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8100 - 8199

8100

The square of 90.

4-cycle : 81002 = 65610000 - 61002 = 37210000 - 21002 = 04410000 - 41002 = 16810000,
(Other examples : 2916 - 5030 - 3009 - 0540 - 2916, 1600 - 5600 - 3600 - 9600 - 1600).

81002 = 93 + 2553 + 3663 = 873 + 2883 + 3453 = 2343 + 2823 + 3123.

Komachi equations: 81002 = 14 / 2344 * 54 * 64 * 784 * 94.

Page of Squares : First Upload June 23, 2008 ; Last Revised November 9, 2010
by Yoshio Mimura, Kobe, Japan

8105

81052 = 65691025, and 6 * 5 * 6 * 9 * 10 / 2 + 5 = 8105.

Page of Squares : First Upload June 23, 2008 ; Last Revised June 23, 2008
by Yoshio Mimura, Kobe, Japan

8106

81062 = (32 + 3)(23402 + 3).

81062 = 13 + 2273 + 3783.

1617k + 8106k + 10290k + 15708k are squares for k = 1,2,3 (1892, 205172, 23456792).
4998k + 6888k + 8106k + 15729k are squares for k = 1,2,3 (1892, 196352, 22080872).
3045k + 3381k + 4305k + 5145k are squares for k = 1,2,3 (1262, 81062, 5318462).

Page of Squares : First Upload June 23, 2008 ; Last Revised January 11, 2014
by Yoshio Mimura, Kobe, Japan

8108

81085 = 35040371352329120768 :
32 + 502 + 402 + 372 + 12 + 352 + 22 + 32 + 292 + 12 + 202 + 72 + 62 + 82 = 8108.

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

8109

81092 = 65755881, and 62 + 52 + 72 + 52 + 52 + 82 + 82 + 12 = 172.

81092 = (59 + 60 + 61 + ... + 109)2 + (110 + 111 + 112 + ... + 160)2.

Page of Squares : First Upload June 23, 2008 ; Last Revised June 23, 2008
by Yoshio Mimura, Kobe, Japan

8110

81102± 3 are primes.

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

8112

81122 = 2603 + 3643 = 1563 + 2863 + 3383.

Page of Squares : First Upload June 23, 2008 ; Last Revised June 23, 2008
by Yoshio Mimura, Kobe, Japan

8116

1 / 8116 = 0.00012321...., 12321 = 1112.

Page of Squares : First Upload June 23, 2008 ; Last Revised June 23, 2008
by Yoshio Mimura, Kobe, Japan

8118

81182 = 65901924, and 6 + 5 + 901 * 9 + 2 - 4 = 6 - 5 + 901 * 9 + 2 * 4 = 8118.

81182 = (12 + 2)(112 + 2)(142 + 2)(302 + 2) = (12 + 2)(22 + 2)(32 + 2)(112 + 2)(522 + 2)
= (12 + 2)(22 + 2)(32 + 2)(82 + 2)(712 + 2) = (12 + 2)(32 + 2)(42 + 2)(112 + 2)(302 + 2)
= (12 + 2)(82 + 2)(112 + 2)(522 + 2) = (112 + 2)(142 + 2)(522 + 2)
= (22 + 2)(32 + 2)(142 + 2)(712 + 2) = (32 + 2)(42 + 2)(112 + 2)(522 + 2)
= (32 + 2)(42 + 2)(82 + 2)(712 + 2) = (52 + 2)(302 + 2)(522 + 2)
= (82 + 2)(142 + 2)(712 + 2).

Page of Squares : First Upload June 23, 2008 ; Last Revised January 11, 2014
by Yoshio Mimura, Kobe, Japan

8119

81192 = 65918161, a zigzag square.

Page of Squares : First Upload June 23, 2008 ; Last Revised June 23, 2008
by Yoshio Mimura, Kobe, Japan

8121

81212 = 93 + 223 + 4043 = 2183 + 2443 + 3453.

Page of Squares : First Upload June 23, 2008 ; Last Revised June 23, 2008
by Yoshio Mimura, Kobe, Japan

8122

81222 = 474 + 634 + 694 + 694.

Page of Squares : First Upload June 23, 2008 ; Last Revised June 23, 2008
by Yoshio Mimura, Kobe, Japan

8123

81232 = 83 + 3123 + 3293.

