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1000 - 1099

1000

(12 + 22 + ... + 122) * (12 + 22 + ... + 1152) = (12 + 22 + ... + 10002),
(12 + 22 + ... + 112) * (12 + 22 + ... + 122) * (12 + 22 + ... + 142) = (12 + 22 + ... + 10002).

10002 = (12 + 4)(22 + 4)(112 + 4)(142 + 4) = (62 + 4)(112 + 4)(142 + 4).

10002 = 163 + 683 + 883 = 353 + 703 + 853.

Komachi equations: 10002 = 16 * 2346 * 56 * 66 / 786 / 96.

Page of Squares : First Upload December 11, 2006 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1001

10012 = 1002001, 1002 + 0 + 0 - 1 = 1002 + 0 * 0 -1 = 1001.

10012 = 1002001, a palindromic square, and a square with 3 kinds of digits.

10012 = (15 + 25 + 35 + ... + 135).

140140k + 185185k + 266266k + 410410k are squares for k = 1,2,3 (10012, 5415412, 3116223112).

Page of Squares : First Upload December 11, 2006 ; Last Revised April 5, 2011
by Yoshio Mimura, Kobe, Japan

1002

10022 = 1004004, a reversible square (4004001 = 20012) and a square with 3 kinds of digits.

1002k + 58116k + 82665k + 109218k are squares for k = 1,2,3 (5012, 1487972, 454311812).

The 4-by-4 magic squares consisting of different squares with constant 1002:

02 42 52312
112192222 62
162242132 12
252 72182 22
     
12 62172262
82292 42 92
192 22212142
242112162 72

Page of Squares : First Upload December 11, 2006 ; Last Revised April 5, 2011
by Yoshio Mimura, Kobe, Japan

1003

10032 = 1006009, a reversible square (9006001 = 30012) and a square with 3 kinds of digits.

Komachi equation: 10032 = 13 * 23 * 33 - 43 + 53 + 673 + 893.

Page of Squares : First Upload December 11, 2006 ; Last Revised July 16, 2010
by Yoshio Mimura, Kobe, Japan

1005

10052 = 1010025, 1010 + 0 * 2 - 5 = 1005.

Komachi equations:
10052 = 12 * 22 / 32 * 42 * 52 * 672 / 82 * 92 = 12 / 22 / 32 / 42 * 52 * 672 * 82 * 92.

Page of Squares : First Upload December 11, 2006 ; Last Revised July 16, 2010
by Yoshio Mimura, Kobe, Japan

1006

10062 = 1012036, 1012 + 0 * 3 - 6 = 1006.

Page of Squares : First Upload December 11, 2006 ; Last Revised December 11, 2006
by Yoshio Mimura, Kobe, Japan

1007

10072 = 583 + 663 + 813.

6042k + 223554k + 310156k + 474297k are squares for k = 1,2,3 (10072, 6092352, 3843245712).

Page of Squares : First Upload June 23, 2008 ; Last Revised April 5, 2011
by Yoshio Mimura, Kobe, Japan

1008

A cubic polynomial:
(X + 7042)(X + 8612)(X + 10082) = X3 + 15012X2 + 12744482X + 6109931522.

A, B, C, A+B, B+C and C+A are squares for (A, B, C, D) = (10082, 11002, 11552).

10082 = (22 - 1)(52 - 1)(82 - 1)(152 - 1) = (82 - 1)(1272 - 1).

Komachi equations:
10082 = 92 * 82 * 72 * 62 * 52 * 42 / 32 / 22 / 102 = 92 * 82 * 72 * 62 / 52 / 42 / 32 * 22 * 102
 = 92 * 82 * 72 / 62 * 52 * 42 * 32 * 22 / 102 = 92 * 82 * 72 / 62 / 52 * 42 * 32 / 22 * 102
 = 982 / 72 * 62 * 52 * 42 * 32 * 22 / 102 = 982 / 72 * 62 / 52 * 42 * 32 / 22 * 102.

10082 = 84 + 164 + 244 + 284 = 124 +244 + 244 + 244.

10082 = (1)(2)(3 + 4 + ... + 11)(12)(13 + 14 + 15)(16),
10082 = (1)(2)(3 + 4 + 5)(6 + 7 + ... + 12)(13 + 14 + 15)(16),
10082 = (1)(2 + 3 + ... + 7)(8)(9 + 10 + ... + 12)(13 + 14 + ... + 19),
10082 = (1)(2 + 3 + 4)(5 + 6 + 7)(8)(9 + 10 + ... + 40),
10082 = (1 + 2)(3)(4)(5 + 6 + ... + 11)(12)(13 + 14 + 15),
10082 = (1 + 2)(3)(4)(5 + 6 + 7)(8 + 9 + ... + 56),
10082 = (1 + 2)(3 + 4)(5 + 6 + ... + 11)(12 + 13 + ... + 15)(16),
10082 = (1 + 2)(3 + 4 + 5)(6)(7)(8)(9 + 10 + ... + 15),
10082 = (1 + 2)(3 + 4 + 5)(6 + 7 + ... + 26)(27 + 28 + 29),
10082 = (1 + 2)(3 + 4 + 5)(6 + 7 + 8)(9 + 10 + ... + 15)(16),
10082 = (1 + 2 + ... + 8)(9 + 10 + ... + 12)(13 + 14 + 15)(16).

