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6800 - 6899

6800

68002 = (12 + 4)(22 + 4)(142 + 4)(762 + 4) = (12 + 4)(22 + 4)(42 + 4)(92 + 4)(262 + 4)
= (12 + 4)(22 + 4)(82 + 4)(92 + 4)(142 + 4) = (12 + 4)(82 + 4)(142 + 4)(262 + 4)
= (22 + 4)(82 + 4)(112 + 4)(262 + 4) = (42 + 4)(62 + 4)(92 + 4)(262 + 4)
= (62 + 4)(142 + 4)(762 + 4) = (62 + 4)(82 + 4)(92 + 4)(142 + 4).

68002 = 204 + 404 + 404 + 804,

Page of Squares : First Upload October 6, 2008 ; Last Revised January 11, 2014
by Yoshio Mimura, Kobe, Japan

6801

68012 = 46253601, a zigzag square.

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

6803

68032 = 813 + 2043 + 3343,

68032 = 46280809, a zigzag square.

Page of Squares : First Upload March 17, 2008 ; Last Revised October 6, 2008
by Yoshio Mimura, Kobe, Japan

6804

68042 = (12 + 5)(32 + 5)(72 + 5)(1012 + 5).

68042 = 723 + 753 + 3573,

68042 = 46294416, and 462 * 9 + 441 * 6 = 6804.

Komachi equations:
68042 = 12 * 22 / 32 / 42 * 5672 * 82 * 92 = 92 * 82 * 72 / 62 * 542 * 32 / 22 */ 12
 = 982 / 72 * 62 * 542 * 32 / 22 */ 12.

68042 = (14 + 15 + 16 + ... + 49)2 + (50 + 51 + 52 + ... + 85)2 + (86 + 87 + 88 + ... + 121)2 + ... + (122 + 123 + 124 + ... + 157)2.

Page of Squares : First Upload March 17, 2008 ; Last Revised January 11, 2014
by Yoshio Mimura, Kobe, Japan

6807

The square root of 6807 is 82.50454...., and 82 = 52 + 02 + 42 + 52 + 42.

68072 = 3053 + 245 + 107.

68072 = 46335249, and 42 + 62 + 32 + 32 + 52 + 22 + 42 + 92 = 142.

Page of Squares : First Upload March 17, 2008 ; Last Revised November 2, 2010
by Yoshio Mimura, Kobe, Japan

6810

The square root of 6810 is 82.5227...., and 82 = 52 + 22 + 22 + 72.

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

6811

68112 = 46389721, a square with different digits.

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

6812

68122 = 244 + 444 + 484 + 784,

Page of Squares : First Upload October 6, 2008 ; Last Revised October 6, 2008
by Yoshio Mimura, Kobe, Japan

6813

68132 = 2093 + 2233 + 2973,

Page of Squares : First Upload October 6, 2008 ; Last Revised October 6, 2008
by Yoshio Mimura, Kobe, Japan

6817

Loop of length 56 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
6817 - 4913 - 2570 - 5525 - ... - 405 - 41 - 1681 - 6817
(Note f(6817) = 682 + 172 = 4913,   f(4913) = 492 + 132 = 2570, etc. See 41)

68172 = 164 + 364 + 364 + 814 = 48 + 68 + 68 + 98,

Page of Squares : First Upload October 6, 2008 ; Last Revised October 9, 2008
by Yoshio Mimura, Kobe, Japan

6818

68182 = 46485124, and 4 + 6 + 4 * 851 / 2 * 4 = 6818.

490k + 1022k + 1274k + 6818k are squares for k = 1,2,3 (982, 70282, 5658522).

Page of Squares : First Upload March 17, 2008 ; Last Revised July 1, 2011
by Yoshio Mimura, Kobe, Japan

6822

68222 = 213 + 643 + 3593,

Page of Squares : First Upload October 6, 2008 ; Last Revised October 6, 2008
by Yoshio Mimura, Kobe, Japan

6823

The square root of 6823 is 82.601452...., and 82 = 62 + 02 + 12 + 42 + 52 + 22.

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

6825

68252 = 46580625, a zigzag square.

4836k + 6825k + 8658k + 17706k are squares for k = 1,2,3 (1952, 214112, 25750532).

Page of Squares : First Upload March 17, 2008 ; Last Revised July 1, 2011
by Yoshio Mimura, Kobe, Japan

6826

68265 = 14819427271011465376 :
142 + 82 + 192 + 42 + 22 + 72 + 22 + 72 + 102 + 112 + 42 + 62 + 52 + 32 + 762
= 12 + 42 + 82 + 12 + 92 + 42 + 22 + 72 + 272 + 12 + 02 + 12 + 12 + 42 + 62 + 52 + 32 + 762
= 12 + 42 + 82 + 12 + 92 + 42 + 272 + 22 + 72 + 12 + 02 + 12 + 12 + 42 + 62 + 52 + 32 + 762 = 6826.

