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5600 - 5699

5600

4-cycle : 56002 = 31360000 - 36002 = 12960000 - 96002 = 92160000 - 16002 = 02560000,
(Other examples : 2916 - 5030 - 3009 - 0540 - 2916, 2100 - 4100 - 8100 - 6100 - 2100.)

56002 = 751 x 752 + 753 x 754 + 755 x 756 + 757 x 758 + 759 x 760+ ... + 847 x 848.

Page of Squares : First Upload December 17, 2007 ; Last Revised December 17, 2007
by Yoshio Mimura, Kobe, Japan

5601

56014 = 984152252182401, and 92 + 82 + 42 + 12 + 522 + 252 + 212 + 82 + 22 + 402 + 12 = 5601.

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

5604

56042 = 43 + 533 + 3153.

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

5605

56052 = S2(352) + S2(369), where S2(n) = 12 + 22 + 32 + ... + n2.

Page of Squares : First Upload December 17, 2007 ; Last Revised December 17, 2007
by Yoshio Mimura, Kobe, Japan

5607

56072 = (351 + 352)2 + (353 + 354)2 + (355 + 356)2 + (357 + 358)2 + ... + (447 + 448)2.

Page of Squares : First Upload December 17, 2007 ; Last Revised December 17, 2007
by Yoshio Mimura, Kobe, Japan

5610

56102 = 342 x 343 + 344 x 345 + 346 x 347 + 348 x 349 + 350 x 351 + ... + 610 x 611.

The integral triangle of sides 2245, 663256, 665499 has square area 56102.

Page of Squares : First Upload December 17, 2007 ; Last Revised October 14, 2011
by Yoshio Mimura, Kobe, Japan

5612

1 / 5612 = 0.00017818959372, 172 + 82 + 182 + 92 + 592 + 372 + 22 = 5612.

Page of Squares : First Upload December 17, 2007 ; Last Revised December 17, 2007
by Yoshio Mimura, Kobe, Japan

5617

Loop of length 5 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
5617 - 3425 - 1781 - 6850 - ... - 1781 - 6850 - 7124 - 5617
(Note f(5617) = 562 + 172 = 3425,   f(3425) = 342 + 252 = 1781, etc. See 1781)

Page of Squares : First Upload October 9, 2008 ; Last Revised October 9, 2008
by Yoshio Mimura, Kobe, Japan

5618

56182 = 534 + 534 + 534 + 534.

56183 = 177314889032, and 12 + 72 + 732 + 12 + 42 + 82 + 82 + 92 + 02 + 32 + 22 = 5618.

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

5620

Loop of length 56 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
5620 - 3536 - 2521 - 1066 - ... - 6260 - 7444 - 7412 - 5620
(Note f(5620) = 562 + 202 = 3536,   f(3536) = 352 + 362 = 2521, etc. See 41)

Page of Squares : First Upload October 9, 2008 ; Last Revised October 9, 2008
by Yoshio Mimura, Kobe, Japan

5622

56222 = 383 + 1273 + 3093.

1 / 5622 = 0.0001778726, 172 + 72+82 + 722 + 62 = 5622.

Page of Squares : First Upload December 17, 2007 ; Last Revised September 18, 2008
by Yoshio Mimura, Kobe, Japan

5624

56242 = 31629376, a zigzag square.

56242 = 31629376, 32 + 12 + 62 + 22 + 92 + 32 + 72 + 62 = 152.

Page of Squares : First Upload December 17, 2007 ; Last Revised December 17, 2007
by Yoshio Mimura, Kobe, Japan

5625

The square of 75.

Page of Squares : First Upload December 17, 2007 ; Last Revised December 17, 2007
by Yoshio Mimura, Kobe, Japan

5628

56282 = (22 + 3)(32 + 3)(82 + 3)(752 + 3) = (82 + 3)(92 + 3)(752 + 3).

56282 = 84 + 84 + 364 + 744.

56282 = (40 + 41 + 42 + ... + 67)2 + (68 + 69 + 70 + ... + 95)2 + (96 + 97 + 98 + ... + 123)2 + ... + (124 + 125 + 126 + ... + 151)2.

994k + 1470k + 2338k + 4802k are squares for k = 1,2,3 (982, 56282, 3573082).

Page of Squares : First Upload December 17, 2007 ; Last Revised December 28, 2013
by Yoshio Mimura, Kobe, Japan

5629

56292 = 31685641, 3 - 1 - 6 - 8 + 5641 = 5629.

Page of Squares : First Upload December 17, 2007 ; Last Revised December 17, 2007
by Yoshio Mimura, Kobe, Japan

5631

56312 = 31708161, a zigzag square.

