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, 2007by 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, 2008by Yoshio Mimura, Kobe, Japan
5604
56042 = 43 + 533 + 3153.
Page of Squares : First Upload September 18, 2008 ; Last Revised September 18, 2008by 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, 2007by 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, 2007by 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, 2011by 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, 2007by 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)
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, 2008by 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)
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, 2008by 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, 2007by Yoshio Mimura, Kobe, Japan
5625
The square of 75.
Page of Squares : First Upload December 17, 2007 ; Last Revised December 17, 2007by 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, 2013by 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, 2007by Yoshio Mimura, Kobe, Japan
5631
56312 = 31708161, a zigzag square.
Page of Squares : First Upload December 17, 2007 ; Last Revised December 17, 2007by 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).
by Yoshio Mimura, Kobe, Japan
5634
56342 = 34 + 154 + 154 + 754.
Page of Squares : First Upload September 18, 2008 ; Last Revised September 18, 2008by 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, 2008by 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)
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, 2008by 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, 2008by 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, 2013by 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)
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, 2007by Yoshio Mimura, Kobe, Japan
5655
56552 = 263 + 2253 + 2743.
Page of Squares : First Upload September 18, 2008 ; Last Revised September 18, 2008by 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, 2008by Yoshio Mimura, Kobe, Japan
5662
56622 = 1003 + 1393 + 3053.
Page of Squares : First Upload September 18, 2008 ; Last Revised September 18, 2008by 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.
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, 2008by 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, 2007by 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.
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, 2011by 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, 2008by Yoshio Mimura, Kobe, Japan
5673
56732 = 253 + 2103 + 2843.
Page of Squares : First Upload September 18, 2008 ; Last Revised September 18, 2008by 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, 2013by 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.
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, 2007by 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, 2007by 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, 2007by Yoshio Mimura, Kobe, Japan
5689
56892 = 333 + 2133 + 2833.
Page of Squares : First Upload September 18, 2008 ; Last Revised September 18, 2008by 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, 2011by 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, 2008by Yoshio Mimura, Kobe, Japan
5693
56932 = 1313 + 2433 + 2513.
Page of Squares : First Upload September 18, 2008 ; Last Revised September 18, 2008by 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, 2007by 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, 2008by 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, 2011by Yoshio Mimura, Kobe, Japan
5699
56992 = 32478601, a square with different digits.
Page of Squares : First Upload December 17, 2007 ; Last Revised December 17, 2007by Yoshio Mimura, Kobe, Japan