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

6829

Loop of length 6 by the function f(N) = b2 + a2 for N = 100b + a:
6829 - 5465 - 7141 - 6722 - 4973 - 7730 - 6829   (Note f(6829) = 682 + 292 = 5465.)

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

6831

68312 = 93 + 183 + 3603 = 93 + 1323 + 3543 = 103 + 2333 + 3243 = 1083 + 2253 + 3243,

Page of Squares : First Upload October 6, 2008 ; Last Revised October 6, 2008
by Yoshio Mimura, Kobe, Japan

6832

68322 = 46676224, and 4 * 6 * 6 + 76 * 22 * 4 = 46 + 6762 + 24 = 6832.

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

6833

68332 = 46689889, and 4 * 6 + 6898 - 89 = 6833.

Loop of length 10 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
6833 - 5713 - 3418 - 1480 - ... - 2228 - 1268 - 4768 - 6833
(Note f(6833) = 682 + 332 = 5713,   f(5713) = 572 + 132 = 3418, etc. See 1268)

Page of Squares : First Upload March 17, 2008 ; Last Revised October 9, 2008
by Yoshio Mimura, Kobe, Japan

6834

68342 = 893 + 1083 + 3553,

68342 = 46703556, and 42 + 62 + 72 + 02 + 32 + 52 + 52 + 62 = 142.

Page of Squares : First Upload March 17, 2008 ; Last Revised October 6, 2008
by Yoshio Mimura, Kobe, Japan

6839

68392 = 853 + 873 + 3573,

Page of Squares : First Upload October 6, 2008 ; Last Revised October 6, 2008
by Yoshio Mimura, Kobe, Japan

6840

68402 = 1703 + 2623 + 2883 = 2083 + 2283 + 2963,

1812 + 6840 = 1992, 1812 - 6840 = 1612.

Page of Squares : First Upload October 6, 2008 ; Last Revised August 5, 2011
by Yoshio Mimura, Kobe, Japan

6842

68422 = 194 + 494 + 494 + 774,

68425 = 14993925520699555232 :
142 + 92 + 92 + 32 + 92 + 22 + 52 + 52 + 202 + 62 + 92 + 92 + 552 + 522 + 32 + 22
= 142 + 92 + 92 + 32 + 92 + 22 + 552 + 202 + 62 + 92 + 92 + 52 + 52 + 522 + 32 + 22 = 6842.

Page of Squares : First Upload October 6, 2008 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

6843

68432 = 46826649, and 42 + 62 + 82 + 22 + 62 + 62 + 42 + 92 = 172.

68432 = 46826649, and 4 * 6 * 8 + 2 + 6649 = 6843.

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

6844

68442 = 46840336, and 4 + 6840 + 3 + 3 - 6 = 4 + 6840 - 3 - 3 + 6 = 6844.

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

6845

68452 = (93 + 94 + 95 + ... + 129)2 + (130 + 131 + 132 + ... + 166)2.

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

6847

68472 = 164 + 344 + 594 + 764,

Page of Squares : First Upload October 6, 2008 ; Last Revised October 6, 2008
by Yoshio Mimura, Kobe, Japan

6848

(68482 + 8) = (12 + 8)(32 + 8)(52 + 8)(82 + 8)(112 + 8).

68482 = (52 + 7)(102 + 7)(1172 + 7).

Page of Squares : First Upload March 17, 2008 ; Last Revised January 11, 2014
by Yoshio Mimura, Kobe, Japan

6850

68502 = 73 + 1493 + 3523,

68502 = (218 + 219 + 220 + 221 + 222)2 + (223 + 224 + 225 + 226 + 227)2 + (228 + 229 + 230 + 231 + 232)2 + ... + (333 + 334 + 335 + 336 + 337)2.

Loop of length 5 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
6850 - 7124 - 5617 - 3425 - ... - 5617 - 3425 - 1781 - 6850
(Note f(6850) = 682 + 502 = 7124,   f(7124) = 712 + 242 = 5617, etc. See 1781)

Page of Squares : First Upload March 17, 2008 ; Last Revised October 9, 2008
by Yoshio Mimura, Kobe, Japan

6851

68512 = (125 + 126 + 127 + ... + 155)2 + (156 + 157 + 158 + ... + 186)2.

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

6852

68522 = (22 + 8)(19782 + 8).

68522 = 543 + 2683 + 3023,

Page of Squares : First Upload October 6, 2008 ; Last Revised January 11, 2014
by Yoshio Mimura, Kobe, Japan

6855

68555 = 15136912331424534375 :
152 + 132 + 692 + 12 + 22 + 32 + 312 + 42 + 242 + 52 + 32 + 42 + 32 + 72 + 52 = 6855.

