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4500 - 4599

4500

45002 = 1203 + 2103 + 2103.

Page of Squares : First Upload September 1, 2008 ; Last Revised September 1, 2008
by Yoshio Mimura, Kobe, Japan

4501

45012 = 20259001, 2 + 0 - 2 + 5 * 900 + 1 = 2 + 0 * 2 + 5 * 900 - 1 = 2 * 0 + 2 + 5 * 900 - 1
  = 2 * 0 * 2 + 5 * 900 + 1 = 20 / 2 * 5 * 90 + 0 + 1 = 20 * 25 * 9 + 0 + 0 + 1
  = 20 * 25 * 9 + 0 * 0 + 1 = 4501.

45015 = 1847332473952522501 : 12 + 82 + 42 + 72 + 32 + 32 + 22 + 42 + 72 + 392 + 52 + 22 + 522 + 22 + 52 + 02 + 12 = 12 + 82 + 42 + 72 + 32 + 32 + 22 + 42 + 72 + 392 + 522 + 52 + 22 + 22 + 52 + 02 + 12 = 4501.

Page of Squares : First Upload September 18, 2007 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

4502

45022 = 20268004, a square with even digits.

Page of Squares : First Upload September 18, 2007 ; Last Revised September 18, 2007
by Yoshio Mimura, Kobe, Japan

4503

45032 = 383 + 1453 + 2583.

Page of Squares : First Upload September 1, 2008 ; Last Revised September 1, 2008
by Yoshio Mimura, Kobe, Japan

4504

The square root of 4504 is 67.1118..., and 67 = 12 + 12 + 12 + 82.

Page of Squares : First Upload September 18, 2007 ; Last Revised September 18, 2007
by Yoshio Mimura, Kobe, Japan

4507

45075 = 1859678158261539307 : 12 + 82 + 52 + 92 + 62 + 72 + 82 + 152 + 82 + 22 + 612 + 52 + 32 + 92 + 32 + 02 + 72 = 4507.

Page of Squares : First Upload December 8, 2008 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

4508

45082 = 633 + 733 + 2703 = 633 + 1773 + 2443 = 144 + 284 + 564 + 564.

45082 = (46 + 47 + 48 + 49 + 50 + 51 + 52)2 + (53 + 54 + 55 + 56 + 57 + 58 + 59)2 + (60 + 61 + 62 + 63 + 64 + 65 + 66)2 + ... + (200 + 201 + 202 + 203 + 204 + 205 + 206)2.

Page of Squares : First Upload September 18, 2007 ; Last Revised September 1, 2008
by Yoshio Mimura, Kobe, Japan

4509

45092 = 453 + 1353 + 2613.

Page of Squares : First Upload September 1, 2008 ; Last Revised September 1, 2008
by Yoshio Mimura, Kobe, Japan

4511

45112 = 1652 + 1672 + 1692 + 1712 + ... + 5012.

Page of Squares : First Upload September 18, 2007 ; Last Revised September 18, 2007
by Yoshio Mimura, Kobe, Japan

4512

45122 = 293 + 1633 + 2523.

The square root of 4512 is 67.1714..., and 67 = 12 + 72 + 12 + 42.

Page of Squares : First Upload September 18, 2007 ; Last Revised September 18, 2007
by Yoshio Mimura, Kobe, Japan

4513

45132 = 263 + 1503 + 2573 = 174 + 284 + 564 + 564.

Page of Squares : First Upload September 1, 2008 ; Last Revised September 1, 2008
by Yoshio Mimura, Kobe, Japan

4514

45142 = 1373 + 1943 + 2193.

Page of Squares : First Upload September 1, 2008 ; Last Revised September 1, 2008
by Yoshio Mimura, Kobe, Japan

4519

45192 = 253 + 393 + 2733.

Page of Squares : First Upload September 1, 2008 ; Last Revised September 1, 2008
by Yoshio Mimura, Kobe, Japan

4521

45212 = (52 + 8)(7872 + 8).

