Squares:400s

400

The smallest squares containing k 400's :
400 = 20²,
4004001 = 2001²,
240080400400 = 489980².

The squares which begin with 400 and end in 400 are
400800400 = 20020², 4004358400 = 63280², 4009422400 = 63320²,
40008000400 = 200020², 40032006400 = 200080²,...

20² = 1 + 7 + 7² + 7³.

20² = (1² + 4)(2² + 4)(4² + 4)(14² + 4) = (4² + 4)(6² + 4)(14² + 4).

Komachi equations:
400² = - 1² * 2² + 34² + 56² * 7² + 8² * 9²,
400² = 9⁴ * 8⁴ / 7⁴ / 6⁴ / 54⁴ * 3⁴ * 210⁴.

20² = 20³ + 30³ + 50³.

4004001 = 2001².

Page of Squares : First Upload anuary 31, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

401

The smallest squares containing k 401's :
2401 = 49²,
401401225 = 20035²,
104010401705401 = 10198549².

The squares which begin with 401 and end in 401 are
401962401 = 20049², 40119689401 = 200299², 40180603401 = 200451²,
401067623401 = 633299², 401260169401 = 633451²,...

The square root of 401 is 20.024..., 20 = 0² + 2² + 4².

401² = 160801, a zigzag square.

Cubic Polynomial : (X + 87²)(X + 144²)(X + 364²) = X³ + 401²X² + 62508²X + 4560192².

Komachi equations:
401² = 9² - 8² + 76² * 5² + 4² * 32² * 1² = 9² - 8² + 76² * 5² + 4² * 32² / 1².

3-by-3 magic squares consisting of different squares with constant 401²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(4, 63, 396, 279, 284, 48, 288, 276, 41),	(24, 140, 375, 265, 276, 120, 300, 255, 76),
(24, 183, 356, 252, 284, 129, 311, 216, 132),	(41, 108, 384, 144, 364, 87, 372, 129, 76),
(84, 228, 319, 256, 279, 132, 297, 176, 204)

401² = 160801, 1 + 6 + 0 + 8 + 0 + 1 = 4²,
401² = 160801, 16 + 0 + 8 + 0 + 1 = 5²,
401² = 160801, 160 + 8 + 0 + 1 = 13²,
401² = 160801, 160 + 801 = 31².

Page of Squares : First Upload anuary 31, 2005 ; Last Revised June 4, 2010
by Yoshio Mimura, Kobe, Japan

402

The smallest squares containing k 402's :
24025 = 155²,
4020194025 = 63405²,
180402640274025 = 13431405².

402² = 161604, a zigzag square.

Komachi equation: 402² = - 1² + 2² * 3² - 4² + 56² * 7² + 89².

The 4-by-4 magic square consisting of different squares with constant 402:

0²	4²	5²	19²
7²	11²	14²	6²
8²	16²	9²	1²
17²	3²	10²	2²

402² = 161604, 16 + 1 + 60 + 4 = 9²,
402² = 161604, 16 + 16 + 0 + 4 = 6²,
402² = 161604, 161 + 60 + 4 = 15².

4620² = 106² + 107² + 108² + 109² + ... + 402².

Page of Squares : First Upload anuary 31, 2005 ; Last Revised June 4, 2010
by Yoshio Mimura, Kobe, Japan

403

The smallest squares containing k 403's :
244036 = 494²,
6403040361 = 80019²,
403940336403289 = 20098267².

403 = (1² + 2² + 3² + ... + 77²) / (1² + 2² + 3² + ... + 10²).

403² = 162409, 1 + 6 + 240 + 9 = 16²,
403² = 162409, 16 + 24 + 0 + 9 = 7².

403² = 162409, a zigzag square with different digits.

403^k + 961^k + 5363^k + 8649^k are squares for k = 1,2,3 (124², 10230², 895652²).

3-by-3 magic squares consisting of different squares with constant 403²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(2, 162, 369, 198, 321, 142, 351, 182, 78),	(6, 63, 398, 177, 358, 54, 362, 174, 33),
(30, 97, 390, 222, 330, 65, 335, 210, 78),	(47, 162, 366, 258, 294, 97, 306, 223, 138),
(78, 159, 362, 182, 342, 111, 351, 142, 138),	(78, 254, 303, 273, 258, 146, 286, 177, 222)

403² + 404² + 405² + 406² + ... + 29554² = 2933420².

