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8600 - 8699

8601

8601 = 102 + 212 + 322 + ... + (11n + 10)2 + ... + 13412.

S2(8601) = S2(3484) + S2(8406), where S2(n) = 12 + 22 + 32 + 42 + ... + n2.

Page of Squares : First Upload July 28, 2008 ; Last Revised July 28, 2008
by Yoshio Mimura, Kobe, Japan

8605

Loop of length 5 : 8605 --- 862 + 052 = 7421 --- 722 + 412 = 5917 --- 592 + 172 = 3770 --- 372 + 702 = 6269 --- 622 + 692 = 8605

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

8607

86074 = 5487912924041601, and
52 + 42 + 872 + 92 + 12 + 292 + 22 + 42 + 02 + 42 + 12 + 62 + 02 + 12 = 8607.

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

8609

86095 = 47289371176810582049, and
42 + 72 + 22 + 82 + 92 + 32 + 72 + 12 + 12 + 762 + 82 + 12 + 02 + 52 + 82 + 22 + 02 + 492 = 8609.

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

8610

86102 = (32 + 5)(62 + 5)(102 + 5)(352 + 5).

86102 = 409 x 410 + 411 x 412 + 413 x 414 + 415 x 416 + ... + 799 x 800.

Page of Squares : First Upload July 28, 2008 ; Last Revised January 11, 2014
by Yoshio Mimura, Kobe, Japan

8614

86142 = 2253 + 3153 + 3163.

86145 = 47426856172874825824, and
42 + 72 + 42 + 262 + 82 + 52 + 62 + 12 + 72 + 282 + 72 + 42 + 82 + 22 + 52 + 822 + 42
= 42 + 72 + 42 + 262 + 82 + 52 + 62 + 12 + 72 + 282 + 72 + 42 + 822 + 52 + 82 + 22 + 42 = 8614.

Page of Squares : First Upload July 28, 2008 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

8616

86162 = 74235456 appears in the decimal expressions of e:
  e = 2.71828•••74235456••• (from the 21530th digit)
  (74235456 is the fourth 8-digit square in the expression of e.)

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

8618

86182 = S2(462) + S2(498), where S2(n) = 12 + 22 + 32 + 42 + ... + n2.

Page of Squares : First Upload July 28, 2008 ; Last Revised July 28, 2008
by Yoshio Mimura, Kobe, Japan

8621

86215 = 47619872192762367101, and
472 + 62 + 12 + 92 + 82 + 722 + 12 + 92 + 272 + 62 + 22 + 32 + 62 + 72 + 102 + 12 = 8621.

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

8623

86232 = 74356129, a square with different digits.

86232 = 253 + 2673 + 3813 = 264 + 314 + 344 + 924.

Page of Squares : First Upload July 28, 2008 ; Last Revised July 28, 2008
by Yoshio Mimura, Kobe, Japan

8624

154k + 8624k + 17633k + 26950k are squares for k = 1,2,3 (2312, 333412, 50692952).

Page of Squares : First Upload July 15, 2011 ; Last Revised July 15, 2011
by Yoshio Mimura, Kobe, Japan

8625

86252 = 453 + 2953 + 3653.

86252 = 74390625, a square with different digits.

86255 = 47730448883056640625, and
42 + 72 + 732 + 02 + 42 + 482 + 82 + 82 + 32 + 02 + 52 + 62 + 62 + 42 + 02 + 62 + 252 = 8625.

Page of Squares : First Upload July 28, 2008 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

8626

1330k + 8417k + 8626k + 10868k are squares for k = 1,2,3 (1712, 162832, 15887612).

Page of Squares : First Upload July 15, 2011 ; Last Revised July 15, 2011
by Yoshio Mimura, Kobe, Japan

8627

86272 = 74425129, and 72 + 42 + 42 + 22 + 52 + 12 + 22 + 92 = 142.

Page of Squares : First Upload July 28, 2008 ; Last Revised July 28, 2008
by Yoshio Mimura, Kobe, Japan

8631

86312 = 683 + 2383 + 3933.

Page of Squares : First Upload July 28, 2008 ; Last Revised July 28, 2008
by Yoshio Mimura, Kobe, Japan

8632

86325 = 47924452309775122432, and
472 + 92 + 22 + 442 + 522 + 302 + 92 + 72 + 72 + 52 + 12 + 22 + 242 + 32 + 22 = 8632.

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

8635

S2(8635) = S2(963) + S2(8631), where S2(n) = 12 + 22 + 32 + 42 + ... + n2.

Page of Squares : First Upload July 28, 2008 ; Last Revised July 28, 2008
by Yoshio Mimura, Kobe, Japan

8637

86372 = 2333 + 3103 + 3183.

