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480 - 489

480

The smallest squares containing k 480's :
248004 = 4982,
4807480896 = 693362,
164803948084801 = 128375992.

262 + 480 = 342, 262 - 480 = 142.

(4802 - 6) = (32 - 6)(52 - 6)(72 - 6)(102 - 6).

4802 = (22 - 1)(32 - 1)(92 - 1)(112 - 1) = (22 - 1)(92 - 1)(312 - 1) = (32 - 1)(42 - 1)(52 - 1)(92 - 1)
= (32 - 1)(92 - 1)(192 - 1) = (52 - 1)(92 - 1)(112 - 1).

(12 + 22 + ... + 422)(432 + 442 + ... + 2802)(2812 + 2822 + ... + 4802) = 23571460502.

Page of Squares : First Upload March 28, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

481

The smallest squares containing k 481's :
3481 = 592,
48177481 = 69412,
164819148152481 = 128381912.

The squares which begin with 481 and end in 481 are
48177481 = 69412,   481407481 = 219412,   48154352481 = 2194412,
481024086481 = 6935592,   481207203481 = 6936912,...

4812 is the first squre which is the sum of 3 fourth powers : 4812 = 124 + 154 + 204.

4813 - 4803 + 4793 - 4783 + 4773 - 4763 + .. + 13 = 74712.

481k + 624k + 1404k + 1716k are squares for k = 1,2,3 (652, 23532, 904152).

3-by-3 magic squares consisting of different squares with constant 4812:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(3, 124, 444, 236, 381, 108, 396, 228, 61),(16, 123, 444, 312, 324, 101, 339, 304, 72),
(72, 196, 411, 264, 357, 124, 371, 216, 168),(84, 236, 387, 259, 348, 156, 372, 189, 196)
(0, 156, 455, 185, 420, 144, 444, 175, 60),(4, 111, 468, 153, 444, 104, 456, 148, 39),
(4, 192, 441, 273, 364, 156, 396, 249, 112),(24, 176, 447, 328, 321, 144, 351, 312, 104),
(39, 104, 468, 216, 423, 76, 428, 204, 81),(39, 276, 392, 312, 284, 231, 364, 273, 156),
(57, 176, 444, 284, 372, 111, 384, 249, 148) 

4812 = 231361, 2 + 3 + 1 + 3 + 6 + 1 = 42,
4812 = 231361, 2 + 3 + 13 + 6 + 1 = 52,
4812 = 231361, 232 + 12 + 32 + 62 + 12 = 242.

Page of Squares : First Upload March 28, 2005 ; Last Revised March 8, 2011
by Yoshio Mimura, Kobe, Japan

482

The smallest squares containing k 482's :
148225 = 3852,
48230748225 = 2196152,
74827482482944 = 86502882.

4822 = 232324, a zigzag square consisting of only 3 kinds of digits.

4822 = 232324, a square pegged by 2.

The sum of (19x + 7)2 is 14432, where x runs over 0, 1,.., 25.

Komachi Square Sum : 4822 = 22 + 32 + 52 + 92 + 612 + 4782.

4822 = 232324, 2 + 3 + 2 + 3 + 2 + 4 = 42.

Page of Squares : First Upload March 28, 2005 ; Last Revised July 27, 2006
by Yoshio Mimura, Kobe, Japan

483

The smallest squares containing k 483's :
164836 = 4062,
4834838089 = 695332,
248312483948356 = 157579342.

4832 = 233289, 233 * 2 + 8 + 9 = 483.

14490k + 47334k + 63273k + 108192k are squares for k = 1,2,3 (4832, 1347572, 403589972).

Komachi equation: 4832 = 9 + 8 / 7 * 6 * 54 * 3 * 210.

3-by-3 magic squares consisting of different squares with constant 4832:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(2, 127, 466, 214, 418, 113, 433, 206, 58),(2, 178, 449, 218, 401, 158, 431, 202, 82),
(2, 314, 367, 337, 262, 226, 346, 257, 218),(17, 110, 470, 310, 358, 95, 370, 305, 58),
(36, 213, 432, 333, 324, 132, 348, 288, 171),(38, 191, 442, 271, 358, 178, 398, 262, 79),
(47, 118, 466, 158, 446, 97, 454, 143, 82),(47, 146, 458, 302, 367, 86, 374, 278, 127),
(50, 242, 415, 310, 335, 158, 367, 250, 190),(62, 262, 401, 289, 302, 242, 382, 271, 118),
(74, 223, 422, 337, 278, 206, 338, 326, 113),(94, 223, 418, 257, 382, 146, 398, 194, 193),
(97, 214, 422, 262, 383, 134, 394, 202, 193) 

(13 + 23 + ... + 773)(783 + 793 + ... + 2303)(2313 + 2323 + ... + 4833) = 90223265452322,
(13 + 23 + ... + 2513)(2523)(2533 + 2543 + ... + 4833) = 142273462383362.

