Squares:550s

550

The smallest squares containing k 550's :
255025 = 505²,
2550755025 = 50505²,
22550655055009 = 4748753².

550² = (2² + 6)(4² + 6)(37² + 6).

Page of Squares : First Upload January 9, 2006 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

551

The smallest squares containing k 551's :
755161 = 869²,
255175512201 = 505149²,
605510551551376 = 24607124².

3-by-3 magic squares consisting of different squares with constant 551²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(3, 74, 546, 294, 462, 61, 466, 291, 42),	(18, 146, 531, 349, 414, 102, 426, 333, 106),
(18, 259, 486, 339, 378, 214, 434, 306, 147),	(29, 174, 522, 234, 477, 146, 498, 214, 99),
(29, 174, 522, 258, 466, 141, 486, 237, 106),	(66, 333, 434, 371, 294, 282, 402, 326, 189),
(74, 195, 510, 330, 426, 115, 435, 290, 174),	(83, 246, 486, 354, 398, 141, 414, 291, 218),
(126, 322, 429, 354, 381, 182, 403, 234, 294)

551 = (1² + 2² + 3² + ... + 28²) / (1² + 2² + 3²).

Page of Squares : First Upload January 9, 2006 ; Last Revised June 1, 2009
by Yoshio Mimura, Kobe, Japan

552

The smallest squares containing k 552's :
55225 = 235²,
552955225 = 23515²,
414355255295524 = 20355718².

1² + 2² + ... + 552² = 56217980, which consists of different digits (the 4th 8-digit sum).

552² = 304704, 3 + 0 + 4 + 70 + 4 = 9²,
552² = 304704, 30 + 47 + 0 + 4 = 9².

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

553

The smallest squares containing k 553's :
65536 = 256²,
55325155369 = 235213²,
553555302395536 = 23527756².

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

3-by-3 magic squares consisting of different squares with constant 553²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(12, 104, 543, 356, 417, 72, 423, 348, 76),	(32, 204, 513, 324, 423, 148, 447, 292, 144),
(33, 216, 508, 328, 417, 156, 444, 292, 153),	(39, 292, 468, 348, 351, 248, 428, 312, 159),
(42, 189, 518, 266, 462, 147, 483, 238, 126),	(48, 193, 516, 383, 384, 108, 396, 348, 167),
(60, 320, 447, 345, 372, 220, 428, 255, 240),	(103, 276, 468, 372, 383, 144, 396, 288, 257)

553² = 305809, 3 + 0 + 5 + 8 + 0 + 9 = 5².

553² = (1² + 6)(209² + 6).

216² + 217² + 218² + 219² + ... + 553² = 7293².

553² + 554² + 555² + 556² + ... + 729² = 8555².

553² = 305809 appears in the decimal expression of e:
e = 2.71828•••305809••• (from the 9623rd digit)
(305809 is the seventh 6-digit square in the expression of e.)

Page of Squares : First Upload May 16, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

554

The smallest squares containing k 554's :
2455489 = 1567²,
554555401 = 23549²,
5548155406655481 = 74485941².

554²± 3 are primes.

554² = 306916, 3 + 0 + 6 + 9 + 1 + 6 = 5²,
554² = 306916, 30³ + 6³ + 9³ + 16³ = 179².

Page of Squares : First Upload January 9, 2006 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

555

The smallest squares containing k 555's :
555025 = 745²,
155565558724 = 394418²,
3405555555551044 = 58357138².

Komachi square sum : 555² = 2² + 78² + 359² + 416².

3-by-3 magic squares consisting of different squares with constant 555²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(0, 180, 525, 333, 420, 144, 444, 315, 108),	(7, 74, 550, 374, 407, 50, 410, 370, 55),
(10, 119, 542, 295, 458, 106, 470, 290, 55),	(10, 199, 518, 230, 470, 185, 505, 218, 74),
(10, 199, 518, 230, 482, 151, 505, 190, 130),	(10, 230, 505, 295, 430, 190, 470, 265, 130),
(10, 266, 487, 295, 410, 230, 470, 263, 134),	(14, 73, 550, 290, 470, 55, 473, 286, 50),
(22, 146, 535, 190, 505, 130, 521, 178, 70),	(25, 70, 550, 134, 535, 62, 538, 130, 41),
(25, 274, 482, 350, 382, 199, 430, 295, 190),	(25, 350, 430, 386, 298, 265, 398, 311, 230),
(45, 228, 504, 360, 396, 147, 420, 315, 180),	(70, 310, 455, 350, 329, 278, 425, 322, 154),
(74, 185, 518, 265, 470, 130, 482, 230, 151),	(74, 218, 505, 265, 470, 130, 482, 199, 190),
(74, 343, 430, 370, 290, 295, 407, 326, 190),	(122, 295, 454, 329, 410, 178, 430, 230, 265),
(130, 265, 470, 290, 442, 169, 455, 206, 242)

(1³ + 2³ + ... + 416³)(417³ + 418² + ... + 555³) = 11067687072².

