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490

The smallest squares containing k 490's :
4900 = 70²,
4904901225 = 70035²,
4905914904900 = 2214930².

490² = (1² + 6)(8² + 6)(22² + 6).

Komachi Square Sums : 490² = 7² + 9² + 32² + 61² + 485² = 1² + 2² + 6² + 9² + 53² + 487².

490^k + 1022^k + 1274^k + 6818^k are squares for k = 1,2,3 (98², 7028², 565852²).
154^k + 490^k + 1708^k + 3577^k are squares for k = 1,2,3 (77², 3997², 225547²).
17^k + 91^k + 299^k + 377^k are squares for k = 1,2,3 (28², 490², 9004²).

(1³ + 2³ + ... + 110³)(111³ + 112³ + ... + 174³)(175³ + 176³ + ... + 490³) = 10160605224000².

Page of Squares : First Upload April 4, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

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491

The smallest squares containing k 491's :
491401 = 701²,
7491594916 = 86554²,
2491491774916 = 1578446².

Komachi Fractions : 243 / 6509187 = (3 / 491)², 432 / 6509187 = (4 / 491)².

1 / 491 = 0.0020366598778004073..., the sum of the squares of its digits is 491.

491² = 241081, 2 + 4 + 1 + 0 + 8 + 1 = 4².

3-by-3 magic squares consisting of different squares with constant 491²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(9, 166, 462, 318, 354, 121, 374, 297, 114),	(14, 138, 471, 327, 354, 94, 366, 311, 102),
(23, 66, 486, 126, 471, 58, 474, 122, 39),	(39, 306, 382, 338, 294, 201, 354, 247, 234),
(42, 121, 474, 201, 438, 94, 446, 186, 87),	(66, 185, 450, 310, 366, 105, 375, 270, 166),
(86, 198, 441, 234, 409, 138, 423, 186, 166)

Page of Squares : First Upload April 4, 2005 ; Last Revised May 11, 2009
by Yoshio Mimura, Kobe, Japan

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492

The smallest squares containing k 492's :
49284 = 222²,
44924922025 = 211955²,
34924920492361 = 5909731².

492² = 242064, a square consisting of even digits.

492²± 5 are primes.

492² = 4! + 5! + 8! + 8! + 8! + 8! + 8! + 8!.

492² = (2² + 8)(142² + 8).

(1³ + 2³ + ... + 203³)(204³)(205³ + 206³ + ... + 492³) = 7207266294144².

492² = 242064, 2 + 4 + 20 + 6 + 4 = 6²,
492² = 242064, 24 + 2 + 0 + 6 + 4 = 6².

Page of Squares : First Upload April 4, 2005 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

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493

The smallest squares containing k 493's :
1493284 = 1222²,
24934936464 = 157908²,
949394934936009 = 30812253².

Komachi Square Sum : 493² = 1² + 3² + 5² + 7² + 62² + 489².

(373 / 493)² = 0.572431896... (Komachic).

(1² + 2² + ... + 57²)(58² + 59² + ... + 275²)(276² + 277³ + ... + 493²) = 3805638535².

(1³ + 2³ + ... + 399³)(400³ + 401³ + ... + 493³) = 7339924200².

3-by-3 magic squares consisting of different squares with constant 493²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(0, 232, 435, 340, 315, 168, 357, 300, 160),	(12, 69, 488, 272, 408, 51, 411, 268, 48),
(12, 187, 456, 216, 408, 173, 443, 204, 72),	(27, 204, 448, 308, 357, 144, 384, 272, 147),
(51, 232, 432, 272, 348, 219, 408, 261, 92)

493² = 243049, 2 + 4 + 30 + 4 + 9 = 7²,
493² = 243049, 243 + 0 + 4 + 9 = 16².

493² = 243049 appears in the decimal expression of e:
  e = 2.71828•••243049••• (from the 19675th digit).

Page of Squares : First Upload April 4, 2005 ; Last Revised May 11, 2009
by Yoshio Mimura, Kobe, Japan

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494

The smallest squares containing k 494's :
44944 = 212²,
4941949401 = 70299²,
494849456924944 = 22245212².

494 = (1² + 2² + 3² + ... + 19²) / (1² + 2²).

(219 / 494)² = 0.196532478... (Komachic).

178^k + 242^k + 494^k + 850^k are squares for k = 1,2,3 (42², 1028², 27468²).

494² = 244036, 2² + 4² + 4² + 0² + 3² + 6² = 9²,
494² = 244036, 2² + 4² + 40² + 36² = 54²,
494² = 244036, 24 + 4 + 0 + 36 = 8²,
494² = 244036, 24 + 40 + 36 = 10².

Page of Squares : First Upload April 4, 2005 ; Last Revised March 8, 2011
by Yoshio Mimura, Kobe, Japan

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495

The smallest squares containing k 495's :
495616 = 704²,
10549549521 = 102711²,
449554953749521 = 21202711².

495 = (1² + 2² + 3² + ... + 27²) / (1² + 2² + 3²).

495² = 44² + 45² + 46² + 47² + ... + 93².

A, B, C, A + B, B + C, and C + A are squares for (A, B, C) = (495², 4888², 8160²).

Komachi equation: 495² = 98 + 7654*32 - 1.