Page of Squares : First Upload December 17, 2007 ; Last Revised December 17, 2007
by Yoshio Mimura, Kobe, Japan

5632

56322 = (12 + 7)(22 + 7)(112 + 7)(532 + 7) = (12 + 7)(22 + 7)(32 + 7)(112 + 7)(132 + 7)
= (12 + 7)(22 + 7)(52 + 7)(92 + 7)(112 + 7) = (12 + 7)(32 + 7)(92 + 7)(532 + 7)
= (12 + 7)(52 + 7)(112 + 7)(312 + 7) = (22 + 7)(32 + 7)(52 + 7)(752 + 7)
= (22 + 7)(92 + 7)(1812 + 7) = (32 + 7)(92 + 7)(112 + 7)(132 + 7) = (312 + 7)(1812 + 7)
= (52 + 7)(132 + 7)(752 + 7) = (92 + 7)(112 + 7)(532 + 7).

Page of Squares : First Upload December 17, 2007 ; Last Revised December 28, 2013
by Yoshio Mimura, Kobe, Japan

5634

56342 = 34 + 154 + 154 + 754.

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

5635

56355 = 5681586223670021875 : 52 + 62 + 82 + 12 + 52 + 82 + 62 + 222 + 32 + 672 + 02 + 02 + 22 + 182 + 72 + 52 = 5635.

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

5636

Loop of length 56 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
5636 - 4432 - 2960 - 4441 - ... - 12161 - 4163 - 5650 - 5636
(Note f(5636) = 562 + 362 = 4432,   f(4432) = 442 + 322 = 2960, etc. See 41)

Page of Squares : First Upload October 9, 2008 ; Last Revised October 9, 2008
by Yoshio Mimura, Kobe, Japan

5638

56382 = 55 + 135 + 255 + 255 + 265.

56382 = 31787044, 3 + 1 * 7 + 8 * 704 - 4 = 3 * 1 + 7 + 8 * 704 - 4 = 5638.

Page of Squares : First Upload December 17, 2007 ; Last Revised September 18, 2008
by Yoshio Mimura, Kobe, Japan

5640

56402 = 1643 + 2083 + 2643.

56402 = 31809600, 3 * 1 * 80 + 9 * 600 = 5640.

Page of Squares : First Upload December 17, 2007 ; Last Revised September 18, 2008
by Yoshio Mimura, Kobe, Japan

5643

56432 = (32 + 2)(52 + 2)(132 + 2)(252 + 2).

56432 = 1023 + 1883 + 2893.

Page of Squares : First Upload September 18, 2008 ; Last Revised December 28, 2013
by Yoshio Mimura, Kobe, Japan

5650

Loop of length 56 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
5650 - 5636 - 4432 - 2960 - ... - 9556 - 12161 - 4163 - 5650
(Note f(5650) = 562 + 502 = 5636,   f(5636) = 562 + 362 = 4432, etc. See 41)

Page of Squares : First Upload October 9, 2008 ; Last Revised October 9, 2008
by Yoshio Mimura, Kobe, Japan

5653

1 / 5653 = 0.00017689...., 17689 = 1332.

56532 = 31956409, 3 + 19 + 5640 - 9 = 31 - 9 + 5640 - 9 = 5653.

Page of Squares : First Upload December 17, 2007 ; Last Revised December 17, 2007
by Yoshio Mimura, Kobe, Japan

5655

56552 = 263 + 2253 + 2743.

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

5657

56575 = 5793365032534520057 : 52 + 72 + 92 + 332 + 62 + 52 + 02 + 322 + 52 + 32 + 42 + 52 + 22 + 02 + 02 + 572 = 572 + 92 + 332 + 62 + 52 + 02 + 322 + 52 + 32 + 42 + 52 + 22 + 02 + 02 + 52 + 72 = 5657.

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

5662

56622 = 1003 + 1393 + 3053.

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

5663

56634 = 1028457255845761,
and 12 + 02 + 22 + 82 + 42 + 52 + 72 + 22 + 52 + 582 + 452 + 72 + 62 + 12 = 5663,
56634 = 1028457255845761,
and 12 + 02 + 22 + 82 + 452 + 72 + 22 + 52 + 582 + 42 + 52 + 72 + 62 + 12 = 5663.

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

5664

56642 = 903 + 1163 + 3103 = 1123 + 1683 + 2963.

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

5666

1 / N = 0.0001764...., where 1764 = 422 (for N = 5666, 5667, 5668).

Page of Squares : First Upload December 17, 2007 ; Last Revised December 17, 2007
by Yoshio Mimura, Kobe, Japan

5670

56702 = (12 + 5)(102 + 5)(112 + 5)(202 + 5) = (12 + 5)(22 + 5)(32 + 5)(102 + 5)(202 + 5)
= (12 + 5)(202 + 5)(1152 + 5) = (12 + 5)(42 + 5)(202 + 5)(252 + 5) = (112 + 5)(202 + 5)(252 + 5)
= (22 + 5)(32 + 5)(202 + 5)(252 + 5) = (22 + 5)(32 + 5)(42 + 5)(52 + 5)(202 + 5)
= (22 + 5)(42 + 5)(52 + 5)(72 + 5)(102 + 5) = (22 + 5)(52 + 5)(172 + 5)(202 + 5)
= (22 + 5)(72 + 5)(102 + 5)(252 + 5) = (32 + 5)(72 + 5)(102 + 5)(202 + 5)
= (42 + 5)(52 + 5)(112 + 5)(202 + 5) = (52 + 5)(102 + 5)(1012 + 5).