Page of Squares : First Upload July 28, 2008 ; Last Revised July 28, 2008
by Yoshio Mimura, Kobe, Japan

8640

86402 = (22 - 1)(32 - 1)(112 - 1)(1612 - 1) = (22 - 1)(312 - 1)(1612 - 1)
= (22 - 1)(42 - 1)(72 - 1)(112 - 1)(172 - 1) = (32 - 1)(192 - 1)(1612 - 1)
= (32 - 1)(42 - 1)(52 - 1)(1612 - 1) = (32 - 1)(72 - 1)(172 - 1)(262 - 1)
= (42 - 1)(72 - 1)(172 - 1)(192 - 1) = (52 - 1)(112 - 1)(1612 - 1).

1022 + 8640 = 1382, 1022 - 8640 = 422.

86402 = 1643 + 3043 + 3483.

Komachi equations:
86402 = 92 * 82 * 72 * 62 * 52 * 42 * 32 / 212 = 92 * 82 / 72 * 62 * 52 * 42 / 32 * 212.

Page of Squares : First Upload July 28, 2008 ; Last Revised January 11, 2014
by Yoshio Mimura, Kobe, Japan

8644

86442 = 113 + 3083 + 3573 = 323 + 2583 + 3863 = 963 + 1703 + 4103.

1 / N = 0.0001156...., and 1156 = 342, for N = 8644, 8645,..., 8650.

Page of Squares : First Upload July 28, 2008 ; Last Revised July 28, 2008
by Yoshio Mimura, Kobe, Japan

8645

86452 = 93 + 1403 + 4163.

86455 = 48286417826561403125, and
482 + 22 + 82 + 62 + 42 + 172 + 82 + 22 + 652 + 62 + 12 + 42 + 02 + 312 + 252
= 482 + 282 + 642 + 172 + 82 + 22 + 62 + 52 + 62 + 12 + 42 + 02 + 312 + 22 + 52 = 8645.

Page of Squares : First Upload July 28, 2008 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

8646

86462± 5 are primes.

86462 = 933 + 2423 + 3913.

Page of Squares : First Upload July 28, 2008 ; Last Revised January 18, 2014
by Yoshio Mimura, Kobe, Japan

8648

86482 = 144 + 624 + 744 + 744.

86482 = 74787904, and 72 + 42 + 72 + 82 + 72 + 92 + 02 + 42 = 182.

Page of Squares : First Upload July 28, 2008 ; Last Revised July 28, 2008
by Yoshio Mimura, Kobe, Japan

8649

The square of 93.

403k + 961k + 5363k + 8649k are squares for k = 1,2,3 (1242, 102302, 8956522).

Page of Squares : First Upload July 28, 2008 ; Last Revised July 15, 2011
by Yoshio Mimura, Kobe, Japan

8652

86522 = (22 + 3)(32 + 3)(102 + 3)(932 + 3) = (92 + 3)(102 + 3)(932 + 3).

86522 = 1263 + 2623 + 3803 = 2023 + 2733 + 3593 = 2253 + 3113 + 3223.

86525 = 48482226238696492032, and
42 + 82 + 42 + 82 + 22 + 22 + 262 + 232 + 82 + 692 + 62 + 492 + 22 + 02 + 32 + 22 = 8652.

Page of Squares : First Upload July 28, 2008 ; Last Revised January 11, 2014
by Yoshio Mimura, Kobe, Japan

8655

86552 = 74909025, and 72 + 42 + 92 + 02 + 92 + 02 + 22 + 52 = 162.

Page of Squares : First Upload July 28, 2008 ; Last Revised July 28, 2008
by Yoshio Mimura, Kobe, Japan

8658

86582 = 74960964, and 7 * 4 + 960 * 9 - 6 - 4 = 8658.

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

Page of Squares : First Upload July 28, 2008 ; Last Revised July 15, 2011
by Yoshio Mimura, Kobe, Japan

8659

86592 = 74978281, a zigzag square.

Page of Squares : First Upload July 28, 2008 ; Last Revised July 28, 2008
by Yoshio Mimura, Kobe, Japan

8661

86612 = 413 + 2503 + 3903.

86615 = 48734912763123359301, and
42 + 82 + 72 + 32 + 42 + 92 + 122 + 762 + 312 + 232 + 32 + 52 + 92 + 302 + 12 = 8661.

Page of Squares : First Upload July 28, 2008 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

8662

86625 = 48763053952327380832, and
42 + 872 + 62 + 32 + 02 + 52 + 32 + 92 + 52 + 22 + 32 + 272 + 32 + 82 + 02 + 82 + 32 + 22 = 8662.

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

8664

952 + 8664 = 1332, 952 - 8664 = 192.

86642 = 683 + 3043 + 3603 = 1523 + 2923 + 3603.

Page of Squares : First Upload July 28, 2008 ; Last Revised August 5, 2011
by Yoshio Mimura, Kobe, Japan

8667

86672 = 883 + 2253 + 3983.

Page of Squares : First Upload July 28, 2008 ; Last Revised July 28, 2008
by Yoshio Mimura, Kobe, Japan

8668

86684 = 5645151616082176, and
52 + 62 + 452 + 152 + 162 + 162 + 02 + 82 + 22 + 12 + 762 = 8668.

