---

500

The smallest squares containing k 500's :
2500 = 50²,
150062500 = 12250²,
250050002500 = 500050².

The squares which begin with 500 and end in 500 are
5005562500 = 70750²,   50019322500 = 223650²,   50064062500 = 223750²,
500061122500 = 707150²,   500202562500 = 707250²,...

1002^k + 58116^k + 82665^k + 109218^k are squares for k = 1,2,3 (501², 148797², 45431181²).

Komachi equation: 500² = 9² * 8² * 7² - 65² - 4² + 3² / 2² * 10².

500² = 50³ + 50³.

500² + 501² + 502² + 503² + ... + 1566² = 35211²,
500² + 501² + 502² + 503² + ... + 156524² = 35753030².

(1³ + 2³ + ... + 48³)(49³ + 50³ + ... + 84³)(85³ + 86³ + ... + 500³) = 496288840320².

Page of Squares : First Upload April 11, 2005 ; Last Revised March 15, 2011
by Yoshio Mimura, Kobe, Japan

---

501

The smallest squares containing k 501's :
50176 = 224²,
35015017129 = 187123²,
5010501501730816 = 70784896².

501² = 251001, 2 + 5 * 100-1 = 501.

501² = 251001, and 25 = 5², 100 = 10², 1 = 1².

501² + 502² + 503² + 504² + ... + 718² = 9047²,
501² + 502² + 503² + 504² + ... + 10100² = 586040²,
501² + 502² + 503² + 504² + ... + 28078² = 2716433².

501² = 251001, 2 + 5 + 1 + 0 + 0 + 1 = 3².

3-by-3 magic squares consisting of different squares with constant 501²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(8, 196, 461, 331, 344, 152, 376, 307, 124),	(9, 282, 414, 342, 306, 201, 366, 279, 198),
(13, 296, 404, 344, 299, 208, 364, 272, 211),	(19, 68, 496, 188, 461, 56, 464, 184, 43),
(19, 128, 484, 352, 341, 104, 356, 344, 77),	(37, 124, 484, 244, 428, 91, 436, 229, 92),
(42, 234, 441, 279, 378, 174, 414, 231, 162),	(44, 268, 421, 293, 356, 196, 404, 229, 188),
(56, 256, 427, 332, 301, 224, 371, 308, 136),	(76, 173, 464, 211, 436, 128, 448, 176, 139),
(76, 296, 397, 328, 331, 184, 371, 232, 244),	(85, 224, 440, 320, 365, 124, 376, 260, 205)

501² = 251001 appears in the decimal expression of π:
  π = 3.14159•••251001••• (from the 16903rd digit),
  (251001 is the tenth 6-digit square in the expression of π.)

Page of Squares : First Upload April 11, 2005 ; Last Revised May 21, 2009
by Yoshio Mimura, Kobe, Japan

---

502

The smallest squares containing k 502's :
65025 = 255²,
9265025025 = 96255²,
3525025025025 = 1877505².

Komachi Square Sums : 502² = 7² + 85² + 263² + 419² = 1² + 2² + 6² + 7² + 83² + 495².

502² = 252004, 2² + 5² + 2² + 0² + 0² + 4² = 7²,
502² = 252004, 25 + 20 + 0 + 4 = 7²,
502² = 252004, 252 + 0 + 0 + 4 = 16².

502² = 252004 appears in the decimal expression of π
  π = 3.14159•••252004••• (from the 71018th digit).

Page of Squares : First Upload April 11, 2005 ; Last Revised August 1, 2006
by Yoshio Mimura, Kobe, Japan

---

503

The smallest squares containing k 503's :
1503076 = 1226²,
5035037764 = 70958²,
503705036503876 = 22443374².

Komachi Fraction : 576 / 9108324 = (4 / 503)².

503² = 2⁴ + 8⁴ + 11⁴ + 22⁴ = 14⁴ + 16⁴ + 16⁴ + 17⁴.

503 is the second integer which is the sum of a square and a primr in 9 ways :
2² + 499, 4² + 487, 6² + 467, 8² + 439, 12² + 359, 14² + 307, 18² + 179, 20² + 103, 22² + 19.

3-by-3 magic squares consisting of different squares with constant 503²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(18, 222, 451, 262, 381, 198, 429, 242, 102),	(35, 222, 450, 330, 350, 147, 378, 285, 170),
(42, 134, 483, 163, 462, 114, 474, 147, 82),	(42, 154, 477, 306, 387, 98, 397, 282, 126),
(46, 243, 438, 318, 354, 163, 387, 262, 186),	(51, 242, 438, 298, 339, 222, 402, 282, 109),
(54, 147, 478, 282, 406, 93, 413, 258, 126),	(82, 291, 402, 318, 282, 269, 381, 298, 138)

503² = 253009, 2 + 53 + 0 + 0 + 9 = 8²,
503² = 253009, 25 + 30 + 0 + 9 = 8².