Page of Squares : First Upload December 11, 2006 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1009

10092 = 1018081, 101 * 80 / 8 - 1 = 1018 + 0 - 8 - 1 = 1009.

10092 = 1018081, a square with 3 kinds of digits.

1009 is the 5th prime for which the Lendre Symbol (a/1009) = 1 for a = 1, 2, ..., 10.

Page of Squares : First Upload December 11, 2006 ; Last Revised November 11, 2006
by Yoshio Mimura, Kobe, Japan

1010

10102 = 1020100, 1020 - 10 +(-) 0 = 1010.

10102 = 1020100, a square with 3 kinds of digits.

10103 = 1030301000, and 12 + 02 + 32 + 02 + 302 + 102 + 02 + 02 = 12 + 02 + 302 + 32 + 02 + 102 + 02 + 02 = 102 + 32 + 02 + 302 + 12 + 02 + 02 + 02 = 102 + 302 + 32 + 02 + 12 + 02 + 02 + 02 = 1010.

1010k + 1690k + 2570k + 2830k are squares for k = 1,2,3 (902, 43002, 2133002).

Page of Squares : First Upload December 11, 2006 ; Last Revised April 5, 2011
by Yoshio Mimura, Kobe, Japan

1011

10112 = 1022121, a reversible square (1212201 = 11012), and a square with 3 kinds of digits.

10112 = 1022121, 1022 - 12 + 1 = 1011.

Komachi equation: 10112 = 1 * 234 * 56 * 78 + 9.

Page of Squares : First Upload December 11, 2006 ; Last Revised July 16, 2010
by Yoshio Mimura, Kobe, Japan

1012

10122 = 1024144, a reversible square (4414201 = 21012).

10122 = (42 + 7)(2112 + 7).

10123 = 1036433728, and 102 + 32 + 62 + 42 + 32 + 32 + 72 + 282 = 1012.

10122 = 522 + 532 + 542 + 552 + 562 + 572 + 582 + ... + 1472.

10122 = 1024144, 1024 = 322, 144 = 122.

Page of Squares : First Upload December 11, 2006 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1013

10132 = 1026169, a reversible square (9616201 = 31012).

10132 = 1026169, 1 + 0 - 2 + 6 * 169 = 1013.

Page of Squares : First Upload December 11, 2006 ; Last Revised December 11, 2006
by Yoshio Mimura, Kobe, Japan

1014

10142 = 1028196, 102 + 8 * 19 * 6 = 1028 + 1 - 9 - 6 = 1014.

10142± 5 are primes.

10142 = 653 + 913.

The 4-by-4 magic squares consisting of different squares with constant 1014:

12 82182252
92222202 72
162212112142
262 52132122
     
22 52162272
92282 72102
202 32222112
232142152 82

Page of Squares : First Upload December 11, 2006 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

1015

1015 = (12 + 22 + 32 + ... + 1152) / (12 + 22 + 32 + ... + 112).

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

1016

10162 = 1032256, 1032 - 2 * 5 - 6 = 1016.

Page of Squares : First Upload December 11, 2006 ; Last Revised December 11, 2006
by Yoshio Mimura, Kobe, Japan

1017

10172 = 1034289, a square with different digits.

10172 = 1034289, 1 * 0 + 3 * 42 * 8 + 9 = 1017.

10172 = 1034289 appears in the decimal expressions of e:
  e = 2.71828•••1034289••• (from the 75148th digit)

Page of Squares : First Upload December 11, 2006 ; Last Revised November 4, 2008
by Yoshio Mimura, Kobe, Japan

1018

10182 = 253 + 433 + 983 = 114 + 114 + 174 + 314.

Page of Squares : First Upload December 11, 2006 ; Last Revised June 23, 2008
by Yoshio Mimura, Kobe, Japan

1019

10192 = 1038361, 1038 - 3 * 6 - 1 = 1019.

10192 = 1742 + 4572 + 8942 = 4982 + 7542 + 4712.

Page of Squares : First Upload December 11, 2006 ; Last Revised September 7, 2013
by Yoshio Mimura, Kobe, Japan

1020

990k + 1020k + 1785k + 1830k are squares for k = 1,2,3 (752, 29252, 1176752).

The integral triangle of sides 1445, 1476, 2281 has square area 10202.

10202 = (1)(2)(3 + 4 + 5)(6 + 7 + ... + 294),
10202 = (1 + 2)(3 + 4 + ... + 7)(8)(9 + 10 + ... + 59),
10202 = (1 + 2)(3 + 4 + ... + 7)(8 + 9)(10 + 11 + ... + 41),
10202 = (1 + 2)(3 + 4 + ... + 7)(8 + 9 + ... + 24)(25 + 26).

