest squares containing k 470's :
47089 = 217²,
22334704704 = 149448²,
47047047037225 = 6859085².

470²± 3 are primes.

1645^k + 54285^k + 56165^k + 108805^k are squares for k = 1,2,3 (470², 133950², 40314250²).
5405^k + 35485^k + 74965^k + 105045^k are squares for k = 1,2,3 (470², 133950², 40314250²).

Komachi equations:
470² = 987² / 6² * 5² * 4² * 3² / 21².

470² = 220900 appears in the decimal expression of e:
  e = 2.71828•••220900••• (from the 124593rd digit).

Page of Squares : First Upload March 22, 2005 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

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471

The smallest squares containing k 471's :
471969 = 687²,
47147173956 = 217134²,
1347147132471009 = 36703503².

1 / 471 = 0.00212314, 21² + 2² + 3² + 1² + 4² = 471.

1 / 471 = 0.00212314225053078556263269..., the sum of the squares of its digits is 471.

(286 / 471)² = 0.368714529... (Komachic).

Komachi equation: 471² = - 12³ - 34³ + 56³ / 7³ * 8³ + 9³.

3-by-3 magic squares consisting of different squares with constant 471²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(10, 125, 454, 154, 430, 115, 445, 146, 50),	(14, 67, 466, 109, 454, 62, 458, 106, 29),
(14, 226, 413, 301, 322, 166, 362, 259, 154),	(19, 182, 434, 238, 371, 166, 406, 226, 77),
(22, 94, 461, 214, 413, 74, 419, 206, 62),	(22, 134, 451, 269, 374, 98, 386, 253, 94),
(29, 122, 454, 274, 374, 83, 382, 259, 94),	(29, 190, 430, 230, 370, 179, 410, 221, 70),
(74, 202, 419, 227, 386, 146, 406, 179, 158),	(106, 298, 349, 323, 206, 274, 326, 301, 158),
(109, 286, 358, 322, 214, 269, 326, 307, 146)

471² = 221841, 2 + 21 + 8 + 4 + 1 = 6²,
471² = 221841, 2 + 218 + 4 + 1 = 15²,
471² = 221841, 22 + 1 + 8 + 4 + 1 = 6²,
471² = 221841, 22 + 18 + 41 = 9².

Page of Squares : First Upload March 22, 2005 ; Last Revised June 8, 2010
by Yoshio Mimura, Kobe, Japan

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472

The smallest squares containing k 472's :
174724 = 418²,
5472004729 = 73973²,
14724725472961 = 3837281².

472² = 8³ + 36³ + 56³.

Komachi equation: 472² = 9³ + 8³ * 7³ - 6³ + 54³ / 3³ * 2³ - 1³.

472² + 473² + 474² + 475² + ... + 24559² = 2222118².

472² = 222784, 2 + 2 + 2 + 7 + 8 + 4 = 5².

Page of Squares : First Upload March 22, 2005 ; Last Revised June 8, 2010
by Yoshio Mimura, Kobe, Japan

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473

The smallest squares containing k 473's :
414736 = 644²,
244734736 = 15644²,
126914734734736 = 11265644².

473² = (4 + 5)² + (6 + 7)² + (8 + 9)² + ... + (68 + 69)².

1 / 473 = 0.002114164, 2² + 1² + 14² + 16² + 4² = 473.

473² = 223729, 22 * 3 * 7 + 2 + 9 = 473.

3-by-3 magic squares consisting of different squares with constant 473²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(9, 68, 468, 292, 369, 48, 372, 288, 49),	(12, 96, 463, 183, 428, 84, 436, 177, 48),
(12, 201, 428, 239, 372, 168, 408, 212, 111),	(15, 252, 400, 320, 300, 177, 348, 265, 180),
(57, 184, 432, 328, 327, 96, 336, 288, 167),	(72, 303, 356, 324, 292, 183, 337, 216, 252),
(76, 177, 432, 252, 384, 113, 393, 212, 156),	(104, 273, 372, 303, 328, 156, 348, 204, 247)

473² + 474² + 475² + 476² + ... + 42553² = 5068063².