Page of Squares : First Upload April 11, 2005 ; Last Revised May 21, 2009
by Yoshio Mimura, Kobe, Japan

---

504

The smallest squares containing k 504's :
5041 = 71²,
1350415504 = 36748²,
15048565045504 = 3879248².

The squares which begin with 504 and end in 504 are
504452221504 = 710248²,   504457903504 = 710252²,   5041138581504 = 2245248²,
5041156543504 = 2245252²,   5043384079504 = 2245748²,...

504² = 254016, a square with different digits.

504² = (2² - 1)(3² - 1)(8² - 1)(13² - 1) = (5² - 1)(8² - 1)(13² - 1).

Cubic Polynomial :
(X + 504²)(X + 999²)(X + 1472²) = X³ + 1849²X² + 1722312²X + 741146112².

504² = 28³ + 29³ + 30³ + 31³ + 32³ + ... + 35³.

Komachi equations:
504² = 12 * 3 / 4 * 56 * 7 * 8 * 9,
504² = 1² * 2² - 3² + 4² + 5² - 6² + 7² * 8² * 9² = 12² - 3² * 4² + 567² * 8² / 9²
 = 12² / 3² - 4² + 567² * 8² / 9² = 12² / 3² / 4² * 567² * 8² / 9²
 = - 1² * 2² + 3² - 4² - 5² + 6² + 7² * 8² * 9² = - 12² + 3² * 4² + 567² * 8² / 9²
 = - 12² / 3² + 4² + 567² * 8² / 9² = 9² * 8² * 7² + 6² - 5² - 4² + 3² - 2² * 1²
 = 9² * 8² * 7² + 6² - 5² - 4² + 3² - 2² / 1² = 9² * 8² * 7² - 6² + 5² + 4² - 3² + 2² * 1²
 = 9² * 8² * 7² - 6² + 5² + 4² - 3² + 2² / 1² = 98² / 7² / 6² * 5² * 432² / 10².

(1 + 2 + 3 + 4 + 5 + 6 + 7)(8)(9 + 10 + 11 + 12)(13 + 14) = 504²,
(1)(2)(3 + 4)(5 + 6 + 7 + 8 + 9 + 10 + 11)(12)(13 + 14) = 504²,
(1 + 2)(3 + 4)(5 + 6 + 7)(8)(9 + 10 + 11 + 12 + 13 + 14 + 15) = 504²,
(1 + 2)(3 + 4 + 5 + 6)(7)(8)(9 + 10 + 11 + 12 + 13 + 14 + 15) = 504²,
(1)(2 + 3 + 4)(5 + 6 + 7 + 8 + 9 + 10 + 11)(12)(13 + 14 + 15) = 504²,
(1 + 2 + 3)(4 + 5 + ... + 12)(13 + 14 + ... + 36) = 504².

(1³ + 2³ + ... + 224³)(225³ + 226³ + ... + 504³) = 3143448000².

504² = 254016, 25 + 4 + 0 + 1 + 6 = 6²,
504² = 254016, 25 + 40 + 16 = 9².

Page of Squares : First Upload April 11, 2005 ; Last Revised December 7, 2013
by Yoshio Mimura, Kobe, Japan

---

505

The smallest squares containing k 505's :
150544 = 388²,
950550561 = 30831²,
5055050595056641 = 71098879².

505² is the fourth square which is the sum of 10 sixth powers.

505² = 18³ + 40³ + 57³.

1² + 9² + 17² + ... + (8x + 1)² + ... + 505² = 2344².

505² = 255025, a square consisting of just 3 kinds of digits.

3-by-3 magic squares consisting of different squares with constant 505²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(0, 100, 495, 303, 396, 80, 404, 297, 60),	(15, 168, 476, 280, 399, 132, 420, 260, 105),
(17, 156, 480, 240, 420, 145, 444, 233, 60),	(28, 96, 495, 180, 465, 80, 471, 172, 60),
(39, 152, 480, 352, 336, 135, 360, 345, 80),	(44, 105, 492, 192, 460, 81, 465, 180, 80),
(72, 271, 420, 305, 360, 180, 396, 228, 215),	(80, 264, 423, 345, 280, 240, 360, 327, 136),
(108, 215, 444, 240, 420, 145, 431, 180, 192)

505² = 255025, 2 + 5 + 50 + 2 + 5 = 8²,
505² = 255025, 2 + 55 + 0 + 2 + 5 = 8²,
505² = 255025, 25 + 50 + 25 = 10².