Page of Squares : First Upload December 11, 2006 ; Last Revised October 4, 2011
by Yoshio Mimura, Kobe, Japan

1021

10212 = 1042441, a reversible square (1442401 = 12012).

Page of Squares : First Upload December 11, 2006 ; Last Revised December 11, 2006
by Yoshio Mimura, Kobe, Japan

1022

10222 = 1044484, a reversible square (4844401 = 22012).

10222 = 1972 + 1982 + 1992 + 2002 + 2012 + 2022 + 2032 + ... + 2202.

10222 = 233 + 453 + 983.

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

Komachi equation: 10222 = 92 * 8762 / 542 / 32 * 212.

The 4-by-4 magic squares consisting of different squares with constant 1022:

02 22172272
112242182 12
152192202 62
262 92 32162
     
02 62192252
112 12242182
152272 22 82
262162 92 32
     
42 62212232
92192162182
142242152 52
272 72102122

Page of Squares : First Upload December 11, 2006 ; Last Revised April 5, 2011
by Yoshio Mimura, Kobe, Japan

1023

10232 = 1046529, a square with different digits.

10232 = 1046529, 1046 - 5 - 2 * 9 = 1023.

87978k + 223014k + 260865k + 474672k are squares for k = 1,2,3 (10232, 5923172, 3694247372).

10232 = 24 + 184 + 224 + 294.

Page of Squares : First Upload December 11, 2006 ; Last Revised April 5, 2011
by Yoshio Mimura, Kobe, Japan

1024

the square of 32.

10242 = (10 + 2 + 4)5.

1 / 1024 = 0.0009765625, where 9765625 = 31252.

322 = 1024 and 10242 = 1048576 are squares with different digits.

10242 = (52 + 7)(1812 + 7).

Komachi equaions:
10242 = 122 * 32 * 42 * 562 / 72 * 82 / 92,
10242 = 125 * 35 * 45 * 565 / 75 / 85 / 95 = 125 * 35 * 45 / 565 * 75 * 85 / 95.

10243 = 1073741824, and 12 + 02 + 72 + 32 + 72 + 42 + 182 + 242 = 1024.

10242 = 86 + 86 + 86 + 86.

Page of Squares : First Upload December 11, 2006 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1025

10252 = 1050625, a zigzag square.

10252 = 15 + 45 + 45 + 165.

Komachi equation: 10252 = - 9 + 8 * 76 * 54 * 32 + 10.

10254 = 1103812890625, and 12 + 12 + 02 + 32 + 82 + 122 + 82 + 92 + 02 + 62 + 252 = 1025.

Page of Squares : First Upload December 11, 2006 ; Last Revised July 16, 2010
by Yoshio Mimura, Kobe, Japan

1026

10262 = (1 + 2)(3 + 4 + ... + 21)(22 + 23 + ... + 59).

10262 = (22 + 2)(132 + 2)(322 + 2) = (22 + 2)(52 + 2)(62 + 2)(132 + 2) = (52 + 2)(62 + 2)(322 + 2).

Komachi equatios:
10262 = 92 * 82 * 762 * 52 / 42 * 32 / 22 / 102 = 92 / 82 * 762 * 52 * 42 * 32 * 22 / 102
 = 92 / 82 * 762 / 52 * 42 * 32 / 22 * 102.

The 4-by-4 magic squares consisting of different squares with constant 1026:

02 12 82312
42272162 52
132142252 62
292102 92 22
   
02 12202252
32302 92 62
212102172142
242 52162132
   
02 32212242
42232152162
72222182132
312 22 62 52
   
02 32212242
82232172122
112222142152
292 22102 92
   
12 22112302
42232202 92
152182212 62
282132 82 32
12 42152282
102 92262132
212202112 82
222232 22 32
     
12 52182262
72292 62102
202 42212132
242122152 92
     
12102212222
122192202112
162232 42152
252 62132142
     
22 92102292
112202192122
152162232 42
262172 62 52

Page of Squares : First Upload December 11, 2006 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1027

10272 = 1054729, a zigzag square with different digits.

Page of Squares : First Upload December 11, 2006 ; Last Revised December 11, 2006
by Yoshio Mimura, Kobe, Japan

1028

10282 = 1056784, a square with different digits.

10282 = 24 + 84 + 84 + 324.

178k + 242k + 494k + 850k are squares for k = 1,2,3 (422, 10282, 274682).

Page of Squares : First Upload December 11, 2006 ; Last Revised April 5, 2011
by Yoshio Mimura, Kobe, Japan

1029

10292 = 493 + 983.

Komachi equations:
10292 = 982 * 72 / 62 * 542 / 32 / 22 * 12 = 982 * 72 / 62 * 542 / 32 / 22 / 12
 = 982 * 72 * 62 / 52 * 42 / 322 * 102,
10292 = 983 + 73 * 63 / 543 * 33 * 213.