473² = 223729, 2 + 2 + 3 + 7 + 2 + 9 = 5².

Page of Squares : First Upload March 22, 2005 ; Last Revised April 27, 2009
by Yoshio Mimura, Kobe, Japan

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474

The smallest squares containing k 474's :
147456 = 384²,
47474744769 = 217887²,
474474474529225 = 21782435².

474² = 224676, 2 / 2 + 467 + 6 = 22 - 4 + 6 * 76 = 474.

474² + 475² + 476² + ... + 855² = 856² + 857² + 858² + ... + 1046².

30^k + 57^k + 168^k + 474^k are squares for k = 1,2,3 (27², 507², 10557²).
30178^k + 44714^k + 51034^k + 98750^k are squares for k = 1,2,3 (474², 123556², 34824780²).

(1³ + 2³ + ... + 87³)(88³ + 89³ + ... + 110³)(111³ + 112³ + ... + 474³) = 2046426061680².

474² = 224676, 2 + 2 + 4 + 67 + 6 = 9²,
474² = 224676, 2² + 2² + 46² + 7² + 6² = 47²,
474² = 224676, 2 + 246 + 76 = 18²,
474² = 224676, 22 + 46 + 7 + 6 = 9²,
474² = 224676, 22 + 46 + 76 = 12²,
474² = 224676, 224 + 676 = 30².

Page of Squares : First Upload March 22, 2005 ; Last Revised September 6, 2011
by Yoshio Mimura, Kobe, Japan

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475

The smallest squares containing k 475's :
47524 = 218²,
14754475024 = 121468²,
47547575475625 = 6895475².

475² = 225625, a square consisting of only 3 kinds of digits.

475² = 225625, 225 = 15², 625 = 25².

Komachi Fractions : 243 / 6091875 = (3 / 475)²,432 / 6091875 = (4 / 475)².

475, 476 and 477 are three consecutive integers having square factors (the 7th case).

3-by-3 magic squares consisting of different squares with constant 475²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(18, 201, 430, 326, 318, 135, 345, 290, 150),	(25, 150, 450, 270, 366, 137, 390, 263, 66),
(25, 150, 450, 270, 375, 110, 390, 250, 105),	(30, 214, 423, 290, 327, 186, 375, 270, 110),
(39, 198, 430, 250, 375, 150, 402, 214, 135),	(70, 135, 450, 162, 434, 105, 441, 138, 110),
(74, 282, 375, 318, 249, 250, 345, 290, 150),	(105, 250, 390, 294, 345, 142, 358, 210, 231)

475² + 476² + 477² + 478² + ... + 1361² = 28384²,
475² + 476² + 477² + 478² + ... + 148699² = 33105765².

475² = 225625, 225 + 6 + 25 = 16².

Page of Squares : First Upload March 22, 2005 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

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476

The smallest squares containing k 476's :
4761 = 69²,
934769476 = 30574²,
23874764764761 = 4886181².

The squares which begin with 476 and end in 476 are
476202125476 = 690074²,   476688061476 = 690426²,   476892449476 = 690574²,
4760801069476 = 2181926²,   4761446941476 = 2182074²,...

The square root of 476 is 21.8174242292714288...,
where 21² = 8² + 1² + 7² + 4² + 2² + 4² + 2² + 2² + 9² + 2² + 7² + 1² + 4² + 2² + 8² + 8².

Komachi equation: 476² = 98² * 765² * 4² / 3² / 210².

476² = 226576, 22 + 65 + 7 + 6 = 10²,
476² = 226576, 226 + 57 + 6 = 17².

476² = 226576 appears in the decimal expression of e:
  e = 2.71828•••226576••• (from the 85773rd digit).