4832 = 233289, 2 + 33 + 289 = 182,
4832 = 233289, 23 + 32 + 89 = 122,
4832 = 233289, 233 + 2 + 89 = 182.

Page of Squares : First Upload March 28, 2005 ; Last Revised March 8, 2011
by Yoshio Mimura, Kobe, Japan

484

The square of 22.

The smallest squares containing k 484's :
484 = 222,
121484484 = 110222,
38924846484484 = 62389782.

The squares which begin with 484 and end in 484 are
484968484 = 220222,   48409680484 = 2200222,   484385376484 = 6959782,
484446624484 = 6960222,   4840096800484 = 22000222,...

(12 + 4)(32 + 4)(72 + 4)(82 + 4) = 4842 + 4.

(13 + 23 + ... + 4833)(4843) = 12446021282.

4842 = 234256, 2 + 34 + 2 + 5 + 6 = 72,
4842 = 234256, 22 + 342 + 22 + 52 + 62 = 352,
4842 = 234256, 2 + 342 + 56 = 202,
4842 = 234256, 23 + 42 + 56 = 112.

Page of Squares : First Upload March 28, 2005 ; Last Revised July 27, 2006
by Yoshio Mimura, Kobe, Japan

485

The smallest squares containing k 485's :
485809 = 6972,
1485485764 = 385422,
4854858548544225 = 696768152.

4852 = 235225, a square consisting of only 3 kinds of digits.

125k + 241k + 305k + 485k are squares for k = 1,2,3 (342, 6342, 125862).

3-by-3 magic squares consisting of different squares with constant 4852:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(0, 291, 388, 325, 288, 216, 360, 260, 195),(3, 204, 440, 280, 360, 165, 396, 253, 120),
(8, 69, 480, 195, 440, 60, 444, 192, 35),(8, 165, 456, 300, 360, 125, 381, 280, 108),
(24, 125, 468, 315, 360, 80, 368, 300, 99),(35, 240, 420, 336, 315, 152, 348, 280, 189),
(60, 168, 451, 195, 424, 132, 440, 165, 120),(60, 235, 420, 260, 372, 171, 405, 204, 172)

(12 + 22)(32 + 42 + ... + 1352)(1362 + 1372 + ... + 4852) = 124388252.

(13 + 23 + ... + 3143)(3153 + 3162 + ... + 4053)(4063 + 4072 + ... + 4853) = 2742744824820002.

4852 = 235225, 2 + 3 + 52 + 2 + 5 = 82,
4852 = 235225, 2 + 35 + 2 + 25 = 82,
4852 = 235225, 2 + 35 + 22 + 5 = 82,
4852 = 235225, 2 + 352 + 2 + 5 = 192,
4852 = 235225, 23 + 52 + 25 = 102.

Page of Squares : First Upload March 28, 2005 ; Last Revised March 8, 2011
by Yoshio Mimura, Kobe, Japan

486

The smallest squares containing k 486's :
94864 = 3082,
148644864 = 121922,
486394486248609 = 220543532.

4862 = 236196 is an exchangeable square, 962361 = 9812.

4862 is the 7th square which is the sum of 4 fifth powers : (9, 9, 9, 9).

4862 = (12 + 2)(22 + 2)(52 + 2)(222 + 2) = (42 + 2)(52 + 2)(222 + 2).

4862 = 95 + 95 + 95 + 95.

4862 is the 3rd square which is the sum of 4 tenth powers : (3, 3, 3, 3).

4862 + 4872 + 4882 + 4892 + ... + 123662 = 7939562.

Komachi equations:
4862 = 92 / 82 / 72 * 62 / 52 * 42 * 32 * 2102 = 92 * 82 * 72 + 62 + 542 * 32 - 2102.

The 4-by-4 magic square consisting of different squares with constant 486:

02 12142172
32162112102
62152122 92
212 22 52 42

4862 = 236196, 2 + 3 + 6 + 19 + 6 = 62,
4862 = 236196, 2 + 3 + 61 + 9 + 6 = 92,
4862 = 236196, 23 + 6 + 196 = 152.