The 4-by-4 magic squares consisting of different squares with constant 1029:

02 22202252
42302 72 82
222 52182142
232102162122
     
02 42222232
72162182202
142262112 62
282 92102 82

Page of Squares : First Upload June 23, 2008 ; Last Revised July 16, 2010
by Yoshio Mimura, Kobe, Japan

1031

10312 = 1062961, a reversible square (1692601 = 13012).

10312 = 1062961 appears in the decimal expressions of e:
  e = 2.71828•••1062961••• (from the 97690th digit)

Page of Squares : First Upload December 11, 2006 ; Last Revised November 4, 2008
by Yoshio Mimura, Kobe, Japan

1032

10322 = (1 + 2 + ... + 128)(129).

10322 = 43 + 165 + 47.

Komachi equation: 10322 = 12 * 232 * 452 - 62 - 782 - 92.

Page of Squares : First Upload December 11, 2006 ; Last Revised January 6, 2011
by Yoshio Mimura, Kobe, Japan

1033

10333 = 1102302937, and 112 + 02 + 22 + 32 + 02 + 292 + 32 + 72 = 1033.

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

1034

462k + 1034k + 2200k + 2233k are squares for k = 1,2,3 (772, 33332, 1516132).
704k + 1034k + 1870k + 6193k are squares for k = 1,2,3 (992, 65892, 4954952).

The 4-by-4 magic squares consisting of different squares with constant 1034:

02 42172272
82242152132
212192142 62
232 92182102
     
02 82212232
132272 62102
172152142182
242 42192 92

Page of Squares : First Upload November 26, 2009 ; Last Revised April 5, 2011
by Yoshio Mimura, Kobe, Japan

1035

10352± 2 are primes.

10352 = (13 + 23 + 33 + ... + 453).

Page of Squares : First Upload December 11, 2006 ; Last Revised December 29, 2013
by Yoshio Mimura, Kobe, Japan

1036

1036 = (12 + 22 + 32 + ... + 552) / (12 + 22 + 32 + 42 + 52).

10362 = 1073296, a square with different digits.

Page of Squares : First Upload December 11, 2006 ; Last Revised November 25, 2008
by Yoshio Mimura, Kobe, Japan

1037

10372 = 1075369, a square with different digits.

Page of Squares : First Upload December 11, 2006 ; Last Revised December 11, 2006
by Yoshio Mimura, Kobe, Japan

1038

The 4-by-4 magic squares consisting of different squares with constant 1038:

02 12192262
52302 82 72
222112172122
232 42182132
     
22 42172272
52252182 82
152192162142
282 62132 72
     
22 42172272
122262132 72
192152162142
232112182 82

Page of Squares : First Upload November 26, 2009 ; Last Revised November 26, 2009
by Yoshio Mimura, Kobe, Japan

1039

Komachi equation: 10392 = 9872 + 62 + 542 + 322 * 102.

10392 = 1942 + 4982 + 8912 = 1982 + 8942 + 4912.

Page of Squares : First Upload July 16, 2010 ; Last Revised September 7, 2013
by Yoshio Mimura, Kobe, Japan

1040

10402 = 183 x 184 + 185 x 186 + 187 x 188 + 189 x 190 + ... + 231 x 232.

10402 = (12 + 4)(22 + 4)(102 + 4)(162 + 4) = (12 + 4)(22 + 4)(32 + 4)(42 + 4)(102 + 4)
= (22 + 4)(102 + 4)(362 + 4) = (22 + 4)(32 + 4)(62 + 4)(162 + 4) = (32 + 4)(42 + 4)(62 + 4)(102 + 4)
= (62 + 4)(102 + 4)(162 + 4).

Komachi equation: 10402 = 122 * 32 * 42 * 52 / 62 * 782 / 92.

Page of Squares : First Upload December 11, 2006 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1041

1 / 1041 = 0.00096061479346781940441,
962 + 0612 + 4792 + 342 + 672 + 8192 + 4042 + 412 = 10412.

142617k + 218610k + 245676k + 476778k are squares for k = 1,2,3 (10412, 5964932, 3695352212).

The 4-by-4 magic squares consisting of different squares with constant 1041:

42 52182262
72282 82122
202 62222112
242142132102

Page of Squares : First Upload December 11, 2006 ; Last Revised April 5, 2011
by Yoshio Mimura, Kobe, Japan

1042

10422 = 1085764, a square with different digits.

Page of Squares : First Upload December 11, 2006 ; Last Revised December 11, 2006
by Yoshio Mimura, Kobe, Japan

1043

10432 = 1087849, a zigzag square.

10432 = 1087849, 1087 - 8 - 4 * 9 = 1043.

