Squares:600s

600

The smallest squares containing k 600's :
1600 = 40²,
60016009 = 7747²,
6002206003600 = 2449940².

The squares which begin with 600 and end in 600 are
6000051600 = 77460², 60005401600 = 244960², 60044601600 = 245040²,
60054403600 = 245060², 60093619600 = 245140²,...

10³ + 600 = 40², 10³ - 600 = 20²,
5⁴ + 600 = 35², 5⁴ - 600 = 5².

A cubic polynomial :
(X + 600²)(X + 1440²)(X + 1547²) = X³ + 2197²X² + 2563320²X + 1336608000².

(1 + 2)(3)(4)(5)(6 + 7 + 8 + 9 + 10)(11 + 12 + 13 + 14) = 600²,
(1)(2)(3 + 4 + 5)(6 + 7 + 8 + 9)(10)(11 + 12 + 13 + 14) = 600².

600² = 360000 appears in the decimal expression of e:
e = 2.71828•••360000••• (from the 89294th digit).

Page of Squares : First Upload June 13, 2005 ; Last Revised July 22, 2011
by Yoshio Mimura, Kobe, Japan

601

The smallest squares containing k 601's :
2601 = 51²,
368601601 = 19199²,
2256016016016 = 1502004².

The squares which begin with 601 and end in 601 are
6014157601 = 77551², 60122549601 = 245199², 60172580601 = 245301²,
601091640601 = 775301², 601321151601 = 775449²,...

601² = 361201, a zigzag square.

3-by-3 magic squares consisting of different squares with constant 601²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(9, 124, 588, 308, 504, 111, 516, 303, 56),	(9, 228, 556, 396, 416, 177, 452, 369, 144),
(16, 108, 591, 207, 556, 96, 564, 201, 52),	(16, 327, 504, 408, 376, 231, 441, 336, 232),
(24, 340, 495, 380, 375, 276, 465, 324, 200),	(36, 119, 588, 336, 492, 79, 497, 324, 96),
(36, 196, 567, 399, 432, 124, 448, 369, 156),	(79, 192, 564, 276, 516, 137, 528, 241, 156)

601² = 361201, 36 + 12 + 0 + 1 = 7².

Page of Squares : First Upload June 13, 2005 ; Last Revised June 29, 2009
by Yoshio Mimura, Kobe, Japan

602

The smallest squares containing k 602's :
126025 = 355²,
6022536025 = 77605²,
160260281766025 = 12659395².

602² = 362404, a zigzag square.

602² = 221 x 222 + 223 x 224 + 225 x 226 + 227 x 228 + ... + 233 x 234.

The 4-by-4 magic square consisting of different squares with constant 602:

0²	4²	15²	19²
9²	1²	18²	14²
11²	21²	2²	6²
20²	12²	7²	3²

602² = 362404, 3² + 6² + 2² + 4² + 0² + 4² = 9²,
602² = 362404, 36 + 24 + 0 + 4 = 8²,
602² = 362404, 36² + 2² + 40² + 4² = 54².

602² = 362404 appears in the decimal expression of π:
π = 3.14159•••362404••• (from the 23938th digit).

Page of Squares : First Upload June 13, 2005 ; Last Revised September 9, 2009
by Yoshio Mimura, Kobe, Japan

603

The smallest squares containing k 603's :
256036 = 506²,
28603603876 = 169126²,
5076036036036 = 2253006².

603² = 1! + 2! + 3! + 6! + 9!

603² = 363609, a zigzag square.

603² = 363609, 3 - 6 - 3 + 609 = 603.

603² = 201² + 402² + 402² : 204² + 204² + 102² = 306².

603, 604 and 605 are three consecutive integers having square factors (the 9th case).

24522^k + 96681^k + 102912^k + 139494^k are squares for k = 1,2,3 (603², 199995²,68722101²).

3-by-3 magic squares consisting of different squares with constant 603²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(2, 154, 583, 374, 457, 122, 473, 362, 94),	(3, 312, 516, 408, 381, 228, 444, 348, 213),
(7, 142, 586, 334, 487, 122, 502, 326, 73),	(10, 247, 550, 278, 490, 215, 535, 250, 122),
(12, 213, 564, 291, 492, 192, 528, 276, 93),	(21, 132, 588, 348, 483, 96, 492, 336, 93),
(23, 382, 466, 418, 346, 263, 434, 313, 278),	(25, 122, 590, 422, 425, 70, 430, 410, 103),
(26, 217, 562, 263, 502, 206, 542, 254, 73),	(38, 137, 586, 313, 506, 98, 514, 298, 103),
(58, 298, 521, 329, 422, 278, 502, 311, 122),	(82, 151, 578, 182, 562, 121, 569, 158, 122),
(82, 226, 553, 346, 473, 142, 487, 298, 194),	(89, 322, 502, 382, 359, 298, 458, 362, 151),
(94, 247, 542, 382, 446, 137, 457, 322, 226),	(94, 362, 473, 382, 359, 298, 457, 322, 226),
(158, 359, 458, 382, 422, 199, 439, 238, 338)

603² = 363609, 3 + 6 + 3 + 60 + 9 = 9²,
603² = 363609, 3 + 63 + 6 + 0 + 9 = 9²,
603² = 363609, 36 + 36 + 0 + 9 = 9².

