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, 2013by 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, 2011by 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:
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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, 2010by 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.
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, 2006by 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, 2011by 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).
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, 2006by 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, 2011by 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, 2010by 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, 2013by 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, 2006by 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:
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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, 2008by Yoshio Mimura, Kobe, Japan
1016
10162 = 1032256, 1032 - 2 * 5 - 6 = 1016.
Page of Squares : First Upload December 11, 2006 ; Last Revised December 11, 2006by 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)
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, 2008by 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, 2013by 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).
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, 2006by 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:
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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, 2011by 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, 2013by 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, 2010by 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:
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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, 2006by 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, 2011by 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:
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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)
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, 2011by 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, 2008by 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:
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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, 2013by 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, 2008by Yoshio Mimura, Kobe, Japan
1037
10372 = 1075369, a square with different digits.
Page of Squares : First Upload December 11, 2006 ; Last Revised December 11, 2006by Yoshio Mimura, Kobe, Japan
1038
The 4-by-4 magic squares consisting of different squares with constant 1038:
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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, 2013by 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, 2013by 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:
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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, 2006by 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, 2006by 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, 2008by 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.
by Yoshio Mimura, Kobe, Japan
1048
10482 = 383 + 683 + 903.
Page of Squares : First Upload June 23, 2008 ; Last Revised June 23, 2008by 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, 2006by 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:
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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).
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, 2006by 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:
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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, 2008by 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.)
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, 2008by Yoshio Mimura, Kobe, Japan
1062
10622 = 103 + 753 + 893.
The 4-by-4 magic squares consisting of different squares with constant 1062:
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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, 2013by 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:
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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).
Page of Squares : First Upload October 9, 2008 ; Last Revised April 5, 2011by Yoshio Mimura, Kobe, Japan
1068
Komachi equation: 10682 = 12 * 22 * 32 / 42 * 562 / 72 * 892.
Page of Squares : First Upload July 16, 2010 ; Last Revised July 16, 2010by Yoshio Mimura, Kobe, Japan
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, 2008by 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.
by Yoshio Mimura, Kobe, Japan
1072
10722 = 403 + 443 + 1003.
10722 = (12 + 7)(3792 + 7).
Page of Squares : First Upload June 23, 2008 ; Last Revised December 14, 2013by Yoshio Mimura, Kobe, Japan
1074
The 4-by-4 magic squares consisting of different squares with constant 1074:
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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, 2010by Yoshio Mimura, Kobe, Japan
1076
10762 = 1157776, a square with odd digits except the last digit 6.
Page of Squares : First Upload August 24, 2013 ; Last Revised August 24, 2013by 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, 2011by 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).
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, 2011by Yoshio Mimura, Kobe, Japan
1082
10822± 3 are primes.
Page of Squares : First Upload January 16, 2014 ; Last Revised January 16, 2014by Yoshio Mimura, Kobe, Japan
1083
10832 = 173 + 763 + 903 = 383 + 733 + 903.
Page of Squares : First Upload June 23, 2008 ; Last Revised June 23, 2008by Yoshio Mimura, Kobe, Japan
1084
10842 = 303 + 803 + 863.
Page of Squares : First Upload June 23, 2008 ; Last Revised June 23, 2008by 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, 2006by 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:
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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, 2008by 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:
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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)
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, 2013by 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, 2006by 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:
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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, 2008by Yoshio Mimura, Kobe, Japan
1098
10982 = 94 + 94 + 94 + 334.
The 4-by-4 magic squares consisting of different squares with constant 1098:
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Page of Squares : First Upload June 23, 2008 ; Last Revised December 3, 2009
by Yoshio Mimura, Kobe, Japan