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450

The smallest squares containing k 450's :
245025 = 495²,
2450745025 = 49505²,
450450450497124 = 21223818².

450² = 15⁴ + 15⁴ + 15⁴ + 15⁴.

A + B, A + C, A + D, B + C, B + D, and C + D are squares
for (A, B, C, D) = (450, 2466, 3775, 5634).

(1 + 2)(3)(4 + 5 + 6 + 7 + 8)(9 + 10 + 11)(12 + 13) = 450²,
(1)(2 + 3 + 4 + 5 + 6)(7 + 8)(9)(10 + 11 + 12 + 13 + 14 + 15) = 450².

450² = (1² + 9)(3² + 9)(4² + 9)(6² + 9) = (1² + 9)(6² + 9)(21² + 9) = (3² + 9)(4² + 9)(21² + 9).

(1³ + 2³ + ... + 290³)(291³ + 292³ + ... + 374³)(375³ + 376³ + ... + 450³) = 173341448280000².

The 4-by-4 magic squares consisting of different squares with constant 450:

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

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451

The smallest squares containing k 451's :
64516 = 254²,
174518404516 = 417754²,
277451451451876 = 16656874².

451² + 452² + 453² + ... + 1508² = 33373².

451² = 3⁰ + 3² + 3⁸ + 3⁹ + 3¹¹.

3-by-3 magic squares consisting of different squares with constant 451²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(6, 159, 422, 303, 314, 114, 334, 282, 111),	(26, 177, 414, 258, 334, 159, 369, 246, 82),
(42, 206, 399, 314, 273, 174, 321, 294, 118),	(63, 134, 426, 186, 399, 98, 406, 162, 111),
(82, 246, 369, 306, 303, 134, 321, 226, 222)

(1³ + 2³ + ... + 132³)(133³ + 134³ + ... + 164³)(165³ + 166³ + ... + 451³) = 9130376166912²,
(1³ + 2³ + ... + 143³)(144³ + 145³ + ... + 164³)(165³ + 166³ + ... + 451³) = 9130376166912²,
(1³ + 2³ + ... + 164³)(165³ + 166³ + ... + 451³) = 1366854720².

451² = 203401, 20 + 3 + 40 + 1 = 8²,
451² = 203401, 2³ + 0³ + 3³ + 4³ + 0³ + 1³ = 10²,
451² = 203401, 20 + 340 + 1 = 19².

Page of Squares : First Upload March 7, 2005 ; Last Revised August 9, 2011
by Yoshio Mimura, Kobe, Japan

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452

The smallest squares containing k 452's :
44521 = 211²,
4520545225 = 67235²,
45245277584521 = 6726461².

1 + 2² + 3² + 4 + 5² + 6² + 7 + 8² + 9² + ... + 94 + 95² + 96² = 452².

452² = 204304, 2 + 0 + 43 + 0 + 4 = 7²,
452² = 204304, 20² + 4² + 3² + 0² + 4² = 21².

Page of Squares : First Upload March 7, 2005 ; Last Revised July 21, 2006
by Yoshio Mimura, Kobe, Japan

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453

The smallest squares containing k 453's :
45369 = 213²,
24453453376 = 156376²,
4445324532453289 = 66673267².

1 / 453 = 0.002207, 2² + 20² + 7² = 453.

3-by-3 magic squares consisting of different squares with constant 453²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(4, 67, 448, 227, 388, 56, 392, 224, 37),	(6, 162, 423, 297, 318, 126, 342, 279, 102),
(8, 172, 419, 248, 349, 148, 379, 232, 88),	(11, 148, 428, 188, 388, 139, 412, 181, 52),
(18, 183, 414, 249, 342, 162, 378, 234, 87),	(28, 157, 424, 184, 392, 133, 413, 164, 88),
(28, 220, 395, 245, 340, 172, 380, 203, 140),	(32, 293, 344, 316, 232, 227, 323, 256, 188),
(52, 221, 392, 259, 308, 208, 368, 248, 91),	(67, 176, 412, 272, 347, 104, 356, 232, 157)

Page of Squares : First Upload March 7, 2005 ; Last Revised April 20, 2009
by Yoshio Mimura, Kobe, Japan

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454

The smallest squares containing k 454's :
374544 = 612²,
14547254544 = 120612²,
4543454245444 = 2131538².

454² = 2³ + 9³ + 59³.

454⁵ = 19287647677024,
1² + 9² + 2² + 8² + 7² + 6² + 4² + 7² + 6² + 7² + 7² + 0² + 2² + 4² = 454.

454² + 455² + 456² + 457² + ... + 1350² = 28106²,
454² + 455² + 456² + 457² + ... + 479² = 2379².

1 / 454 = 0.00220264317180616740088..., the sum of the squares of its digits is 454.

454² = 206116, 2 + 0 + 6 + 1 + 1 + 6 = 4²,
454² = 206116, 2 + 0 + 6 + 1 + 16 = 5²,
454² = 206116, 2 + 0 + 6 + 11 + 6 = 5².

Page of Squares : First Upload March 7, 2005 ; Last Revised July 21, 2006
by Yoshio Mimura, Kobe, Japan

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455

The smallest squares containing k 455's :
455625 = 675²,
4553145529 = 67477²,
45545545577536 = 6748744².

455² = 207025, 2 + 0 + 7 + 0 + 2 + 5 = 4²,
455² = 207025, 2³ + 0³ + 7³ + 0³ + 2³ + 5³ = 22²,
455² = 207025, 20² + 70² + 2² + 5² = 73².