Page of Squares : First Upload March 28, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

487

The smallest squares containing k 487's :
487204 = 6982,
174875348761 = 4181812,
248764879487524 = 157722822.

4872 = 237169, a square with different digits.

4872 = 44 + 44 + 74 + 224.

4872 = 237169, 23716 = 1542, 9 = 32.

4872 = 30 + 35 + 36 + 310 + 311.

Komachi Square Sums : 4872 = 12 + 32 + 52 + 72 + 692 + 4822 = 22 + 52 + 72 + 92 + 612 + 4832.

3-by-3 magic squares consisting of different squares with constant 4872:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(6, 98, 477, 243, 414, 82, 422, 237, 54),(6, 138, 467, 258, 397, 114, 413, 246, 78),
(6, 282, 397, 307, 306, 222, 378, 253, 174),(62, 141, 462, 222, 422, 99, 429, 198, 118),
(62, 210, 435, 285, 370, 138, 390, 237, 170),(93, 282, 386, 334, 243, 258, 342, 314, 147)

1992 + 2002 + 2012 + 2022 + ... + 4872 = 60012.

4872 = 237169, 2 + 37 + 16 + 9 = 82,
4872 = 237169, 233 + 73 + 13 + 63 + 93 = 1162,
4872 = 237169, 23 + 7 + 1 + 69 = 102.

4872 = 237169 appears in the decimal expressions of π and e:
  π = 3.14159•••237169••• (from the 29008th digit),
  e = 2.71828•••237169••• (from the 46473rd digit).

Page of Squares : First Upload March 28, 2005 ; Last Revised August 26, 2011
by Yoshio Mimura, Kobe, Japan

488

The smallest squares containing k 488's :
14884 = 1222,
4882934884 = 698782,
303488488488241 = 174209212.

Komachi Square Sum : 4882 = 72 + 92 + 532 + 622 + 4812.

488 is the first integer which is the sum of a square and a prime in 9 ways:
12 + 487, 32 + 479, 52 + 463, 72 + 439, 112 + 367, 152 + 263, 172 + 199, 192 + 127, 212 + 47.

4882 = 64 + 64 + 64 + 224.

4882 = 238144, 23 + 33 + 83 + 13 + 43 + 43 = 262,
4882 = 238144, 22 + 32 + 82 + 142 + 42 = 172,
4882 = 238144, 2 + 38 + 1 + 4 + 4 = 72,
4882 = 238144, 23 + 8 + 14 + 4 = 72,
4882 = 238144, 23 + 814 + 4 = 292,
4882 = 238144, 238 + 14 + 4 = 162.

Page of Squares : First Upload March 28, 2005 ; Last Revised July 27, 2006
by Yoshio Mimura, Kobe, Japan

489

The smallest squares containing k 489's :
4489 = 672,
844890489 = 290672,
138348948925489 = 117621832.

The squares which begin with 489 and end in 489 are
4890624489 = 699332,   48921919489 = 2211832,   48981214489 = 2213172,
489044266489 = 6993172,   489206521489 = 6994332,...

3-by-3 magic squares consisting of different squares with constant 4892:

A2B2C2
D2E2F2
G2H2K2
where (A, B, C, D, E, F, G, H, K) = 
(1, 112, 476, 196, 436, 103, 448, 191, 44),(4, 257, 416, 332, 304, 191, 359, 284, 172),
(4, 268, 409, 304, 319, 212, 383, 256, 164),(25, 164, 460, 220, 415, 136, 436, 200, 95),
(31, 184, 452, 296, 353, 164, 388, 284, 89),(32, 151, 464, 311, 352, 136, 376, 304, 73),
(44, 217, 436, 289, 364, 152, 392, 244, 161),(54, 294, 387, 333, 306, 186, 354, 243, 234),
(56, 208, 439, 343, 296, 184, 344, 329, 112),(124, 287, 376, 308, 344, 161, 359, 196, 268)

4892 = 43 + 93 + 623.

(13 + 23 + ... + 563)(573 + 583 + ... + 843)(853 + 863 + ... + 4893) = 6103313874002.

4892 = 239121, 2 + 3 + 9 + 1 + 21 = 62,
4892 = 239121, 23 + 9 + 1 + 2 + 1 = 62,
4892 = 239121, 22 + 32 + 92 + 12 + 22 + 12 = 102.

Page of Squares : First Upload March 28, 2005 ; Last Revised May 11, 2009
by Yoshio Mimura, Kobe, Japan