The square root of 8123 is 90.1276...., and 90 = 12 + 22 + 72 + 62.

Page of Squares : First Upload June 23, 2008 ; Last Revised June 23, 2008
by Yoshio Mimura, Kobe, Japan

8125

81252 = 56 + 106 + 106 + 206.

81254 = 4358062744140625,
and 432 + 52 + 82 + 02 + 622 + 72 + 42 + 42 + 12 + 402 + 62 + 252 = 8125.

Loop of length 14 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
8125 - 7186 - 12437 - 1946 - ... - 10993 - 8731 - 8530 - 8125
(Note f(8125) = 812 + 252 = 7186,   f(7186) = 712 + 862 = 12437, etc. See 1946)

Page of Squares : First Upload June 23, 2008 ; Last Revised December 1, 2008
by Yoshio Mimura, Kobe, Japan

8126

81262 = 1333 + 1513 + 3923.

Page of Squares : First Upload June 23, 2008 ; Last Revised June 23, 2008
by Yoshio Mimura, Kobe, Japan

8127

81272 = (42 + 5)(342 + 5)(522 + 5).

Komachi equations: 81272 = 92 / 82 * 72 * 62 / 52 * 432 * 22 * 102.

Page of Squares : First Upload November 9, 2010 ; Last Revised January 11, 2014
by Yoshio Mimura, Kobe, Japan

8128

81282 = 943 + 2213 + 3793.

Page of Squares : First Upload June 23, 2008 ; Last Revised June 23, 2008
by Yoshio Mimura, Kobe, Japan

8129

81292 = 66080641, and 6 + 60 + 8064 - 1 = 66 + 0 + 8064 - 1 = 8129.

Page of Squares : First Upload June 23, 2008 ; Last Revised June 23, 2008
by Yoshio Mimura, Kobe, Japan

8131

81312 = 66113161, a square consisting of 3 kinds of digits.

81312 = 66113161, and 62 + 62 + 12 + 12 + 32 + 12 + 62 + 12 = 112.

Page of Squares : First Upload June 23, 2008 ; Last Revised June 23, 2008
by Yoshio Mimura, Kobe, Japan

8136

Loop of length 56 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
8136 - 7857 - 9333 - 9738 - ... - 7976 - 12017 - 690 - 8136
(Note f(8136) = 812 + 362 = 7857,   f(7857) = 782 + 572 = 9333, etc. See 41)

81362 = 1853 + 2923 + 3273.

Page of Squares : First Upload June 23, 2008 ; Last Revised October 9, 2008
by Yoshio Mimura, Kobe, Japan

8140

The square root of 8140 is 90.2219...., and 90 = 22 + 22 + 12 + 92.

81402 = 1832 + 1842 + 1852 + 1862 + 1872 + 1882 + ... + 5892.

Page of Squares : First Upload June 23, 2008 ; Last Revised June 23, 2008
by Yoshio Mimura, Kobe, Japan

8142

81425 = 35781248505531179232 :
32 + 52 + 782 + 12 + 22 + 42 + 82 + 52 + 02 + 52 + 52 + 312 + 172 + 92 + 232 + 22 = 8142.

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

8143

81432 = 1612 + 1632 + 1652 + 1672 + 1692 + 1712 + ... + 7372.

Page of Squares : First Upload June 23, 2008 ; Last Revised June 23, 2008
by Yoshio Mimura, Kobe, Japan

8144

81442 = 1003 + 1723 + 3923.

Page of Squares : First Upload June 23, 2008 ; Last Revised June 23, 2008
by Yoshio Mimura, Kobe, Japan

8145

81452 = 1823 + 2683 + 3453.

Page of Squares : First Upload June 23, 2008 ; Last Revised June 23, 2008
by Yoshio Mimura, Kobe, Japan

8146

81462 = 2023 + 2853 + 3273.

Page of Squares : First Upload June 23, 2008 ; Last Revised June 23, 2008
by Yoshio Mimura, Kobe, Japan

8147

81472 = 66373609, and 62 + 62 + 32 + 72 + 32 + 62 + 02 + 92 = 162.

Page of Squares : First Upload June 23, 2008 ; Last Revised June 23, 2008
by Yoshio Mimura, Kobe, Japan

8148

81482 = 66389904, and 6 - 6 / 3 + 8 + 9 * 904 = 6 * 6 - 3 * 8 + 9 * 904
= 6 / 6 + 3 + 8 + 9 * 904 = 8148.