Page of Squares : First Upload December 11, 2006 ; Last Revised December 11, 2006
by Yoshio Mimura, Kobe, Japan

1044

10442 = 1089936, 10 * 8 * 9 + 9 * 36 = 1089 - 9 - 36 = 1044.

A cubic polynomial :
(X + 10442)(X + 26392)(X + 43682) = X3 + 52092X2 + 126988682X + 120343466882.

10442 = 123 + 303 + 1023.

Page of Squares : First Upload December 11, 2006 ; Last Revised June 23, 2008
by Yoshio Mimura, Kobe, Japan

1045

10452 = (34 + 35 + 36 + 37 + 38)2 + (39 + 40 + 41 + 42 + 43)2 + (44 + 45 + 46 + 47 + 48)2 + ... + (84 + 85 + 86 + 87 + 88)2,
10452 = (79 + 80)2 + (81 + 82)2 + (83 + 84)2 + ... + (127 + 128)2.

Page of Squares : First Upload December 11, 2006 ; Last Revised December 11, 2006
by Yoshio Mimura, Kobe, Japan

1048

10482 = 383 + 683 + 903.

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

1049

10492 = 1100401, a square with 3 kinds of digits.

Page of Squares : First Upload December 11, 2006 ; Last Revised December 11, 2006
by Yoshio Mimura, Kobe, Japan

1050

10502 = 83 + 213 + 1033 = 623 + 633 + 853.

10502 = (12 + 6)(32 + 6)(82 + 6)(122 + 6) = (22 + 6)(122 + 6)(272 + 6) = (62 + 6)(1622 + 6)
= (22 + 6)(32 + 6)(62 + 6)(132 + 6) = (32 + 6)(122 + 6)(222 + 6) = (62 + 6)(122 + 6)(132 + 6).

126k + 538k + 1050k + 1422k are squares for k = 1,2,3 (562, 18522, 647362).

The 4-by-4 magic squares consisting of different squares with constant 1050:

02 82192252
112212222 22
202162132152
232172 62142
     
02112202232
132222192 62
162212 82172
252 22152142
     
12 32162282
72292122 42
102142232152
302 22112 52
     
12 32162282
72292122 42
182142192132
262 22172 92
12 72102302
82242172112
122162252 52
292132 62 22
     
12 72182262
112272142 22
122162192172
282 42132 92
     
22 32192262
62252172102
132202162152
292 42122 72

10502 = (1)(2)(3 + 4)(5)(6 + 7 + ... + 9)(10 + 11)(12 + 13),
10502 = (1)(2)(3 + 4)(5)(6 + 7 + 8)(9 + 10 + 11)(12 + 13),
10502 = (1)(2)(3 + 4 + ... + 17)(18 + 19 + ... + 24)(25),
10502 = (1)(2)(3 + 4 + ... + 17)(18 + 19 + ... + 87),
10502 = (1)(2)(3 + 4 + ... + 7)(8 + 9 + ... + 22)(23 + 24 + ... + 26),
10502 = (1)(2 + 3)(4)(5)(6 + 7 + ... + 12)(13 + 14 + ... + 22),
10502 = (1)(2 + 3)(4)(5)(6 + 7 + ... + 19)(20 + 21 + 22),
10502 = (1)(2 + 3)(4)(5)(6 + 7 + 8)(9 + 10 + ... + 33),
10502 = (1)(2 + 3)(4 + 5 + ... + 11)(12 + 13 + ... + 86),
10502 = (1)(2 + 3)(4 + 5 + ... + 17)(18 + 19 + ... + 57),
10502 = (1)(2 + 3)(4 + 5 + ... + 31)(32 + 33 + ... + 43),
10502 = (1)(2 + 3)(4 + 5 + ... + 8)(9 + 10 + ... + 12)(13 + 14 + ... + 22),
10502 = (1)(2 + 3)(4 + 5 + 6)(7)(8 + 9 + ... + 12)(13 + 14 + 15),
10502 = (1)(2 + 3)(4 + 5 + 6)(7 + 8 + ... + 13)(14)(15),
10502 = (1 + 2 + ... + 14)(15)(16 + 17 + ... + 40),
10502 = (1 + 2 + ... + 20)(21 + 22 + ... + 104),
10502 = (1 + 2 + ... + 4)(5)(6 + 7 + ... + 9)(10 + 11 + ... + 39),
10502 = (1 + 2 + ... + 5)(6 + 7 + ... + 15)(16 + 17 + ... + 40),
10502 = (1 + 2 + ... + 5)(6 + 7 + ... + 19)(20)(21),
10502 = (1 + 2 + ... + 5)(6 + 7 + ... + 9)(10)(11 + 12 + ... + 24),
10502 = (1 + 2 + ... + 5)(6 + 7 + 8)(9 + 10 + ... + 16)(17 + 18),
10502 = (1 + 2 + ... + 6)(7 + 8)(9 + 10 + ... + 16)(17 + 18).

Page of Squares : First Upload December 11, 2006 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1053

The square root of 1053 is 32.44 ..., and 32 = 42 + 42.