555² = 308025, 3 + 0 + 8 + 0 + 25 = 6².

Page of Squares : First Upload May 16, 2005 ; Last Revised June 1, 2009
by Yoshio Mimura, Kobe, Japan

556

The smallest squares containing k 556's :
27556 = 166²,
5562325561 = 74581²,
55635562155625 = 7458925².

The squares which begin with 556 and end in 556 are
55617675556 = 235834², 556017783556 = 745666², 556268355556 = 745834²,
556763699556 = 746166², 5560946883556 = 2358166²,...

556² = 309136, 3 + 0 + 9 + 1 + 36 = 7²,
556² = 309136, 30 + 9 + 1 + 3 + 6 = 7².

556² + 557² + 558² + 559² + ... + 804² = 10790²,
556² + 557² + 558² + 559² + ... + 2091² = 54704²,
556² + 557² + 558² + 559² + ... + 17155² = 1297290².

556² = 309136 appears in the decimal expression of e:
e = 2.71828•••309136••• (from the 22067th digit)

Page of Squares : First Upload May 16, 2005 ; Last Revised August 9, 2006
by Yoshio Mimura, Kobe, Japan

557

The smallest squares containing k 557's :
1557504 = 1248²,
55745571025 = 236105²,
55765557557956 = 7467634².

557² = 310249, a square with different digits.

Cubic polynomial :
(X + 288²)(X + 316²)(X + 357²) = X³ + 557²X² + 177708²X + 32489856².

3-by-3 magic squares consisting of different squares with constant 557²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(8, 168, 531, 261, 468, 152, 492, 251, 72),	(8, 237, 504, 324, 408, 197, 453, 296, 132),
(12, 219, 512, 379, 372, 168, 408, 352, 141),	(27, 208, 516, 252, 456, 197, 496, 243, 72),
(35, 132, 540, 300, 460, 93, 468, 285, 100),	(72, 316, 453, 384, 357, 188, 397, 288, 264),
(93, 188, 516, 244, 483, 132, 492, 204, 163)

557² = 310249, 31 + 0 + 24 + 9 = 8²,
557² = 310249, 310 + 2 + 49 = 19².

Page of Squares : First Upload May 16, 2005 ; Last Revised June 1, 2009
by Yoshio Mimura, Kobe, Japan

558

The smallest squares containing k 558's :
558009 = 747²,
35586558736 = 188644²,
1558877755855876 = 39482626².

15810^k + 70494^k + 87978^k + 137082^k are squares for k = 1,2,3 (558², 178188²,60093252²).

The 4-by-4 magic squares consisting of different squares with constant 558:

0²	5²	7²	22²
6²	17²	13²	8²
9²	12²	18²	3²
21²	10²	4²	6¹

0²	6²	9²	21²
10²	20²	3²	7²
13²	1²	18²	8²
17²	11²	12²	2²

558² = 311364, 3⁸ + 1⁸ + 1⁸ + 3⁸ + 6⁸ + 4⁸ = 1326²,
558² = 311364, 3 + 1 + 136 + 4 = 12²,
558² = 311364, 3 + 11 + 3 + 64 = 9²,
558² = 311364, 3 + 1 + 13 + 64 = 9²,
558² = 311364, 311 + 3 + 6 + 4 = 18².

Page of Squares : First Upload January 9, 2006 ; Last Revised March 15, 2011
by Yoshio Mimura, Kobe, Japan

559

The smallest squares containing k 559's :
559504 = 748²,
559559025 = 23655²,
455955904559376 = 21353124².

1³ - 2³ + 3³ - 4³ + 5³ - 6³ + ... + 83³ - 84³ + 85³ = 559².

3-by-3 magic squares consisting of different squares with constant 559²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(18, 219, 514, 261, 458, 186, 494, 234, 117),	(21, 314, 462, 366, 357, 226, 422, 294, 219),
(26, 102, 549, 234, 501, 82, 507, 226, 66),	(26, 117, 546, 378, 406, 69, 411, 366, 98),
(26, 234, 507, 318, 411, 206, 459, 298, 114),	(37, 114, 546, 294, 469, 78, 474, 282, 91),
(54, 107, 546, 138, 534, 91, 539, 126, 78),	(54, 213, 514, 242, 474, 171, 501, 206, 138),
(66, 270, 485, 390, 325, 234, 395, 366, 150)

559² = 312481, 3 + 12 + 4+81 = 10²,
559² = 312481, 3 + 12 + 48+1 = 8²,
559² = 312481, 31 + 24 + 8+1 = 8²,
559² = 312481, 312 + 48 + 1 = 19².

(1³ + 2³ + ... + 9³)(10³ + 11³ + ... + 34³)(35³ + 36³ + ... + 559³) = 4178790000².

Page of Squares : First Upload May 16, 2005 ; Last Revised June 1, 2009
by Yoshio Mimura, Kobe, Japan