Page of Squares : First Upload June 13, 2005 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

604

The smallest squares containing k 604's :
9604 = 98²,
876041604 = 29598²,
604604266044804 = 24588702².

The squares which begin with 604 and end in 604 are
60467793604 = 245902², 604353869604 = 777402², 604658649604 = 777598²,
6041282241604 = 2457902², 6042245777604 = 2458098²,...

604² = 364816, a zigzag square.

604² = 364816 is an exchangeable square, where 481636 = 694².

A cubic polynomial :
(X + 288²)(X + 604²)(X + 627²) = X³ + 917²X² + 454188²X + 109067904².

604² + 605² + 606² + ... + 16627² = 1237854².

604² = 364816, 3 + 6 + 48 + 1 + 6 = 8²,
604² = 364816, 36 + 4 + 8 + 16 = 8²,
604² = 364816, 3 + 6 + 4 + 81 + 6 = 10²,
604² = 364816, 36 + 48 + 16 = 10².

Page of Squares : First Upload June 13, 2005 ; Last Revised August 21, 2006
by Yoshio Mimura, Kobe, Japan

605

The smallest squares containing k 605's :
60516 = 246²,
26050605604 = 161402²,
605360506605625 = 24604075².

(578 / 605)² = 0.912735468... (Komachic).

605² = (28 + 29 + 30 + ... + 38)² + (39 + 40 + 41 + ... + 49)².

3-by-3 magic squares consisting of different squares with constant 605²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(12, 216, 565, 309, 488, 180, 520, 285, 120),	(20, 75, 600, 240, 552, 61, 555, 236, 48),
(20, 240, 555, 420, 405, 160, 435, 380, 180),	(20, 300, 525, 420, 371, 228, 435, 372, 196),
(21, 228, 560, 272, 504, 195, 540, 245, 120),	(36, 245, 552, 405, 420, 160, 448, 360, 189),
(51, 268, 540, 380, 435, 180, 468, 324, 205),	(56, 267, 540, 360, 420, 245, 483, 344, 120),
(72, 160, 579, 204, 555, 128, 565, 180, 120),	(75, 300, 520, 344, 408, 285, 492, 331, 120),
(75, 300, 520, 376, 432, 195, 468, 299, 240)

605² = 366025, 36 + 6 + 0 + 2 + 5 = 7²,
605² = 366025, 36 + 60 + 25 = 11².

(1³ + ... + 50³)(51³ + ... + 374³)(375³ + ... + 605³) = 15140941200000²,
(1³ + ... + 164³)(165³ + ... + 374³)(375³ + ... + 605³) = 157679193672000²,
(1³ + 2³ + ... + 267³)(268³ + ... + 374³)(375³ + ... + 605³) = 365473764089280².

Page of Squares : First Upload June 13, 2005 ; Last Revised June 29, 2009
by Yoshio Mimura, Kobe, Japan

606

The smallest squares containing k 606's :
606841 = 779²,
360696064 = 18992²,
166061676066064 = 12886492².

606² = 367236, 3 + 67 / 2 * 3 * 6 = 606.

606² = 4³ + 21³ + 71³.

210^k + 606^k + 762^k + 1338^k are squares for k = 1,2,3 (54², 1668², 55404²).

The 4-by-4 magic square consisting of different squares with constant 606:

1²	3²	14²	20²
5²	23²	4²	²
16²	2²	15²	11²
18²	8²	13²	7²

606² = 367236, 3 + 67 + 2 + 3 + 6 = 9²,
606² = 367236, 36 + 7 + 2 + 36 = 9²,
606² = 367236, 36 + 72 + 36 = 12².

Page of Squares : First Upload June 13, 2005 ; Last Revised March 17, 2011
by Yoshio Mimura, Kobe, Japan

607

The smallest squares containing k 607's :
1607824 = 1268²,
19607560729 = 140027²,
607607660790601 = 24649699².

607² = 6⁴ + 11⁴ + 12⁴ + 24⁴.