Page of Squares : First Upload December 28, 2013 ; Last Revised December 28, 2013
by Yoshio Mimura, Kobe, Japan

4522

45222 = 174 + 174 + 194 + 674.

45222 = 20448484, a square with even digits.

45222 = 20448484, 22 + 02 + 42 + 42 + 82 + 42 + 82 + 42 = 142.

Page of Squares : First Upload September 18, 2007 ; Last Revised September 1, 2008
by Yoshio Mimura, Kobe, Japan

4523

The square root of 4523 is 67.25325..., and 67 = 22 + 52 + 32 + 22 + 52.

Page of Squares : First Upload September 18, 2007 ; Last Revised September 18, 2007
by Yoshio Mimura, Kobe, Japan

4524

609k + 4524k + 6960k + 8932k are squares for k = 1,2,3 (1452, 122092, 10689112).
3190k + 4524k + 13137k + 20358k are squares for k = 1,2,3 (2032, 248532, 32908332).

Page of Squares : First Upload June 14, 2011 ; Last Revised June 14, 2011
by Yoshio Mimura, Kobe, Japan

4525

1/4525 = 0.0002209... and 2209 = 472. (the same for 45262).

Page of Squares : First Upload September 18, 2007 ; Last Revised September 18, 2007
by Yoshio Mimura, Kobe, Japan

4527

45272 = 64 + 244 + 334 + 664 = 424 + 484 + 484 + 514.

Page of Squares : First Upload September 1, 2008 ; Last Revised September 1, 2008
by Yoshio Mimura, Kobe, Japan

4531

A cubic polynomial:
(X + 8282)(X + 26882)(X + 45312) = X3 + 53332X2 + 129369482X + 100844835842.

Page of Squares : First Upload September 18, 2007 ; Last Revised September 18, 2007
by Yoshio Mimura, Kobe, Japan

4533

45335 = 1913941217714857893 : 192 + 132 + 92 + 42 + 122 + 12 + 72 + 72 + 142 + 82 + 572 + 82 + 92 + 32 = 4533.

Page of Squares : First Upload December 8, 2008 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

4535

45355 = 1918167184754084375 : 12 + 92 + 12 + 82 + 12 + 62 + 72 + 12 + 82 + 472 + 52 + 42 + 02 + 82 + 432 + 72 + 52 = 4535.

Page of Squares : First Upload December 8, 2008 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

4536

45362 = 1983 + 2343 = 33 + 1893 + 2403 = 123 + 143 + 2743 = 723 + 2103 + 2223.

Komachi equations:
45362 = 122 * 32 / 42 + 5672 * 82 - 92 = 122 * 32 / 42 * 5672 * 82 / 92
 = - 122 * 32 / 42 + 5672 * 82 + 92.

45364 = 423342805487616, and 42 + 22 + 32 + 32 + 422 + 82 + 02 + 52 + 482 + 72 + 62 + 162 = 4536,
45364 = 423342805487616, and 422 + 32 + 32 + 42 + 22 + 82 + 02 + 52 + 482 + 72 + 62 + 162 = 4536.

Page of Squares : First Upload September 1, 2008 ; Last Revised October 13, 2010
by Yoshio Mimura, Kobe, Japan

4537

45372 = 283 + 603 + 2733.

45372 = 20584369, a square with different digits.

45374 = 423716247128161, and 422 + 32 + 72 + 12 + 62 + 22 + 472 + 122 + 82 + 162 + 12 = 4537,
45374 = 423716247128161, and 422 + 32 + 72 + 162 + 22 + 472 + 122 + 82 + 12 + 62 + 12 = 4537.

Loop of length 35 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
4537 - 3394 - 9925 - 10426 - ... - 5493 - 11565 - 4451 - 4537
(Note f(4537) = 452 + 372 = 3394,   f(3394) = 332 + 942 = 9925, etc. See 37)

Page of Squares : First Upload September 18, 2007 ; Last Revised December 1, 2008
by Yoshio Mimura, Kobe, Japan

4539

Komachi equation: 45392 = 122 * 342 / 562 * 72 * 892.