Page of Squares : First Upload anuary 31, 2005 ; Last Revised March 4, 2011
by Yoshio Mimura, Kobe, Japan

404

The smallest squares containing k 404's :
10404 = 102²,
404090404 = 20102²,
194404404638404 = 13942898².

The squares which begin with 404 and end in 404 are
404090404 = 20102², 4045214404 = 63602², 40442014404 = 201102²,
404366266404 = 635898², 404625754404 = 636102²,...

The square root of 404 is 20.099751242...,
and 20² = 0² + 9² + 9² + 7² + 5² + 12² + 4² + 2².

404² = 163216, 16 + 32 + 16 = 8².

Page of Squares : First Upload anuary 31, 2005 ; Last Revised July 13, 2006
by Yoshio Mimura, Kobe, Japan

405

The smallest squares containing k 405's :
405769 = 637²,
8405405761 = 91681²,
974054050405921 = 31209839².

The square root of 405 is 20.1246117974981...,
20² = 1² + 2² + 4² + 6² + 1² + 1² + 7² + 9² + 7² + 4² + 9² + 8² + 1².

405² = 164025, a square with different digits.

405² = 9³ + 18³ + 54³.

405² = (23 + 24 + 25 + ... + 31)² + (32 + 33 + 34 + ... + 40)².

Loop of length 56 by the function f(N) = ... + c² + b² + a² where N = ... + 100²c + 100b + a:
405 - 41 - 1681 - 6817 - ... - 7684 - 12832 - 1809 - 405
(Note f(405) = 4² + 05² = 41, f(41) = 41² = 1681, etc. See 41)

Komachi Cubic Sum : 405² = 9³ + 18³ + 27³ + 36³ + 45³.

1² + 2² + ... + 405² = 22225455, which consists of 3 kinds of digits.

3-by-3 magic squares consisting of different squares with constant 405²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(5, 116, 388, 220, 325, 100, 340, 212, 59),	(5, 140, 380, 220, 320, 115, 340, 205, 80),
(11, 200, 352, 248, 275, 164, 320, 220, 115),	(16, 88, 395, 115, 380, 80, 388, 109, 40),
(28, 160, 371, 275, 280, 100, 296, 245, 128),	(30, 105, 390, 150, 366, 87, 375, 138, 66),
(30, 150, 375, 210, 327, 114, 345, 186, 102),	(40, 173, 364, 245, 280, 160, 320, 236, 77),
(40, 245, 320, 268, 224, 205, 301, 232, 140),	(80, 205, 340, 235, 304, 128, 320, 172, 179),
(80, 235, 320, 269, 208, 220, 292, 256, 115)

405² = 164025, 1 + 6 + 4 + 0 + 25 = 6²,
405² = 164025, 16 + 40 + 25 = 9².

(1³ + 2³ + ... + 314³)(315³ + 316³ + ... + 350³)(351³ + 352³ + ... + 405³) = 98457506532000².

Page of Squares : First Upload anuary 31, 2005 ; Last Revised April 6, 2009
by Yoshio Mimura, Kobe, Japan

406

The smallest squares containing k 406's :
140625 = 375²,
40640625 = 6375²,
9024064064064 = 3004008².

406² = 164836, a zigzag square.

406² + 407² + 408² + ... + 420² = 421² + 422² + 423² + ... + 434².

11165^k + 24157^k + 46893^k + 82621^k are squares for k = 1,2,3 (406², 98658², 26126506²).

406² = 164836, 1 + 6 + 4 + 83 + 6 = 10²,
406² = 164836, 1 + 6 + 48 + 3 + 6 = 8²,
406² = 164836, 1² + 648² + 36² = 649²,
406² = 164836, 16 + 4 + 8 + 36 = 8²,
406² = 164836, 16 + 48 + 36 = 10².

(1³ + 2³ + ... + 231³)(232³ + 233³ + ... + 406³) = 2094241380².

Page of Squares : First Upload anuary 31, 2005 ; Last Revised September 6, 2011
by Yoshio Mimura, Kobe, Japan

407

The smallest squares containing k 407's :
407044 = 638²,
3407407129 = 58373²,
407873407407376 = 20195876².

The square root of 407 is 20.1742410018320143988...,
20² = 1² + 7² + 4² + 2² + 4² + ... + 4² + 3² + 9² + 8² + 8².

