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2100 - 2199

2100

21002 = (22 + 6)(62 + 6)(82 + 6)(122 + 6).

4-cycle of squares:
2016,2024,2028,2030,2040,2048,2052,2058,2070,2080, 21002 = 04410000 -- 41002 = 16810000 -- 81002 = 65610000 -- 61002 = 37210000,
(other examples: 2916-5030-3009-0540-2916, 1600-5600-3600-9600-1600).

21002 = (8 + 9 + 10 + ... + 27)2 + (28 + 29 + 30 + ... + 47)2 + (48 + 49 + 50 + ... + 67)2 + ... + (68 + 69 + 70 + ... + 87)2.

21002 = 7! + 7! + 7! + 8! + 9! + 9! + 10!.

Page of Squares : First Upload March 19, 2007 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

2101

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

21012 = 303 + 1223 + 1373.

Page of Squares : First Upload March 19, 2007 ; Last Revised July 17, 2008
by Yoshio Mimura, Kobe, Japan

2102

21022 = 4418404, a reversible square (4048144 = 20122).

Page of Squares : First Upload March 19, 2007 ; Last Revised March 19, 2007
by Yoshio Mimura, Kobe, Japan

2103

21032 = 4422609, and 4 * 4 * 22 * 6 + 0 - 9 = 4 * 4 * 22 * 6 - 0 - 9 = 2103.

Page of Squares : First Upload March 19, 2007 ; Last Revised March 19, 2007
by Yoshio Mimura, Kobe, Japan

2106

21062 = 603 + 1093 + 1433 = 723 + 1173 + 1353.

21062 = 3! + 3! + 4! + 8! + 8! + 9! + 9! + 10!.

Komachi equations:
21062 = 12 * 2342 * 562 / 72 / 82 * 92 = 12 * 2342 / 562 * 72 * 82 * 92.

Page of Squares : First Upload March 19, 2007 ; Last Revised September 14, 2010
by Yoshio Mimura, Kobe, Japan

2107

21072 = 4439449, a square with just 3 kinds of digits.

Page of Squares : First Upload March 19, 2007 ; Last Revised March 19, 2007
by Yoshio Mimura, Kobe, Japan

2108

21082 = 4443664, a square with just 3 kinds of digits.

21082 = 4443664, and 44 * 4 / 3 * 6 * 6 - 4 = 4 * 44 / 3 * 6 * 6 - 4 = 2108.

Page of Squares : First Upload March 19, 2007 ; Last Revised March 19, 2007
by Yoshio Mimura, Kobe, Japan

2109

21092 = 103 + 333 + 1643.

Page of Squares : First Upload July 17, 2008 ; Last Revised July 17, 2008
by Yoshio Mimura, Kobe, Japan

2111

21112 = 4456321, a reversible square (1236544 = 11122).

Page of Squares : First Upload March 19, 2007 ; Last Revised March 19, 2007
by Yoshio Mimura, Kobe, Japan

2112

21122 = 403 + 963 + 1523.

A cubic polynomial :
(X + 18042)(X + 21122)(X + 25832) = X3 + 37932X2 + 81234122X + 98413539842.

Page of Squares : First Upload March 19, 2007 ; Last Revised July 17, 2008
by Yoshio Mimura, Kobe, Japan

2113

21132 = 55 + 55 + 55 + 135 + 215.

Page of Squares : First Upload July 17, 2008 ; Last Revised July 17, 2008
by Yoshio Mimura, Kobe, Japan

2114

21142 = 42 x 43 + 43 x 44 + 44 x 45 + 45 x 46 + 46 x 47 + ... + 237 x 238.

Page of Squares : First Upload March 19, 2007 ; Last Revised March 19, 2007
by Yoshio Mimura, Kobe, Japan

2115

21152 = 253 + 363 + 1643 = 45 + 75 + 135 + 215.