505² = 255025 appears in the decimal expression of π:
  π = 3.14159•••255025••• (from the 1743rd digit),
  (255025 is the first 6-digit square in the expression of π.)

Page of Squares : First Upload April 11, 2005 ; Last Revised May 21, 2009
by Yoshio Mimura, Kobe, Japan

---

506

The smallest squares containing k 506's :
50625 = 225²,
29506650625 = 171775²,
150607506450625 = 12272225².

506 = 1² + 2² + 3²+ ... + 11².

506 = (1² + 2² + 3² + ... + 115²) / (1² + 2² + 3² + ... + 14²).

506² = 256036, 25*60/3 + 6 = 506.

506² = 256036, 256 = 16² and 36 = 6².

506² = 256036, 2 + 5 + 6 + 0 + 36 = 7²,
506² = 256036, 25 + 60 + 36 = 11².

Page of Squares : First Upload April 11, 2005 ; Last Revised November 25, 2008
by Yoshio Mimura, Kobe, Japan

---

507

The smallest squares containing k 507's :
75076 = 274²,
8050755076 = 89726²,
1150750796507524 = 33922718².

507² = 257049, a square with different digits.

30^k + 57^k + 168^k + 474^k are squares for k = 1,2,3 (27², 507², 10557²).

3-by-3 magic squares consisting of different squares with constant 507²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(2, 71, 502, 281, 418, 58, 422, 278, 41),	(7, 130, 490, 190, 455, 118, 470, 182, 55),
(7, 262, 434, 314, 343, 202, 398, 266, 167),	(14, 182, 473, 302, 377, 154, 407, 286, 98),
(26, 167, 478, 247, 422, 134, 442, 226, 103),	(26, 218, 457, 247, 394, 202, 442, 233, 86),
(26, 313, 398, 338, 286, 247, 377, 278, 194),	(36, 117, 492, 168, 468, 99, 477, 156, 72),
(48, 189, 468, 324, 372, 117, 387, 288, 156),	(50, 218, 455, 343, 350, 130, 370, 295, 182),
(62, 233, 446, 313, 334, 218, 394, 302, 103),	(106, 322, 377, 343, 326, 182, 358, 217, 286)

507² = 257049, 2 + 5 + 7049 = 84²,
507² = 257049, 25 + 7 + 0 + 49 = 9²,
507² = 257049, 25 + 70 + 49 = 12².

Page of Squares : First Upload April 11, 2005 ; Last Revised March 15, 2011
by Yoshio Mimura, Kobe, Japan

---

508

The smallest squares containing k 508's :
508369 = 713²,
9508005081 = 97509²,
4508748750845089 = 67147217².

1 / 508 = 0.00196..., and 196 = 14².

508²± 3 are primes.

508² = 258064, a square with different digits.

508² = 258064, 2 + 5 + 8 + 0 + 6 + 4 = 5²,
508² = 258064, 25 + 80 + 64 = 13².

Page of Squares : First Upload April 11, 2005 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

---

509

The smallest squares containing k 509's :
509796 = 714²,
2509509025 = 50095²,
575090950950916 = 23981054².

(375 / 509)² = 0.542783916... (Komachic).

509² = 259081, a square with different digits.

509² is the 8th square which is the sum of 4 fifth powers : 2⁵ + 9⁵ + 10⁵ + 10⁵.

1 / 509 = 0.00196..., and 196 = 14².

3-by-3 magic squares consisting of different squares with constant 509²:

A2	B2	C2
D2	E2	F2
G2	H2	K2

where (A, B, C, D, E, F, G, H, K) =
(21, 248, 444, 324, 336, 203, 392, 291, 144),	(24, 189, 472, 276, 392, 171, 427, 264, 84),
(24, 288, 419, 356, 309, 192, 363, 284, 216),	(76, 237, 444, 339, 356, 132, 372, 276, 211),
(108, 301, 396, 339, 252, 284, 364, 324, 147)

509² = 259081, 2 + 5 + 9 + 0 + 8 + 1 = 5²,
509² = 259081, 25 + 90 + 81 = 14².

Page of Squares : First Upload April 11, 2005 ; Last Revised May 21, 2009
by Yoshio Mimura, Kobe, Japan

---