(42 - 9)(72 - 9)(82 - 9)(92 - 9) = 10532 - 9.

10532 = (1 + 2)(3)(4 + 5)(6 + 7)(8 + 9 + ... + 46).

Page of Squares : First Upload December 11, 2006 ; Last Revised December 11, 2006
by Yoshio Mimura, Kobe, Japan

1054

10542 = 1110916, 12 + 12 + 12 + 02 + 92 + 12 + 62 = 112.

10542± 3 are primes.

125426k + 247690k + 340442k + 397358k are squares for k = 1,2,3 (10542, 5923482, 3454948762).

The 4-by-4 magic squares consisting of different squares with constant 1054:

12 42142292
112222202 72
162232132102
262 52172 82

Page of Squares : First Upload December 11, 2006 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

1055

10552 = 1242 + 1252 + 1262 + 1272 + 1282 + 1292 + 1302 + ... + 1732.

The quadratic polynomial -1055X2 + 52690X - 48719 takes the values 542, 2292, 3162, 3812, 4342, 4792 at X = 1, 2,..., 6.

Page of Squares : First Upload January 22, 2007 ; Last Revised December 15, 2008
by Yoshio Mimura, Kobe, Japan

1056

10562 = 1115136, a square with odd digits except the last digit 6.

10562 = 45 + 85 + 85 + 165.

A cubic polynomial :
(X + 10562)(X + 11482)(X + 14492) = X3 + 21292X2 + 25647722X + 17566053122.

10562 = 1115136 appears in the decimal expressions of e:
  e = 2.71828•••1115136••• (from the 1761st digit)
  (1115136 is the first 7-digit square in the expression of e.)

Page of Squares : First Upload December 11, 2006 ; Last Revised August 24, 2013
by Yoshio Mimura, Kobe, Japan

1058

10582 = 1119364, 1119 + 3 - 64 = 1058.

10582 = 233 + 693 + 923 = 234 + 234 + 234 +234.

Page of Squares : First Upload December 11, 2006 ; Last Revised June 23, 2008
by Yoshio Mimura, Kobe, Japan

1062

10622 = 103 + 753 + 893.

The 4-by-4 magic squares consisting of different squares with constant 1062:

02 12102312
32242212 62
182172202 72
272142112 42
     
02 32182272
52242192102
142212162132
292 62112 82
     
02 52192262
72202182172
222212112 42
232142162 92

Page of Squares : First Upload June 23, 2008 ; Last Revised December 3, 2009
by Yoshio Mimura, Kobe, Japan

1064

10642= 57 x 58 + 58 x 59 + 59 x 60 +...+ 152 x 153.

10642= (12 + 3)(42 + 3)(52 + 3)(232 + 3).

Page of Squares : First Upload December 11, 2006 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1065

1065 = (12 + 22 + 32 + ... + 352) / (12 + 22 + 32).

10652 = 13 + 743 + 903 = 163 + 413 + 1023.

The quadratic polynomial 1065X2 - 5670X + 8449 takes the values 622, 372, 322, 532, 822, 1132 at X = 1, 2,..., 6,

1065k + 3360k + 6180k + 7620k are squares for k = 1,2,3 (1352, 104252, 8471252).
1065k + 8094k + 14058k + 22152k are squares for k = 1,2,3 (2132, 274772, 37656272).

The 4-by-4 magic squares consisting of different squares with constant 1065:

02 42 52322
102232202 62
172142242 22
262182 82 12

Page of Squares : First Upload December 11, 2006 ; Last Revised April 5, 2011
by Yoshio Mimura, Kobe, Japan

1066

Loop of length 56 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
1066 - 4456 - 5072 - 7684 - ... - 5620 - 3536 - 2521 - 1066
(Note f(1066) = 102 + 662 = 4456,   f(4456) = 442 + 562 = 5072, etc. See 41)

116194k + 216398k + 308074k + 495690k are squares for k = 1,2,3 (10662, 6332042, 4034063802).

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by Yoshio Mimura, Kobe, Japan

1068

Komachi equation: 10682 = 12 * 22 * 32 / 42 * 562 / 72 * 892.

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1070

10705 = 1402551730700000 : 12 + 42 + 02 + 22 + 52 + 52 + 12 + 72 + 302 + 72 + 02 + 02 + 02 + 02 + 02 = 1070.

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

1071

10712± 2 are primes.

10713 = 1228480911, and 12 + 22 + 282 + 42 + 82 + 02 + 92 + 112 = 1071.

10712 = (1)(2 + 3 + ... + 52)(53 + 54 + ... + 66),
10712 = (1 + 2 + ... + 17)(18 + 19 + ... + 24)(25 + 26),
10712 = (14 + 15 + 16 + ... + 34)2 + (35 + 36 + 37 + ... + 55)2 + (56 + 57 + 58 + ... + 76)2 + ... + (35 + 36 + 37 + ... + 55)2.