Page of Squares : First Upload July 17, 2008 ; Last Revised July 17, 2008
by Yoshio Mimura, Kobe, Japan

2116

The square of 46.

21162 = 693 + 1153 + 1383.

Page of Squares : First Upload March 19, 2007 ; Last Revised July 17, 2008
by Yoshio Mimura, Kobe, Japan

2117

21172 = (44 + 45)2 + (46 + 47)2 + (48 + 49)2 + (50 + 51)2 + ... + (188 + 189)2.

Page of Squares : First Upload March 19, 2007 ; Last Revised March 19, 2007
by Yoshio Mimura, Kobe, Japan

2118

21184 = 20123514133776, and 202 + 122 + 352 + 142 + 12 + 32 + 32 + 72 + 72 + 62 = 2118.

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

2120

21202 = (12 + 4)(22 + 4)(72 + 4)(462 + 4) = (62 + 4)(72 + 4)(462 + 4).

Page of Squares : First Upload December 21, 2013 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

2121

21212 = 4498641, a reversible square (1468944 = 12122).

Page of Squares : First Upload March 19, 2007 ; Last Revised March 19, 2007
by Yoshio Mimura, Kobe, Japan

2123

21232 = 4507129, a square with different digits.

21234 = 20314211822641, and 22 + 02 + 32 + 142 + 22 + 112 + 82 + 22 + 22 + 62 + 412 = 2123.

Page of Squares : First Upload March 19, 2007 ; Last Revised December 1, 2008
by Yoshio Mimura, Kobe, Japan

2124

21242 = (22 + 8)(132 + 8)(462 + 8).

A cubic polynomial :
(X + 6082)(X + 13112)(X + 21242) = X3 + 25692X2 + 31712522X + 16930149122.

Page of Squares : First Upload March 19, 2007 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

2125

21252 = 43 + 1213 + 1403.

Page of Squares : First Upload July 17, 2008 ; Last Revised July 17, 2008
by Yoshio Mimura, Kobe, Japan

2126

21262 = 4519876, a square with different digits.

21262 = 4519876, and 4 * 519 + 8 * 7 - 6 = 4 * 519 + 8 + 7 * 6 = 2126.

Page of Squares : First Upload March 19, 2007 ; Last Revised March 19, 2007
by Yoshio Mimura, Kobe, Japan

2127

21272 = 493 + 1263 + 1343.

Page of Squares : First Upload July 17, 2008 ; Last Revised July 17, 2008
by Yoshio Mimura, Kobe, Japan

2128

21282 = 4528384, a zigzag square.

21282 = 4528384, and 45 * 2 * 8 * 3 - 8 * 4 = 2128.

Komachi equation: 21282 = 982 * 762 * 52 * 42 * 32 / 2102.

Page of Squares : First Upload March 19, 2007 ; Last Revised September 14, 2010
by Yoshio Mimura, Kobe, Japan

2129

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

21292 = 4532641, and 4 * 532 + 6 - 4 - 1 = 2129.

Page of Squares : First Upload March 19, 2007 ; Last Revised March 19, 2007
by Yoshio Mimura, Kobe, Japan

2132

21322 = 4545424, a square with just 3 kinds of digits.

21322 = 4545424, a square pegged by 4.

Page of Squares : First Upload March 19, 2007 ; Last Revised March 19, 2007
by Yoshio Mimura, Kobe, Japan

2133

21332± 2 are primes.

21332 = 75 + 125 + 155 + 165 + 195.

Page of Squares : First Upload July 17, 2008 ; Last Revised December 29, 2013
by Yoshio Mimura, Kobe, Japan

2134

341 + 21341 + 28731 = 712, 342 + 21342 + 28732 = 35792, 343 + 21343 + 28733 = 1828452  (See 71).

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

2136

21362 = (42 + 8)(4362 + 8).

21362 = 533 + 1203 + 1393.

Page of Squares : First Upload July 17, 2008 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

2139

21392 = 1663 + 45 + 17.