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

8670

86702 = 93 + 3223 + 3473.

Page of Squares : First Upload July 28, 2008 ; Last Revised July 28, 2008
by Yoshio Mimura, Kobe, Japan

8671

86712 = 75186241, and 72 + 52 + 12 + 82 + 62 + 22 + 42 + 12 = 142.

Page of Squares : First Upload July 28, 2008 ; Last Revised July 28, 2008
by Yoshio Mimura, Kobe, Japan

8672

86722 = 213 + 353 + 4223 = 1203 + 3203 + 3443.

Page of Squares : First Upload July 28, 2008 ; Last Revised July 28, 2008
by Yoshio Mimura, Kobe, Japan

8673

86732 = 494 + 564 + 564 + 844.

86732 = 11002 + 11012 + 11022 + 11032 + ... + 11582.

Page of Squares : First Upload July 28, 2008 ; Last Revised July 28, 2008
by Yoshio Mimura, Kobe, Japan

8676

86762 = 75272976, and 75 * 2 + 7 * 29 * 7 * 6 = 8676.

Page of Squares : First Upload July 28, 2008 ; Last Revised July 28, 2008
by Yoshio Mimura, Kobe, Japan

8677

86774 = 5668633640928241, and
52 + 62 + 62 + 82 + 62 + 32 + 32 + 62 + 402 + 92 + 22 + 822 + 42 + 12 = 8677.

86775 = 49186734102334347157, and
492 + 12 + 82 + 62 + 72 + 342 + 102 + 232 + 32 + 432 + 472 + 152 + 72
= 492 + 182 + 62 + 72 + 32 + 412 + 02 + 232 + 342 + 32 + 472 + 152 + 72 = 8677.

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

8679

86792 = 13 + 2183 + 4023 = 633 + 773 + 4213.

Page of Squares : First Upload July 28, 2008 ; Last Revised July 28, 2008
by Yoshio Mimura, Kobe, Japan

8681

86812 = 2423 + 2723 + 3453.

Page of Squares : First Upload July 28, 2008 ; Last Revised July 28, 2008
by Yoshio Mimura, Kobe, Japan

8683

86835 = 49357028460152043643, and
492 + 32 + 52 + 702 + 282 + 42 + 62 + 02 + 12 + 52 + 202 + 42 + 32 + 62 + 42 + 32 = 8683.

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

8685

86852± 2 are primes.

Loop of length 35 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
8685 - 14621 - 2558 - 3989 - ... - 9925 - 10426 - 693 - 8685
(Note f(8685) = 862 + 852 = 14621,   f(14621) = 12 + 462 + 212 = 2558, etc. See 37)

Page of Squares : First Upload October 9, 2008 ; Last Revised January 11, 2014
by Yoshio Mimura, Kobe, Japan

8686

86862± 3 are primes.

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

8687

86872 = 322 + 332 + 342 + 352 + 362 + ... + 6092.

Page of Squares : First Upload July 28, 2008 ; Last Revised July 28, 2008
by Yoshio Mimura, Kobe, Japan

8688

86882 = 75481344, and 72 + 52 + 42 + 82 + 12 + 32 + 42 + 42 = 142.

Page of Squares : First Upload July 28, 2008 ; Last Revised July 28, 2008
by Yoshio Mimura, Kobe, Japan

8689

86892 = 75498721, and 72 + 52 + 42 + 92 + 82 + 72 + 22 + 12 = 172.

Page of Squares : First Upload July 28, 2008 ; Last Revised July 28, 2008
by Yoshio Mimura, Kobe, Japan

8694

86942 = (12 + 5)(22 + 5)(32 + 5)(82 + 5)(382 + 5) = (12 + 5)(22 + 5)(312 + 5)(382 + 5)
= (12 + 5)(22 + 5)(42 + 5)(82 + 5)(312 + 5) = (12 + 5)(82 + 5)(112 + 5)(382 + 5)
= (22 + 5)(172 + 5)(1692 + 5) = (22 + 5)(32 + 5)(42 + 5)(1692 + 5)
= (22 + 5)(82 + 5)(112 + 5)(312 + 5) = (32 + 5)(382 + 5)(612 + 5)
= (32 + 5)(42 + 5)(82 + 5)(612 + 5) = (32 + 5)(72 + 5)(82 + 5)(382 + 5)
= (42 + 5)(112 + 5)(1692 + 5) = (42 + 5)(312 + 5)(612 + 5) = (42 + 5)(72 + 5)(82 + 5)(312 + 5)
= (72 + 5)(312 + 5)(382 + 5) = (82 + 5)(172 + 5)(612 + 5).

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

8696

86963 = 657595137536, and 62 + 52 + 752 + 92 + 52 + 12 + 32 + 72 + 532 + 62 = 8696.

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

8697

86975 = 49756216673337223257, and
42 + 92 + 752 + 62 + 22 + 162 + 62 + 72 + 32 + 32 + 372 + 22 + 232 + 252 + 72 = 8697.

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