Page of Squares : First Upload December 11, 2006 ; Last Revised December 29, 2013
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1072

10722 = 403 + 443 + 1003.

10722 = (12 + 7)(3792 + 7).

Page of Squares : First Upload June 23, 2008 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1074

The 4-by-4 magic squares consisting of different squares with constant 1074:

02 12 72322
162192212 42
172262102 32
232 62222 52
     
02 12172282
72302102 52
202132192122
252 22182112
     
02 72 82312
112182232102
132262152 22
282 52162 32
     
12 62192262
82232202 92
152222132142
282 52122112

Page of Squares : First Upload December 3, 2009 ; Last Revised December 3, 2009
by Yoshio Mimura, Kobe, Japan

1075

Komachi equation: 10752 = 983 - 73 - 63 + 53 * 43 * 33 - 23 - 103.

Page of Squares : First Upload July 16, 2010 ; Last Revised July 16, 2010
by Yoshio Mimura, Kobe, Japan

1076

10762 = 1157776, a square with odd digits except the last digit 6.

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by Yoshio Mimura, Kobe, Japan

1078

10782 = 1162084, 1162 + 0 - 84 = 1078.

924k + 1078k + 1617k + 2310k are squares for k = 1,2,3 (772, 31572, 1363672).

Page of Squares : First Upload December 11, 2006 ; Last Revised April 5, 2011
by Yoshio Mimura, Kobe, Japan

1080

10802 = 413 + 763 + 873.

10802 = (22 - 1)(42 - 1)(1612 - 1).

392 + 1080 = 512, 392 - 1080 = 212.

10802 = (1)(2)(3)(4)(5)(6 + 7 + ... + 21)(22 + 23),
10802 = (1)(2)(3)(4 + 5)(6)(7 + 8)(9 + 10 + ... + 23),
10802 = (1)(2)(3)(4 + 5)(6 + 7 + ... + 10)(11 + 12 + ... + 34),
10802 = (1)(2)(3)(4 + 5)(6 + 7 + ... + 14)(15)(16),
10802 = (1)(2)(3 + 4 + ... + 6)(7 + 8 + 9)(10)(11 + 12 + ... + 19),
10802 = (1)(2)(3 + 4 + 5)(6)(7 + 8)(9)(10 + 11 + ... + 14),
10802 = (1)(2 + 3)(4)(5 + 6 + 7)(8 + 9 + ... + 19)(20),
10802 = (1)(2 + 3)(4 + 5)(6 + 7 + ... + 10)(11 + 12 + ... + 37),
10802 = (1)(2 + 3)(4 + 5)(6 + 7 + ... + 21)(22 + 23 + ... + 26),
10802 = (1)(2 + 3 + ... + 13)(14 + 15 + ... + 31)(32),
10802 = (1)(2 + 3 + ... + 28)(29 + 30 + 31)(32),
10802 = (1)(2 + 3 + ... + 7)(8)(9 + 10 + ... + 16)(17 + 18 + 19),
10802 = (1)(2 + 3 + 4)(5)(6 + 7 + ... + 10)(11 + 12 + ... + 37),
10802 = (1)(2 + 3 + 4)(5)(6 + 7 + ... + 21)(22 + 23 + ... + 26),
10802 = (1)(2 + 3 + 4)(5 + 6 + ... + 10)(11 + 12 + 13)(14 + 15 + ... + 18),
10802 = (1 + 2)(3)(4 + 5 + ... + 8)(9 + 10 + 11)(12 + 13 + ... + 20),
10802 = (1 + 2)(3)(4 + 5 + 6)(7 + 8 + 9)(10)(11 + 12 + 13),
10802 = (1 + 2)(3 + 4 + ... + 6)(7 + 8 + 9)(10 + 11 + ... + 14)(15),
10802 = (1 + 2)(3 + 4 + 5)(6)(7 + 8)(9 + 10 + 11)(12),
10802 = (1 + 2)(3 + 4 + 5)(6 + 7 + ... + 14)(15 + 16 + ... + 30),
10802 = (1 + 2 + ... + 5)(6 + 7 + ... + 14)(15 + 16 + 17)(18),
10802 = (1 + 2 + 3)(4)(5)(6 + 7 + ... + 21)(22 + 23),
10802 = (1 + 2 + 3)(4 + 5)(6)(7 + 8)(9 + 10 + ... + 23),
10802 = (1 + 2 + 3)(4 + 5)(6 + 7 + ... + 10)(11 + 12 + ... + 34),
10802 = (1 + 2 + 3)(4 + 5)(6 + 7 + ... + 14)(15)(16).

Page of Squares : First Upload December 11, 2006 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1081

10812 = (13 + 23 + 33... + 463).

10812 + 10822 + 10832 + ... + 11042 = 11052 + 11062 + 11072 + ... + 11272.

Page of Squares : First Upload December 11, 2006 ; Last Revised September 9, 2011
by Yoshio Mimura, Kobe, Japan

1082

10822± 3 are primes.