Komachi equations:
607² = 9² + 8² * 76² - 54² / 3² * 2² * 1² = 9² + 8² * 76² - 54² / 3² * 2² / 1².

3-by-3 magic squares consisting of different squares with constant 607²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(6, 318, 517, 347, 426, 258, 498, 293, 186),	(22, 171, 582, 309, 498, 158, 522, 302, 69),
(22, 171, 582, 342, 498, 59, 501, 302, 162),	(22, 258, 549, 342, 459, 202, 501, 302, 162),
(27, 346, 498, 402, 363, 274, 454, 342, 213),	(30, 315, 518, 357, 10, 270, 490, 318, 165),
(59, 162, 582, 342, 491, 102, 498, 318, 139),	(102, 238, 549, 342, 459, 202, 491, 318, 162),
(117, 238, 546, 294, 507, 158, 518, 234, 213),	(126, 373, 462, 402, 294, 347, 437, 378, 186)

607² = 368449, 36 + 84 + 49 = 13².

Page of Squares : First Upload January 25, 2006 ; Last Revised June 22, 2010
by Yoshio Mimura, Kobe, Japan

608

The smallest squares containing k 608's :
6084 = 78²,
6086089 = 2467²,
160896083608324 = 12684482².

A cubic polynomial :
(X + 608²)(X + 1311²)(X + 2124²) = X³ + 2569²X² + 3171252²X + 1693014912².

Komachi equations:
608² = 9² + 8² * 76² - 54² / 3² / 2² */ 1² = 9² * 8² * 76² / 54² * 3² * 2² */ 1²
= - 9² + 8² * 76² + 54² / 3² / 2² */ 1²,
608² = - 12³ / 3³ * 4³ + 56³ / 7³ + 8³ * 9³.

608² = 369664, 3 * 6 + 9 * 66 - 4 = 608,
608² = 369664, 36 + 96 * 6 - 4 = 608.

6086089 = 2467².

608² = 369664, 3 + 6 + 96 + 64 = 13²,
608² = 369664, 36 + 96 + 64 = 14².

Page of Squares : First Upload June 13, 2005 ; Last Revised June 22, 2010
by Yoshio Mimura, Kobe, Japan

609

The smallest squares containing k 609's :
10609 = 103²,
4609609 = 2147²,
26609059609609 = 5158397².

The squares which begin with 609 and end in 609 are
60936403609 = 246853², 60958128609 = 246897², 609019477609 = 780397²,
609341043609 = 780603², 609409738609 = 780647²,...

609²± 2 are primes.

4609609 = 2147².

609² = 4⁴ + 15⁴ + 20⁴ + 20⁴.

609² = 203² + 406² + 406² : 604² + 604² + 302² = 906².

32² + 33² + 34² + 35² + 36² + ... + 609² = 8687².

609^k + 4524^k + 6960^k + 8932^k are squares for k = 1,2,3 (145², 12209², 1068911²).

Komachi equation: 609² = 9² * 87² * 6² / 54² / 3² * 21².

3-by-3 magic squares consisting of different squares with constant 609²:

A²	B²	C²
D²	E²	F²
G²	H²	K²

where (A, B, C, D, E, F, G, H, K) =
(4, 176, 583, 209, 548, 164, 572, 199, 64),	(8, 289, 536, 376, 424, 223, 479, 328, 184),
(16, 185, 580, 340, 484, 145, 505, 320, 116),	(17, 76, 604, 124, 592, 71, 596, 121, 32),
(25, 184, 580, 316, 500, 145, 520, 295, 116),	(44, 167, 584, 428, 424, 89, 431, 404, 148),
(47, 116, 596, 244, 551, 88, 556, 232, 89),	(52, 236, 559, 391, 416, 212, 464, 377, 116),
(54, 258, 549, 309, 486, 198, 522, 261, 174),	(68, 239, 556, 281, 508, 184, 536, 236, 167),
(69, 234, 558, 306, 498, 171, 522, 261, 174),	(71, 256, 548, 388, 404, 239, 464, 377, 116),
(76, 263, 544, 292, 496, 199, 529, 236, 188),	(76, 376, 473, 401, 388, 244, 452, 281, 296),
(121, 332, 496, 416, 316, 313, 428, 401, 164),	(124, 344, 487, 412, 409, 184, 431, 292, 316),
(152, 319, 496, 344, 464, 193, 479, 232, 296)

609² = 370881 appears in the decimal expression of e:
e = 2.71828•••370881••• (from the 103913rd digit).

Page of Squares : First Upload June 13, 2005 ; Last Revised December 29, 2013
by Yoshio Mimura, Kobe, Japan