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

6856

68565 = 15147956335281995776 :
12 + 512 + 472 + 92 + 52 + 62 + 32 + 352 + 22 + 82 + 192 + 92 + 52 + 72 + 72 + 62 = 6856.

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

6858

68582 = 194 + 594 + 594 + 694,

Page of Squares : First Upload October 6, 2008 ; Last Revised October 6, 2008
by Yoshio Mimura, Kobe, Japan

6859

68592 = 573 + 1903 + 3423 = 1183 + 2253 + 3243,

Page of Squares : First Upload October 6, 2008 ; Last Revised October 6, 2008
by Yoshio Mimura, Kobe, Japan

6860

68605 = 15192196831817600000 :
12 + 52 + 192 + 212 + 92 + 62 + 82 + 32 + 12 + 82 + 12 + 762 + 02 + 02 + 02 + 02 + 02 = 6860.

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

6862

68625 = 15214355808258756832 :
12 + 52 + 22 + 12 + 432 + 52 + 52 + 82 + 02 + 82 + 22 + 52 + 82 + 72 + 52 + 682 + 32 + 22 = 6862.

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

6863

68635 = 15225444988405910543 : 12 + 52 + 22 + 252 + 42 + 42 + 492 + 82 + 82 + 42 + 02 + 592 + 102 + 52 + 42 + 32 = 6863.

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

6864

68642 = 47114496, and 47 * 1 * 144 + 96 = 6864.

68642 = (122 - 1)(232 - 1)(252 - 1).

1452 + 6864 = 1672, 1452 - 6864 = 1192.

6864 = (12 + 22 + 32 + ... + 3512) / (12 + 22 + 32 + ... + 182).

Page of Squares : First Upload March 17, 2008 ; Last Revised January 11, 2014
by Yoshio Mimura, Kobe, Japan

6866

68662 = 573 + 673 + 3603,

68662 = 47141956, a zigzag square.

68662 = 47141956, and 42 + 72 + 12 + 42 + 12 + 92 + 52 + 62 = 152.

Page of Squares : First Upload March 17, 2008 ; Last Revised October 6, 2008
by Yoshio Mimura, Kobe, Japan

6868

68682 = 303 + 1083 + 3583,

Page of Squares : First Upload October 6, 2008 ; Last Revised October 6, 2008
by Yoshio Mimura, Kobe, Japan

6871

68712 = 24 + 54 + 504 + 804,

Page of Squares : First Upload October 6, 2008 ; Last Revised October 6, 2008
by Yoshio Mimura, Kobe, Japan

6872

68725 = 15325538876401221632 :
152 + 32 + 252 + 52 + 32 + 82 + 82 + 762 + 42 + 02 + 12 + 22 + 22 + 12 + 62 + 32 + 22 = 6872.

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

6875

68752 = 1453 + 1703 + 3403,

68752 = 47265625, a zigzag square.

The square root of 6875 is 82.91...., and 82 = 92 + 12.

Page of Squares : First Upload March 17, 2008 ; Last Revised October 6, 2008
by Yoshio Mimura, Kobe, Japan

6876

68762 = 1503 + 1883 + 3343,

68764 = 2235339394949376,
and 22 + 22 + 352 + 32 + 392 + 392 + 42 + 92 + 492 + 32 + 72 + 62 = 6876,
and 22 + 22 + 352 + 32 + 392 + 392 + 492 + 42 + 92 + 32 + 72 + 62 = 6876.

68765 = 15370193679671909376 :
152 + 32 + 72 + 02 + 12 + 92 + 32 + 62 + 72 + 92 + 672 + 192 + 02 + 92 + 372 + 62
= 152 + 32 + 72 + 02 + 12 + 92 + 32 + 672 + 92 + 62 + 72 + 192 + 02 + 92 + 372 + 62
= 152 + 32 + 72 + 02 + 192 + 32 + 62 + 72 + 92 + 672 + 12 + 92 + 02 + 92 + 372 + 62
= 152 + 32 + 72 + 02 + 192 + 32 + 672 + 92 + 62 + 72 + 12 + 92 + 02 + 92 + 372 + 62
= 152 + 372 + 02 + 12 + 92 + 32 + 62 + 72 + 92 + 672 + 192 + 02 + 92 + 32 + 72 + 62
= 152 + 372 + 02 + 12 + 92 + 32 + 672 + 92 + 62 + 72 + 192 + 02 + 92 + 32 + 72 + 62
= 152 + 372 + 02 + 192 + 32 + 62 + 72 + 92 + 672 + 12 + 92 + 02 + 92 + 32 + 72 + 62
= 152 + 372 + 02 + 192 + 32 + 672 + 92 + 62 + 72 + 12 + 92 + 02 + 92 + 32 + 72 + 62 = 6876.