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

4542

45422 = 20629764, 2 + 0 + 6 * 2 * 9 * 7 * 6 + 4 = 2 + 0 + 6 + 4 * 7 * 9 * 3 * 6 = 2 + 0 + 6 * 4 * 7 * 9 * 3 + 6 = 4544.

Page of Squares : First Upload September 18, 2007 ; Last Revised September 18, 2007
by Yoshio Mimura, Kobe, Japan

4544

45442 = 643 + 643 + 2723 = 963 + 1443 + 2563.

45445 = 1937276514257076224 : 192 + 32 + 72 + 272 + 62 + 512 + 42 + 22 + 52 + 72 + 02 + 72 + 62 + 22 + 242 = 4544.

Page of Squares : First Upload September 1, 2008 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

4545

45452 = 483 + 1293 + 2643.

45452 = 20657025, 2 + 0 + 65 * 70 - 2 - 5 = 20 + 65 * 70 - 25 = 4545.

Page of Squares : First Upload September 18, 2007 ; Last Revised September 1, 2008
by Yoshio Mimura, Kobe, Japan

4550

45502 = (123 + 124 + 125 + 126 + 127)2 + (128 + 129 + 130 + 131 + 132)2 + (133 + 134 + 135 + 136 + 137)2 + ... + (238 + 239 + 240 + 241 + 242)2.

Page of Squares : First Upload September 18, 2007 ; Last Revised September 18, 2007
by Yoshio Mimura, Kobe, Japan

4553

45532 = 1333 + 1493 + 2473.

Page of Squares : First Upload September 1, 2008 ; Last Revised September 1, 2008
by Yoshio Mimura, Kobe, Japan

4554

45542 = 20738916, a square with different digits.

3553k + 4554k + 5742k + 6600k are squares for k = 1,2,3 (1432, 104832, 7849272).
667k + 2507k + 2553k + 2737k are squares for k = 1,2,3 (922, 45542, 2306442).

Page of Squares : First Upload September 18, 2007 ; Last Revised June 14, 2011
by Yoshio Mimura, Kobe, Japan

4556

45562 = 20757136 appears in the decimal expressions of π:
  π = 3.14159•••20757136••• (from the 12943rd digit)
  (20757136 is the second 8-digit square in the expression of π.)

Page of Squares : First Upload November 4, 2008 ; Last Revised November 4, 2008
by Yoshio Mimura, Kobe, Japan

4558

344k + 1505k + 4558k + 10234k are squares for k = 1,2,3 (1292, 113092, 10816652).

Page of Squares : First Upload June 14, 2011 ; Last Revised June 14, 2011
by Yoshio Mimura, Kobe, Japan

4559

45592 = 1523 + 1853 + 2223.

45592 = 20784481, 2 * 0 + 78 + 4481 = 2 + 0 - 84 + 83 * 56 = 4566.

Page of Squares : First Upload September 18, 2007 ; Last Revised September 1, 2008
by Yoshio Mimura, Kobe, Japan

4560

Komachi equations: 45602 = 92 * 82 * 762 * 52 / 42 / 32 * 22 */ 12.

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

4562

45622 = 30 + 33 + 34 + 35 + 38 + 39 + 310 + 313 + 314 + 315.

Page of Squares : First Upload August 30, 2011 ; Last Revised August 30, 2011
by Yoshio Mimura, Kobe, Japan

4563

45632 = 783 + 2733 = 573 + 1173 + 2673 = 1213 + 1563 + 2483.

Page of Squares : First Upload September 1, 2008 ; Last Revised September 1, 2008
by Yoshio Mimura, Kobe, Japan

4568

45682 = 1403 + 1983 + 2183.

45682 = 20866624, a square with even digits.

45682 = 20866624, 22 + 02 + 82 + 62 + 62 + 62 + 22 + 42 = 142.