407 = (1² + 2² + 3² + ... + 55²) / (1² + 2² + 3² + ... + 7²).

The square root of 407 is 20.17424100183...,
20² = 17² + 4² + 2² + 4² + 1² + 0² + 0² + 1² + 8² + 3².

407² = 165649, a zigzag square.

The sum of (19x + 12)² for x = 0,1,2,...,10 is 407².

A + B, A + C, A + D, B + C, B + D, C + D are squares for A = 407, B = 3314, C = 4082, D = 5522.

253^k + 295^k + 341^k + 407^k are squares for k = 1,2,3 (36², 658², 12204²).

3-by-3 magic squares consisting of different squares with constant 407²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(2, 126, 387, 234, 317, 102, 333, 222, 74),	(3, 102, 394, 178, 354, 93, 366, 173, 42),
(3, 186, 362, 254, 282, 147, 318, 227, 114),	(18, 61, 402, 142, 378, 51, 381, 138, 38),
(18, 115, 390, 285, 282, 70, 290, 270, 93),	(45, 250, 318, 282, 210, 205, 290, 243, 150),
(74, 222, 333, 243, 294, 142, 318, 173, 186)

407² = 165649, 1 + 65 + 6 + 49 = 11²,
407² = 165649, 16 + 56 + 49 = 11².

Page of Squares : First Upload anuary 31, 2005 ; Last Revised March 4, 2011
by Yoshio Mimura, Kobe, Japan

408

The smallest squares containing k 408's :
40804 = 202²,
20408408164 = 142858²,
264084080440896 = 16250664².

408² = 166464, a square with 3 kinds of digits.

408² = 166464, 1 * 6 * 64 + 6 * 4 = 16 * 6 * 4 + 6 * 4 = 408.

408² = (3² + 8)(4² + 8)(20² + 8) = (8² + 8)(48² + 8).

408^k + 1156^k + 2601^k + 3060^k are squares for k = 1,2,3 (85², 4199², 218773²).
6^k + 408^k + 690^k + 921^k are squares for k = 1,2,3 (45², 1221², 34317²).

408² = 166464, 1 + 6 + 6 + 4 + 64 = 9²,
408² = 166464, 1 + 6 + 64 + 6 + 4 = 9²,
408² = 166464, 1 + 66 + 4 + 6 + 4 = 9²,
408² = 166464, 1 + 664 + 64 = 27²,
408² = 166464, 16 + 6 + 4 + 6 + 4 = 6²,
408² = 166464, 16 + 64 + 64 = 12².

408² = 166464, with 16 = 4² and 64 = 8².

408² = 10³ + 20³ + 54³ = 10³ + 38³ + 48³.

408² = 166464 is an exchangeable square, 646416 = 804².

408² = 2 x 3 x 4 + 4 x 5 x 6 + 6 x 7 x 8 + ... + 32 x 33 x 34.

Page of Squares : First Upload anuary 31, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

409

The smallest squares containing k 409's :
4096 = 64²,
40998409 = 6403²,
8409437409409 = 2899903².

The squares which begin with 409 and end in 409 are
40998409 = 6403², 40944308409 = 202347², 40966974409 = 202403²,
409084322409 = 639597², 409155960409 = 639653²,...

409² + 410² + 411² + ... + 226224² = 62122486².

409 is the seventh prime for which the Legendre Symbol (a/409) = 1 for a = 1, 2,,..,6.

(1² + 2² + ... + 189²) + (1² + 2² + ... + 395²) = (1² + 2² + ... + 409²).

3-by-3 magic squares consisting of different squares with constant 409²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

(12, 176, 369, 201, 324, 148, 356, 177, 96),	(36, 111, 392, 183, 356, 84, 364, 168, 81)
(49, 132, 384, 228, 329, 84, 336, 204, 113),	(76, 201, 348, 264, 292, 111, 303, 204, 184)

409² = 167281, 1 + 6 + 7 + 2 + 8 + 1 = 5²,
409² = 167281, 16 + 72 + 81 = 13²,
409² = 167281, 167 + 28 + 1 = 14²,
409² = 167281, 1672 + 8 + 1 = 41².

(1³ + 2³ + ... + 120³)(121³ + 122³ + ... + 164³)(165³ + 166³ + ... + 409³) = 6858756135000².

Page of Squares : First Upload anuary 31, 2005 ; Last Revised april 6, 2009
by Yoshio Mimura, Kobe, Japan