81482 = (12 + 3)(22 + 3)(92 + 3)(1682 + 3) = (22 + 3)(32 + 3)(52 + 3)(1682 + 3)
= (52 + 3)(92 + 3)(1682 + 3).

Page of Squares : First Upload June 23, 2008 ; Last Revised January 11, 2014
by Yoshio Mimura, Kobe, Japan

8150

81502 = (262 + 263 + 264 + 265 + 266)2 + (267 + 268 + 269 + 270 + 271)2 + (272 + 273 + 274 + 275 + 276)2 + ... + (382 + 383 + 384 + 385 + 386)2.

Page of Squares : First Upload June 23, 2008 ; Last Revised June 23, 2008
by Yoshio Mimura, Kobe, Japan

8151

1 / 8151 = 0.000122684333210, 12 + 222 + 682 + 432 + 332 + 22 + 102 = 8151,
1 / 8151 = 0.00012268433321, 122 + 22 + 682 + 432 + 332 + 212 = 8151.

81512 = 773 + 813 + 4033.

Page of Squares : First Upload June 23, 2008 ; Last Revised June 23, 2008
by Yoshio Mimura, Kobe, Japan

8154

81542 = 253 + 2673 + 3623 = 154 + 254 + 614 + 854.

8154 = (12 + 22 + 32 + ... + 16602) / (12 + 22 + 32 + ... + 822).

Page of Squares : First Upload June 23, 2008 ; Last Revised November 25, 2008
by Yoshio Mimura, Kobe, Japan

8157

1 / N = 0.0001225...., and 1225 = 352, where N = 8157, 8158,..., 8163.

Page of Squares : First Upload June 23, 2008 ; Last Revised June 23, 2008
by Yoshio Mimura, Kobe, Japan

8160

81602 = (162 - 1)(5112 - 1) = (22 - 1)(92 - 1)(162 - 1)(332 - 1).

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

8162

81622 = (42 + 6)(102 + 6)(1692 + 6).

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

8163

81632 = 163 + 2543 + 3693 = 1343 + 2173 + 3783.

Page of Squares : First Upload June 23, 2008 ; Last Revised June 23, 2008
by Yoshio Mimura, Kobe, Japan

8164

1 / 8164 = 0.0001224889759, 12 + 22 + 22 + 482 + 82 + 92 + 752 + 92 = 8164.

Page of Squares : First Upload June 23, 2008 ; Last Revised June 23, 2008
by Yoshio Mimura, Kobe, Japan

8165

81655 = 36289496730323103125 :
362 + 22 + 82 + 92 + 42 + 92 + 672 + 32 + 02 + 32 + 232 + 12 + 02 + 312 + 252 = 8165.

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

8166

81665 = 36311724768814740576 :
362 + 32 + 12 + 172 + 242 + 72 + 62 + 82 + 82 + 142 + 742 + 02 + 52 + 72 + 62 = 8166.

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

8167

81672 = 66699889, a square with 3 kinds of digits.

81672 = 44 + 344 + 674 + 824.

Page of Squares : First Upload June 23, 2008 ; Last Revised June 23, 2008
by Yoshio Mimura, Kobe, Japan

8169

81692 = 66732561, and 62 + 62 + 72 + 32 + 22 + 52 + 62 + 12 = 142.

Page of Squares : First Upload June 23, 2008 ; Last Revised June 23, 2008
by Yoshio Mimura, Kobe, Japan

8170

8170k + 9374k + 20210k + 28810k are squares for k = 1,2,3 (2582, 373242, 57910682).

Page of Squares : First Upload July 12, 2011 ; Last Revised July 12, 2011
by Yoshio Mimura, Kobe, Japan

8171

The square root of 8171 is 90.39...., 90 = 32 + 92.

81712 = 2583 + 2863 + 2973.

Page of Squares : First Upload June 23, 2008 ; Last Revised June 23, 2008
by Yoshio Mimura, Kobe, Japan

8172

The square root of 8172 is 90.39...., 90 = 32 + 92.

Page of Squares : First Upload June 23, 2008 ; Last Revised June 23, 2008
by Yoshio Mimura, Kobe, Japan

8173

81732 = 1853 + 365 + 27.