Page of Squares : First Upload September 18, 2007 ; Last Revised September 1, 2008
by Yoshio Mimura, Kobe, Japan

4569

45692 = 20875761, 20 + 8 + 757 * 6 - 1 = 4569.

Page of Squares : First Upload September 18, 2007 ; Last Revised September 18, 2007
by Yoshio Mimura, Kobe, Japan

4570

830k + 2010k + 4570k + 4690k are squares for k = 1,2,3 (1102, 69002, 4553002).

Page of Squares : First Upload June 14, 2011 ; Last Revised June 14, 2011
by Yoshio Mimura, Kobe, Japan

4572

45722 = 463 + 1343 + 2643 = 723 + 1993 + 2333.

45722 = 20903184, a zigzag square.

Page of Squares : First Upload September 18, 2007 ; Last Revised September 1, 2008
by Yoshio Mimura, Kobe, Japan

4573

85k + 1649k + 4097k + 4573k are squares for k = 1,2,3 (1022, 63582, 4109582).

Page of Squares : First Upload June 14, 2011 ; Last Revised June 14, 2011
by Yoshio Mimura, Kobe, Japan

4575

45752 = 1053 + 1303 + 2603.

Page of Squares : First Upload September 1, 2008 ; Last Revised September 1, 2008
by Yoshio Mimura, Kobe, Japan

4577

45772 = 933 + 1253 + 2633.

Page of Squares : First Upload September 1, 2008 ; Last Revised September 1, 2008
by Yoshio Mimura, Kobe, Japan

4578

45782± 5 are primes.

45782 = 13 + 2103 + 2273.

45782 = 20958084, a zigzag square.

2793k + 4578k + 13062k + 15288k are squares for k = 1,2,3 (1892, 208112, 24329972).

Page of Squares : First Upload September 18, 2007 ; Last Revised January 18, 2014
by Yoshio Mimura, Kobe, Japan

4579

45792 = 20967241, a zigzag square.

Page of Squares : First Upload September 18, 2007 ; Last Revised September 18, 2007
by Yoshio Mimura, Kobe, Japan

4584

45842 = 823 + 923 + 2703 = 104 + 504 + 504 + 544.

Page of Squares : First Upload September 1, 2008 ; Last Revised September 1, 2008
by Yoshio Mimura, Kobe, Japan

4588

45882 = (12 + 3)(112 + 3)(2062 + 3).

Page of Squares : First Upload December 28, 2013 ; Last Revised December 28, 2013
by Yoshio Mimura, Kobe, Japan

4589

45894 = 443478153684241, and 442 + 32 + 472 + 82 + 152 + 32 + 62 + 82 + 42 + 22 + 42 + 12 = 4589.

Page of Squares : First Upload December 1, 2008 ; Last Revised December 1, 2008
by Yoshio Mimura, Kobe, Japan

4590

45902 = (32 + 9)(92 + 9)(1142 + 9) = (52 + 9)(62 + 9)(92 + 9)(122 + 9) = (92 + 9)(122 + 9)(392 + 9).

A 4x4 perfect magic square consisting of squares with the constant 4590.

12582282212
482112412222
292322422312
382 92192522

Page of Squares : First Upload September 18, 2007 ; Last Revised December 28, 2013
by Yoshio Mimura, Kobe, Japan

4591

45912 = 125 + 145 + 155 + 255 + 255.

Page of Squares : First Upload September 1, 2008 ; Last Revised September 1, 2008
by Yoshio Mimura, Kobe, Japan

4598

45982 = 933 + 1743 + 2473 = 993 + 1213 + 2643.

Page of Squares : First Upload September 1, 2008 ; Last Revised September 1, 2008
by Yoshio Mimura, Kobe, Japan

4599

45992 = 33 + 933 + 2733.

45992 = (134 + 135 + 136 + ... + 154)2 + (155 + 156 + 157 + ... + 175)2.

Page of Squares : First Upload September 18, 2007 ; Last Revised September 1, 2008
by Yoshio Mimura, Kobe, Japan