Page of Squares : First Upload March 22, 2005 ; Last Revised June 8, 2010
by Yoshio Mimura, Kobe, Japan

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477

The smallest squares containing k 477's :
477481 = 691²,
477247716 = 21846²,
4774771580477284 = 69099722².

33^k + 205^k + 441^k + 477^k are squares for k = 1,2,3 (34², 682², 14246²).
18762^k + 48336^k + 57081^k + 103350^k are squares for k = 1,2,3 (477², 128949², 37542285²).

3-by-3 magic squares consisting of different squares with constant 477²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(11, 68, 472, 152, 448, 61, 452, 149, 32),	(16, 67, 472, 212, 424, 53, 427, 208, 44),
(16, 163, 448, 277, 368, 124, 388, 256, 107),	(28, 107, 464, 296, 368, 67, 373, 284, 88),
(32, 152, 451, 332, 331, 88, 341, 308, 128),	(42, 258, 399, 303, 294, 222, 366, 273, 138),
(53, 212, 424, 248, 376, 157, 404, 203, 152),	(56, 292, 373, 317, 256, 248, 352, 277, 164),
(66, 177, 438, 222, 402, 129, 417, 186, 138),	(100, 248, 395, 280, 355, 152, 373, 200, 220)

477² + 478² + 479² + 480² + ... + 3259² = 107272²,
477² + 478² + 479² + 480² + ... + 6454² = 299327².

(1² + 2² + ... + 15²)(16² + 17² + ... + 29²)(30² + 31² + ... + 477²) = 18141200².

477² = 227529, 2275 + 29 = 48².

Page of Squares : First Upload March 22, 2005 ; Last Revised March 8, 2011
by Yoshio Mimura, Kobe, Japan

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478

The smallest squares containing k 478's :
478864 = 692²,
12314784784 = 110972²,
110544784784784 = 10514028².

478² = 1⁴ + 11⁴ + 17⁴ + 19⁴.

478² = 228484, a square consisting of just 3 kinds of digits.

186^k + 234^k + 258^k + 478^k are squares for k = 1,2,3 (34², 620², 12068²).

478² = 228484, 2 + 2 + 8 + 4 + 84 = 10²,
478² = 228484, 2 + 2 + 8 + 48 + 4 = 8²,
478² = 228484, 2 + 2 + 84 + 8 + 4 = 10².

Page of Squares : First Upload March 22, 2005 ; Last Revised March 8, 2011
by Yoshio Mimura, Kobe, Japan

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479

The smallest squares containing k 479's :
47961 = 219²,
1479479296 = 38464²,
479047947962896 = 21887164².

454² + 455² + 456² + 457² + ... + 479² = 2379².

479² + 480² + 481² + 482² + ... + 1519² = 33659².

479 is the first prime for which the Legendre Symbol (a/479) = 1 for a = 1, 2,..., 12,
(the 9th prime for a = 1, 2,..., 6. the 2nd prime for a =1, 2,..., 10).

3-by-3 magic squares consisting of different squares with constant 479²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(2, 69, 474, 309, 362, 54, 366, 306, 43),	(2, 219, 426, 246, 366, 187, 411, 218, 114),
(11, 126, 462, 294, 363, 106, 378, 286, 69),	(21, 218, 426, 322, 309, 174, 354, 294, 133),
(54, 146, 453, 222, 411, 106, 421, 198, 114),	(54, 267, 394, 331, 306, 162, 342, 254, 219),
(70, 246, 405, 315, 330, 146, 354, 245, 210),	(78, 261, 394, 299, 282, 246, 366, 286, 117)

479² = 229441, 22 + 94 + 4 + 1 = 11².

479² = 229441 appears in the decimal expression of e:
  e = 2.71828•••229441••• (from the 135119th digit).

Page of Squares : First Upload March 22, 2005 ; Last Revised April 27, 2009
by Yoshio Mimura, Kobe, Japan

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