Page of Squares : First Upload November 9, 2010 ; Last Revised November 9, 2010
by Yoshio Mimura, Kobe, Japan

8175

81752 = 233 + 1113 + 4033.

Page of Squares : First Upload June 23, 2008 ; Last Revised June 23, 2008
by Yoshio Mimura, Kobe, Japan

8176

81762 = 843 + 2063 + 3863 = 923 + 1803 + 3923.

Page of Squares : First Upload June 23, 2008 ; Last Revised June 23, 2008
by Yoshio Mimura, Kobe, Japan

8177

8177 = (12 + 22 + 32 + ... + 1102) / (12 + 22 + 32 + 42 + 52).

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

8180

81802 = 1283 + 1543 + 3943.

Page of Squares : First Upload June 23, 2008 ; Last Revised June 23, 2008
by Yoshio Mimura, Kobe, Japan

8181

81812 = 903 + 2973 + 3423.

1 / 8181 = 0.00012223444566, 122 + 222 + 342 + 42 + 452 + 662 = 8181.

Page of Squares : First Upload June 23, 2008 ; Last Revised June 23, 2008
by Yoshio Mimura, Kobe, Japan

8184

The integral triangle of sides 5057, 30880, 33759 (or 5185, 31713, 34976) has square area 81842.

Page of Squares : First Upload October 18, 2011 ; Last Revised October 18, 2011
by Yoshio Mimura, Kobe, Japan

8189

81892 = 1983 + 2763 + 3373 = 2563 + 2813 + 3043.

Page of Squares : First Upload June 23, 2008 ; Last Revised June 23, 2008
by Yoshio Mimura, Kobe, Japan

8190

81902 = (11 + 12 + 13 + 14 + 15 + 16)2 + (17 + 18 + 19 + 20 + 21 + 22)2 + (23 + 24 + 25 + 26 + 27 + 28)2 + ... + (317 + 318 + 319 + 320 + 321 + 322)2.

The integral triangle of sides 3825, 36946, 38321 has square area 81902.

Page of Squares : First Upload June 23, 2008 ; Last Revised October 18, 2011
by Yoshio Mimura, Kobe, Japan

8191

81912 = 67092481, a square with different digits.

81912 = 163 + 3213 + 3243 = 573 + 2763 + 3583.

Page of Squares : First Upload June 23, 2008 ; Last Revised June 23, 2008
by Yoshio Mimura, Kobe, Japan

8192

81922 = (32 + 7)(112 + 7)(1812 + 7).

81922 = 644 + 644 + 644 + 644 = 325 + 325 = 166 + 166 + 166 + 166 = 88 + 88 + 88 + 88.

Page of Squares : First Upload June 23, 2008 ; Last Revised January 11, 2014
by Yoshio Mimura, Kobe, Japan

8193

81932 = 44 + 274 + 684 + 824.

81932 = 40 + 47 + 413.

81935 = 36916011643785224193 :
32 + 62 + 92 + 162 + 02 + 12 + 12 + 642 + 372 + 82 + 52 + 222 + 412 + 92 + 32 = 8193.

Page of Squares : First Upload June 23, 2008 ; Last Revised August 30, 2011
by Yoshio Mimura, Kobe, Japan

8194

81942 = 67141636, a zigzag square.

81945 = 36938546139294728224 :
362 + 92 + 32 + 82 + 542 + 62 + 12 + 392 + 292 + 42 + 72 + 282 + 22 + 242 = 8194.

Page of Squares : First Upload June 23, 2008 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

8196

81962 = 1973 + 2743 + 3393.

Page of Squares : First Upload June 23, 2008 ; Last Revised June 23, 2008
by Yoshio Mimura, Kobe, Japan

8197

81972 = 67190809, a zigzag square.

1 / N = 0.000121...., and 121 = 112 for N = 8197, 8198,..., 8264.

1 / 8197 = 0.000121995852141, 12 + 22 + 192 + 92 + 582 + 522 + 12 + 412 = 8197.

Page of Squares : First Upload June 23, 2008 ; Last Revised June 23, 2008
by Yoshio Mimura, Kobe, Japan

8198

81982 = 23 + 3073 + 3373.

Page of Squares : First Upload June 23, 2008 ; Last Revised June 23, 2008
by Yoshio Mimura, Kobe, Japan

8199

81992± 2 are primes.

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