3-by-3 magic squares consisting of different squares with constant 495²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(5, 70, 490, 238, 430, 59, 434, 235, 38),	(10, 155, 470, 197, 430, 146, 454, 190, 53),
(10, 155, 470, 245, 410, 130, 430, 230, 85),	(10, 158, 469, 245, 406, 142, 430, 235, 70),
(10, 245, 430, 283, 350, 206, 406, 250, 133),	(14, 250, 427, 277, 350, 214, 410, 245, 130),
(15, 120, 480, 192, 444, 105, 456, 183, 60),	(15, 120, 480, 312, 375, 84, 384, 300, 87),
(43, 274, 410, 326, 293, 230, 370, 290, 155),	(50, 101, 482, 130, 470, 85, 475, 118, 74),
(50, 181, 458, 325, 358, 106, 370, 290, 155),	(50, 302, 389, 325, 314, 202, 370, 235, 230),
(70, 235, 430, 290, 370, 155, 395, 230, 190),	(70, 235, 430, 298, 370, 139, 389, 230, 202),
(70, 235, 430, 298, 386, 85, 389, 202, 230),	(85, 178, 454, 230, 421, 122, 430, 190, 155),
(85, 230, 430, 298, 370, 139, 386, 235, 202),	(91, 262, 410, 338, 266, 245, 350, 325, 130),
(118, 251, 410, 274, 382, 155, 395, 190, 230)

495² = 245025, 2 + 4 + 5 + 0 + 25 = 6²,
495² = 245025, 2 + 4 + 50 + 25 = 9²,
495² = 245025, 24 + 5 + 0 + 2 + 5 = 6²,
495² = 245025, 24 + 50 + 2 + 5 = 9²,
495² = 245025, 24³ + 50³ + 25³ = 393².

Page of Squares : First Upload April 4, 2005 ; Last Revised May 11, 2009
by Yoshio Mimura, Kobe, Japan

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496

The smallest squares containing k 496's :
18496 = 136²,
374964496 = 19364²,
39496649652496 = 6284636².

The squares which begin with 496 and end in 496 are
49668362496 = 222864²,   496128644496 = 704364²,   496511892496 = 704636²,
496833258496 = 704864²,   4960134762496 = 2227136²,...

496² = 246016, 2 * 4 * 60 + 16 = 496.

496^k + 5270^k + 7161^k + 11098^k are squares for k = 1,2,3 (155², 14229², 1371347²).

496² = 246016, 2 + 46 + 0 + 16 = 8²,
496² = 246016, 24 + 60 + 16 = 10².

Page of Squares : First Upload April 4, 2005 ; Last Revised March 8, 2011
by Yoshio Mimura, Kobe, Japan

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497

The smallest squares containing k 497's :
49729 = 223²,
10749749761 = 103681²,
135074974974976 = 11622176².

497 = (1² + 2² + 3² + ... + 35²) / (1² + 2² + 3² + 4²).

3-by-3 magic squares consisting of different squares with constant 497²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(12, 223, 444, 304, 348, 183, 393, 276, 128),	(21, 182, 462, 322, 357, 126, 378, 294, 133),
(33, 108, 484, 276, 407, 72, 412, 264, 87),	(33, 164, 468, 204, 432, 137, 452, 183, 96),
(36, 192, 457, 268, 393, 144, 417, 236, 132),	(47, 192, 456, 312, 344, 177, 384, 303, 88),
(48, 201, 452, 228, 412, 159, 439, 192, 132),	(52, 279, 408, 348, 312, 169, 351, 268, 228)

497² = 247009, 247 + 0 + 0 + 9 = 16².

Page of Squares : First Upload April 4, 2005 ; Last Revised May 11, 2009
by Yoshio Mimura, Kobe, Japan

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498

The smallest squares containing k 498's :
498436 = 706²,
54986498064 = 234492²,
1149849849838096 = 33909436².

498² = 248004, a square consisting of even digits.

498² = 25³ + 30³ + 59³.

The square root of 498 is 22.315913, 22² = 3² + 15² + 9² + 13².

498^k + 942^k + 3138^k + 3522^k are squares for k = 1,2,3 (90², 4836², 274860²).
222^k + 354^k + 370^k + 498^k are squares for k = 1,2,3 (38², 748², 15148²).

Komachi equation: 498² = 1² - 23² * 4² * 5² + 678² - 9².

The 4-by-4 magic square consisting of different squares with constant 498:

498² = 248004, 2² + 4² + 8² + 0² + 0² + 4² = 10²,
498² = 248004, 2⁴ + 4⁴ + 8⁴ + 0⁴ + 0⁴ + 4⁴ = 68²,
498² = 248004, 2⁶ + 4⁶ + 8⁶ + 0⁶ + 0⁶ + 4⁶ = 520²,
498² = 248004, 2⁸ + 4⁸ + 8⁸ + 0⁸ + 0⁸ + 4⁸ = 4112²,
498² = 248004, 24 + 8 + 0 + 0 + 4 = 6²,
498² = 248004, 24³ + 8³ + 0³ + 0³ + 4³ = 120².

Page of Squares : First Upload April 4, 2005 ; Last Revised March 8, 2011
by Yoshio Mimura, Kobe, Japan

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499

The smallest squares containing k 499's :
499849 = 707²,
5049949969 = 71063²,
499774997499904 = 22355648².

499² = 249001, 2 + 4 + 9 + 0 + 0 + 1 = 4²,
499² = 249001, 24 + 9001 = 95².

3-by-3 magic squares consisting of different squares with constant 499²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(6, 113, 486, 239, 426, 102, 438, 234, 49),	(6, 129, 482, 274, 402, 111, 417, 266, 66),
(6, 207, 454, 321, 346, 162, 382, 294, 129),	(22, 111, 486, 351, 342, 94, 354, 346, 63),
(31, 258, 426, 294, 354, 193, 402, 239, 174),	(50, 174, 465, 255, 410, 126, 426, 225, 130),
(78, 321, 374, 346, 234, 273, 351, 302, 186)

Page of Squares : First Upload April 4, 2005 ; Last Revised May 11, 2009
by Yoshio Mimura, Kobe, Japan

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