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by Yoshio Mimura, Kobe, Japan

1083

10832 = 173 + 763 + 903 = 383 + 733 + 903.

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

1084

10842 = 303 + 803 + 863.

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

1085

1 / 1085 = 0.0009216 ...., and 9216 = 962.

(12 + 22 + 32 + ... + 2192) + (12 + 22 + 32 + ... + 10822) = (12 + 22 + 32 + ... + 10852).

10852 = 1177225, 1 + 1 + 7 * 7 * 22 + 5 = 11 * 7 * 7 * 2 + 2 + 5 = 1085.

Page of Squares : First Upload December 11, 2006 ; Last Revised December 11, 2006
by Yoshio Mimura, Kobe, Japan

1086

10862 = 1179396, 117 * 9 + 3 * 9 + 6 = 117 * 9 + 39 - 6 = 1179 + 3 - 96 = 1086.

10862 = 1179396, a square with odd digits except the last digit 6.

138k + 570k + 1086k + 1122k are squares for k = 1,2,3 (542, 16682, 536762).

The 4-by-4 magic squares consisting of different squares with constant 1086:

02 22112312
142282 92 52
192172202 62
232 32222 82
     
12 32202262
52312 62 82
222 42192152
242102172112
     
12 52 62322
102242192 72
162142252 32
272172 82 22

Page of Squares : First Upload December 11, 2006 ; Last Revised August 24, 2013
by Yoshio Mimura, Kobe, Japan

1088

10882 = 84 + 164 + 164 + 324.

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

1089

the square of 33.

332 = 1089, a reversible square (9801 = 992).

(12 + 22 + 32 + ... + 1162) + (12 + 22 + 32 + ... + 1252) = 10892.

The 4-by-4 magic squares consisting of different squares with constant 1089:

02 42172282
62302122 32
182132202142
272 22162102

Page of Squares : First Upload December 11, 2006 ; Last Revised December 3, 2009
by Yoshio Mimura, Kobe, Japan

1090

Loop of length 35 by the function f(N) = ... + c2 + b2 + a2 where N = ... + 1002c + 100b + a:
1090 - 8200 - 6724 - 5065 - ... - 1369 - 4930 - 3301 - 1090
(Note f(1090) = 102 + 902 = 8200,   f(8200) = 822 + 002 = 6724, etc. See 37)

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

1092

10922 = (12 + 3)(122 + 3)(452 + 3) = (12 + 3)(22 + 3)(62 + 3)(332 + 3) = (62 + 3)(92 + 3)(192 + 3)
= (12 + 3)(62 + 3)(72 + 3)(122 + 3) = (22 + 3)(32 + 3)(62 + 3)(192 + 3)
= (22 + 3)(62 + 3)(72 + 3)(92 + 3) = (22 + 3)(92 + 3)(452 + 3) = (52 + 3)(62 + 3)(332 + 3).

10922 = (1)(2)(3 + 4 + 5)(6 + 7)(8 + 9 + ... + 20)(21),
10922 = (1)(2 + 3 + ... + 5)(6 + 7)(8)(9)(10 + 11 + ... + 16),
10922 = (1)(2 + 3 + ... + 50)(51 + 52 + ... + 66),
10922 = (1 + 2)(3 + 4 + ... + 10)(11 + 12 + ... + 17)(18 + 19 + ... + 21).

78k + 1092k + 1950k + 10569k are squares for k = 1,2,3 (1172, 108032, 10905572).

The integral triangle of sides 939, 2548, 2785 (or 1192, 10985, 12159) has square area 10922.

Page of Squares : First Upload December 11, 2006 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1093

21092 - 1 is congruent to 0 modulo 10932.

Page of Squares : First Upload December 11, 2006 ; Last Revised December 11, 2006
by Yoshio Mimura, Kobe, Japan

1095

Komachi equation: 10952 = - 94 - 84 + 74 - 64 + 54 * 44 + 324 + 14.

41610k + 179580k + 358065k + 619770k are squares for k = 1,2,3 (10952, 7391252, 5383622252).

The 4-by-4 magic squares consisting of different squares with constant 1095:

32102192252
112212222 72
172232 92142
262 52132152
     
32112172262
142 72252152
192222 92132
232212102 52

Page of Squares : First Upload December 3, 2009 ; Last Revised April 5, 2011
by Yoshio Mimura, Kobe, Japan

1097

10972 = 193 + 473 + 1033.

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

1098

10982 = 94 + 94 + 94 + 334.

The 4-by-4 magic squares consisting of different squares with constant 1098:

02 42112312
122282132 12
152172222102
272 32182 62
     
02 52 72322
122192232 82
152262142 12
272 62182 32
     
02 92212242
122232192 82
152222102172
272 22142132

Page of Squares : First Upload June 23, 2008 ; Last Revised December 3, 2009
by Yoshio Mimura, Kobe, Japan