56702 = (5 + 6 + 7 + ... + 13)2 + (14 + 15 + 16 + ... + 22)2 + (23 + 24 + 25 + ... + 31)2 + ... + (212 + 213 + 214 + ... + 220)2.

Komachi equations:
56702 = 12 * 22 * 32 * 42 * 52 * 62 * 72 / 82 * 92 = 12 / 22 * 32 / 42 * 52 * 62 * 72 * 82 * 92
 = 12 / 22 * 32 * 452 / 62 * 72 * 82 * 92 = 92 * 82 * 72 * 62 * 52 / 42 * 32 / 22 */ 12
 = 92 / 82 * 72 * 62 * 52 * 42 * 32 * 22 */ 12.

Page of Squares : First Upload December 17, 2007 ; Last Revised October 25, 2010
by Yoshio Mimura, Kobe, Japan

5671

56712 + 56722 + 56732 + ... + 57242 = 57252 + 57262 + 57272 + ... + 57772.

Page of Squares : First Upload September 16, 2011 ; Last Revised September 16, 2011
by Yoshio Mimura, Kobe, Japan

5672

56722 = 32171584, 32 + 22 + 12 + 72 + 12 + 52 + 82 + 42 = 132.

56725 = 5870581354415685632 : 52 + 82 + 72 + 02 + 52 + 82 + 12 + 32 + 52 + 442 + 152 + 62 + 82 + 562 + 32 + 22 = 5672.

Page of Squares : First Upload December 17, 2007 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

5673

56732 = 253 + 2103 + 2843.

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

5676

56762 = (22 + 8)(62 + 8)(2472 + 8).

56762 = 1702 + 1712 + 1722 + 1732 + 1742 + ... + 4662.

The integral triangle of sides 3180, 20339, 20857 has square area 56762.

Page of Squares : First Upload December 17, 2007 ; Last Revised December 28, 2013
by Yoshio Mimura, Kobe, Japan

5678

56782± 3 are primes.

Komachi equations:
56782 = 122 * 32 / 42 + 56782 - 92 = 122 * 32 / 42 * 56782 / 92
 = - 122 * 32 / 42 + 56782 + 92.

Page of Squares : First Upload October 25, 2010 ; Last Revised January 18, 2014
by Yoshio Mimura, Kobe, Japan

5681

56812 = S2(325) + S2(396), where S2(n) = 12 + 22 + 32 + ... + n2.

56812 = 1682 + 1692 + 1702 + 1712 + 1722 + ... + 4662.

Page of Squares : First Upload December 17, 2007 ; Last Revised December 17, 2007
by Yoshio Mimura, Kobe, Japan

5683

56832 = 32296489, 3 + 2 - 2 * 9 + 64 * 89 = 5683.

Page of Squares : First Upload December 17, 2007 ; Last Revised December 17, 2007
by Yoshio Mimura, Kobe, Japan

5684

56842 = 32307856, 32 + 22 + 32 + 02 + 72 + 82 + 52 + 62 = 142.

Page of Squares : First Upload December 17, 2007 ; Last Revised December 17, 2007
by Yoshio Mimura, Kobe, Japan

5689

56892 = 333 + 2133 + 2833.

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

5690

1330k + 3610k + 5690k + 6270k are squares for k = 1,2,3 (1302, 93002, 6929002).

Page of Squares : First Upload June 21, 2011 ; Last Revised June 21, 2011
by Yoshio Mimura, Kobe, Japan

5692

56922 = 1043 + 2463 + 2543.

56922 = 32398864, 3 * 23 - 9 + 88 * 64 = 3 * 239 * 8 - 8 * 6 + 4 = 5692.

Page of Squares : First Upload December 17, 2007 ; Last Revised September 18, 2008
by Yoshio Mimura, Kobe, Japan

5693

56932 = 1313 + 2433 + 2513.

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

5696

1 / 5696 = 0.000175561...., 175561 = 4192.

56962 = 32444416, 32 * 4 * 44 + 4 * 16 = 32 * 44 * 4 + 4 * 16 = 5696.

Page of Squares : First Upload December 17, 2007 ; Last Revised December 17, 2007
by Yoshio Mimura, Kobe, Japan

5697

56972 = 1533 + 2323 + 2543.

56972 = (78 + 79 + 80)2 + (81 + 82 + 83)2 + (84 + 85 + 86)2 + ... + (318 + 319 + 320)2.

Page of Squares : First Upload December 17, 2007 ; Last Revised September 18, 2008
by Yoshio Mimura, Kobe, Japan

5698

5698k + 6622k + 8338k + 10318k are squares for k = 1,2,3 (1762, 158842, 14674882).

Page of Squares : First Upload June 21, 2011 ; Last Revised June 21, 2011
by Yoshio Mimura, Kobe, Japan

5699

56992 = 32478601, a square with different digits.

Page of Squares : First Upload December 17, 2007 ; Last Revised December 17, 2007
by Yoshio Mimura, Kobe, Japan