Page of Squares : First Upload October 6, 2008 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

6877

68772 = 1843 + 3453,

1841 + 3451 = 232, 1842 + 3452 = 3912, 1843 + 3453 = 68772  (See 23).

68775 = 15381373628049378157 :
12 + 532 + 82 + 12 + 372 + 32 + 62 + 282 + 02 + 42 + 92 + 372 + 82 + 152 + 72 = 6877.

Page of Squares : First Upload October 6, 2008 ; Last Revised January 6, 2011
by Yoshio Mimura, Kobe, Japan

6878

6878 = (12 + 22 + 32 + ... + 1802) / (12 + 22 + 32 + ... + 92).

68782 = 47306884, and 4 - 7 - 3 + 0 + 6884 = 6878.

Page of Squares : First Upload March 17, 2008 ; Last Revised November 25, 2008
by Yoshio Mimura, Kobe, Japan

6880

68802 = 173 + 1023 + 3593,

Page of Squares : First Upload October 6, 2008 ; Last Revised October 6, 2008
by Yoshio Mimura, Kobe, Japan

6882

68822 = 47361924, a zigzag square.

68824 = 2243151844981776,
and 22 + 242 + 32 + 152 + 12 + 82 + 42 + 42 + 92 + 82 + 12 + 72 + 762 = 6882.

1798k + 6882k + 25110k + 27714k are squares for k = 1,2,3 (2482, 380682, 61196482).

Page of Squares : First Upload March 17, 2008 ; Last Revised July 1, 2011
by Yoshio Mimura, Kobe, Japan

6885

68852 = 1263 + 2253 + 3243,

Komachi equations:
68852 = 92 * 82 * 7652 * 42 / 322 */ 12 = 92 / 82 * 7652 / 42 * 322 */ 12.

Page of Squares : First Upload October 6, 2008 ; Last Revised November 2, 2010
by Yoshio Mimura, Kobe, Japan

6886

68864 = 2248371509664016,
and 22 + 22 + 482 + 32 + 72 + 12 + 52 + 02 + 92 + 662 + 42 + 02 + 12 + 62 = 6886.

68865 = 15482286215546414176 :
12 + 52 + 482 + 22 + 22 + 82 + 622 + 12 + 52 + 52 + 42 + 62 + 42 + 142 + 172 + 62 = 6886.

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

6887

68872 = 47430769, and 42 + 72 + 42 + 32 + 02 + 72 + 62 + 92 = 162.

68875 = 15493531338703283207 :
12 + 52 + 492 + 32 + 532 + 12 + 32 + 382 + 72 + 02 + 32 + 22 + 82 + 32 + 22 + 02 + 72 = 6887.

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

6888

68882 = 464 + 464 + 544 + 744,

68882 = 47444544, a square with just 3 kinds of digits.

68882 = 47444544.

4998k + 6888k + 8106k + 15729k are squares for k = 1,2,3 (1892, 196352, 22080872).

Page of Squares : First Upload March 17, 2008 ; Last Revised July 1, 2011
by Yoshio Mimura, Kobe, Japan

6889

The square of 83.

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

6890

68902 = (12 + 9)(22 + 9)(162 + 9)(372 + 9) = (112 + 9)(162 + 9)(372 + 9).

6890k + 43810k + 46670k + 54730k are squares for k = 1,2,3 (3902, 845002, 187083002).

Page of Squares : First Upload July 1, 2011 ; Last Revised January 11, 2014
by Yoshio Mimura, Kobe, Japan

6894

68945 = 15572430287447104224 :
12 + 52 + 572 + 22 + 42 + 302 + 22 + 82 + 72 + 442 + 72 + 12 + 02 + 42 + 22 + 242
= 12 + 52 + 572 + 22 + 42 + 302 + 282 + 72 + 42 + 42 + 72 + 12 + 02 + 422 + 22 + 42
= 12 + 52 + 572 + 242 + 302 + 22 + 82 + 72 + 442 + 72 + 12 + 02 + 42 + 22 + 22 + 42 = 6894.

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

6897

68972 = (12 + 2)(32 + 2)(192 + 2)(632 + 2) = (52 + 8)(72 + 8)(1592 + 8).

1 / N = 0.000144... and 144 = 122 for N = 6897, 6898, ..., 6944.

Page of Squares : First Upload March 17, 2008 ; Last Revised January 11, 2014
by Yoshio Mimura, Kobe, Japan

6898

68982 = 563 + 2673 + 3053,

68982 = 47582404, a zigzag square.

Page of Squares : First Upload March 17, 2008 ; Last Revised October 6, 2008
by Yoshio Mimura, Kobe, Japan

6899

68992 = 47596201, a square with different digits.

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