3-by-3 magic squares consisting of different squares with constant 455²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(0, 175, 420, 273, 336, 140, 364, 252, 105),	(6, 67, 450, 310, 330, 45, 333, 306, 50),
(13, 66, 450, 234, 387, 50, 390, 230, 45),	(13, 234, 390, 270, 310, 195, 366, 237, 130),
(22, 150, 429, 285, 330, 130, 354, 275, 78),	(30, 90, 445, 130, 429, 78, 435, 122, 54),
(30, 130, 435, 211, 390, 102, 402, 195, 86),	(30, 157, 426, 195, 390, 130, 410, 174, 93),
(30, 195, 410, 270, 338, 141, 365, 234, 138),	(30, 270, 365, 302, 285, 186, 339, 230, 198),
(45, 174, 418, 310, 318, 99, 330, 275, 150),	(45, 230, 390, 310, 270, 195, 330, 285, 130),
(78, 221, 390, 285, 330, 130, 346, 222, 195),	(90, 270, 355, 291, 310, 162, 338, 195, 234)

Page of Squares : First Upload March 7, 2005 ; Last Revised April 20, 2009
by Yoshio Mimura, Kobe, Japan

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456

The smallest squares containing k 456's :
13456 = 116²,
774564561 = 27831²,
456456979456 = 675616².

The squares which begin with 456 and end in 456 are
45631795456 = 213616²,   456143547456 = 675384²,   456456979456 = 675616²,
456819181456 = 675884²,   4560855699456 = 2135616²,...

456² = 207936, a square with different digits.

456²± 5 are primes.

456² = (1² + 3)(3² + 3)(4² + 3)(15² + 3) = (4² + 8)(7² + 8)(12² + 8).

456² = 2³ + 46³ + 48³.

Komachi equations:
456² = 9² * 8² * 76² * 5² / 4² / 3² * 2² / 10² = 9² * 8² * 76² / 5² / 4² / 3² / 2² * 10²
= 9² / 8² * 76² / 5² * 4² / 3² * 2² * 10².

456² + 457² + 458² + 459² + ... + 466² = 1529²,
456² + 457² + 458² + 459² + ... + 1033² = 18343².

456² = 207936, 207 + 9 + 3 + 6 = 15².

456² = 207936 appears in the decimal expression of e:
  e = 2.71828•••207936••• (from the 105801st digit).

Page of Squares : First Upload March 7, 2005 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

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457

The smallest squares containing k 457's :
45796 = 214²,
13345794576 = 115524²,
1645764571264576 = 40568024².

1 / 457 = 0.00218818, 2² + 1² + 8² + 8² + 18² = 457,
1 / 457 = 0.00218818, 2² + 18² + 8² + 1² + 8² = 457.

3-by-3 magic squares consisting of different squares with constant 457²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(7, 108, 444, 228, 384, 97, 396, 223, 48),	(24, 63, 452, 92, 444, 57, 447, 88, 36),
(36, 263, 372, 288, 276, 223, 353, 252, 144),	(48, 183, 416, 297, 304, 168, 344, 288, 87),
(92, 249, 372, 312, 308, 129, 321, 228, 232)

(1³ + 2³ + ... + 152³)(153³ + 154³ + ... + 457³) = 1209370140²,
(1³ + 2³ + ... + 409³)(410³ + 411³ + ... + 457³) = 5251044660².

457² = 208849, 20 + 8 + 8 + 4 + 9 = 7²,
457² = 208849, 20 + 8 + 84 + 9 = 11²,
457² = 208849, 20 + 88 + 4 + 9 = 11²,
457² = 208849, 20² + 8² + 8² + 4² + 9² = 25².

Page of Squares : First Upload March 7, 2005 ; Last Revised April 20, 2009
by Yoshio Mimura, Kobe, Japan

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458

The smallest squares containing k 458's :
458329 = 677²,
206458458129 = 454377²,
458458798545801 = 21411651².

458² = 209764, a square with different digits.

(1³ + 2³ + ... + 216³)(217³ + 218³ + ... + 458³) = 2401369740².

458² = 209764, 20 + 9 + 7 + 64 = 10²,
458² = 209764, 209 + 76 + 4 = 17².

Page of Squares : First Upload March 7, 2005 ; Last Revised July 21, 2006
by Yoshio Mimura, Kobe, Japan

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459

The smallest squares containing k 459's :
34596 = 186²,
459459225 = 21435²,
145964599459561 = 12081581².

459² = (1² + 2)(265² + 2) = (1² + 8)(3² + 8)(37² + 8).

3-by-3 magic squares consisting of different squares with constant 459²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(2, 134, 439, 286, 343, 106, 359, 274, 82),	(7, 214, 406, 266, 329, 178, 374, 238, 119),
(23, 106, 446, 314, 329, 62, 334, 302, 89),	(26, 247, 386, 281, 314, 182, 362, 226, 169),
(34, 103, 446, 238, 386, 71, 391, 226, 82),	(34, 119, 442, 233, 386, 86, 394, 218, 89),
(34, 170, 425, 295, 334, 110, 350, 265, 134),	(34, 238, 391, 313, 274, 194, 334, 281, 142),
(51, 204, 408, 264, 348, 141, 372, 219, 156),	(55, 266, 370, 310, 295, 166, 334, 230, 215),
(71, 166, 422, 202, 394, 121, 406, 167, 134),	(86, 169, 418, 194, 398, 121, 407, 154, 146)

459² = 210681, 21 + 0 + 6 + 8 + 1 = 6²,
459² = 210681, 210 + 6 + 8 + 1 = 15².

459² = 210681 appears in the decimal expressions of π and e:
  π = 3.14159•••210681••• (from the 77898th digit),
  e = 2.71828•••210681••• (from the 1019th digit),
  (210681 is the second 6-digit square in the expression of e.)

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

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