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, 2008by 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, 2008by Yoshio Mimura, Kobe, Japan
8607
86074 = 5487912924041601, and
52 + 42 + 872 + 92 + 12 + 292 + 22 + 42 + 02 + 42 + 12 + 62 + 02 + 12 = 8607.
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.
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, 2014by 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.
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.)
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, 2008by 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.
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, 2008by 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, 2011by 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.
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, 2011by 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, 2008by Yoshio Mimura, Kobe, Japan
8631
86312 = 683 + 2383 + 3933.
Page of Squares : First Upload July 28, 2008 ; Last Revised July 28, 2008by Yoshio Mimura, Kobe, Japan
8632
86325 = 47924452309775122432, and
472 + 92 + 22 + 442 + 522 + 302 + 92 + 72 + 72 + 52 + 12 + 22 + 242 + 32 + 22 = 8632.
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, 2008by Yoshio Mimura, Kobe, Japan
8637
86372 = 2333 + 3103 + 3183.
Page of Squares : First Upload July 28, 2008 ; Last Revised July 28, 2008by 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.
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, 2008by 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.
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, 2014by 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, 2008by 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, 2011by 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.
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, 2008by 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, 2011by Yoshio Mimura, Kobe, Japan
8659
86592 = 74978281, a zigzag square.
Page of Squares : First Upload July 28, 2008 ; Last Revised July 28, 2008by 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.
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.
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, 2011by Yoshio Mimura, Kobe, Japan
8667
86672 = 883 + 2253 + 3983.
Page of Squares : First Upload July 28, 2008 ; Last Revised July 28, 2008by Yoshio Mimura, Kobe, Japan
8668
86684 = 5645151616082176, and
52 + 62 + 452 + 152 + 162 + 162 + 02 + 82 + 22 + 12 + 762 = 8668.
by Yoshio Mimura, Kobe, Japan
8670
86702 = 93 + 3223 + 3473.
Page of Squares : First Upload July 28, 2008 ; Last Revised July 28, 2008by 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, 2008by Yoshio Mimura, Kobe, Japan
8672
86722 = 213 + 353 + 4223 = 1203 + 3203 + 3443.
Page of Squares : First Upload July 28, 2008 ; Last Revised July 28, 2008by 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, 2008by 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, 2008by 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.
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, 2008by Yoshio Mimura, Kobe, Japan
8681
86812 = 2423 + 2723 + 3453.
Page of Squares : First Upload July 28, 2008 ; Last Revised July 28, 2008by 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.
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)
by Yoshio Mimura, Kobe, Japan
8686
86862± 3 are primes.
Page of Squares : First Upload January 18, 2014 ; Last Revised January 18, 2014by Yoshio Mimura, Kobe, Japan
8687
86872 = 322 + 332 + 342 + 352 + 362 + ... + 6092.
Page of Squares : First Upload July 28, 2008 ; Last Revised July 28, 2008by 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, 2008by 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, 2008by 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).
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, 2008by Yoshio Mimura, Kobe, Japan
8697
86975 = 49756216673337223257, and
42 + 92 + 752 + 62 + 22 + 162 + 62 + 72 + 32 + 32 + 372 + 22 + 232 + 252 + 72 = 8697.
by Yoshio Mimura, Kobe, Japan