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

2142

The integral triangle of sides 9265, 19656, 28913 has square area 21422.

Page of Squares : First Upload October 7, 2011 ; Last Revised October 7, 2011
by Yoshio Mimura, Kobe, Japan

2143

21432 = 4592449, and 42 + 52 + 92 + 22 + 442 + 92.

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

2144

21442 = (52 + 7)(3792 + 7).

Page of Squares : First Upload December 21, 2013 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

2147

21472 = 4609609 appears in the decimal expressions of e:
  e = 2.71828•••4609609••• (from the 108711st digit)

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

2153

21532 = 65 + 65 + 175 + 205.

1 / 2153 = 0.00046446818..., 42 + 62 + 442 + 62 + 82 + 12 + 82 = 2153.

Page of Squares : First Upload March 19, 2007 ; Last Revised July 17, 2008
by Yoshio Mimura, Kobe, Japan

2155

21552 = 413 + 923 + 1563.

Page of Squares : First Upload July 17, 2008 ; Last Revised July 17, 2008
by Yoshio Mimura, Kobe, Japan

2156

21562 = 4648336, and 4 * 64 * 8 + 3 * 36 = 2156.

385k + 1232k + 2156k + 2156k are squares for k = 1,2,3 (772, 33112, 1482252).

Page of Squares : First Upload March 19, 2007 ; Last Revised May 13, 2011
by Yoshio Mimura, Kobe, Japan

2157

21572 = 353 + 493 + 1653.

21574 = 21647142717201, and 212 + 62 + 42 + 72 + 142 + 272 + 172 + 202 + 12 = 2157.

Page of Squares : First Upload July 17, 2008 ; Last Revised December 1, 2008
by Yoshio Mimura, Kobe, Japan

2158

2158 = (12 + 22 + 32 + ... + 140272) / (12 + 22 + 32 + ... + 10852).

21582 = 4656964, a square pegged by 6.

Page of Squares : First Upload March 19, 2007 ; Last Revised November 25, 2008
by Yoshio Mimura, Kobe, Japan

2160

21602 = 4665600, and 4 * 6 * 65 + 600 = 2160.

512 + 2160 = 692, 512 - 2160 = 212.

21602 = (22 - 1)(32 - 1)(172 - 1)(262 - 1) = (22 - 1)(42 - 1)(172 - 1)(192 - 1)
= (52 - 1)(172 - 1)(262 - 1).

21602 + 21612 + 21622 + ... + 38922 = 38932 + 38942 + 38952 + ... + 47602.

Komachi equation: 21602 = 92 * 82 / 72 / 62 * 52 * 42 * 32 * 212.

21602 = 933 + 1233 + 1263.

Page of Squares : First Upload March 19, 2007 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

2161

21612 = 273 + 1173 + 1453.

42k + 66k + 108k + 145k are squares for k = 1,2,3 (192, 1972, 21612).

Page of Squares : First Upload July 17, 2008 ; Last Revised May 13, 2011
by Yoshio Mimura, Kobe, Japan

2162

21622 = 74 + 314 + 374 + 374.

21622 = S2(162) + S2(213), where S2(n) = 12 + 22 + 32 + 42 + ... + n2.

Page of Squares : First Upload March 19, 2007 ; Last Revised July 17, 2008
by Yoshio Mimura, Kobe, Japan

2163

21632 = 4678569, and 467856 = 6842, 9 = 32.

21632 = 4678569, and 4 + 6 - 7 + 8 * 5 * 6 * 9 = 2163.

21632 = (42 + 5)(4722 + 5).

Page of Squares : First Upload March 19, 2007 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

2166

21662 = 563 + 573 + 1633 = 953 + 1143 + 1333.

Page of Squares : First Upload July 17, 2008 ; Last Revised July 17, 2008
by Yoshio Mimura, Kobe, Japan

2167

21672 = 4695889, and 46 * 9 * 5 + 88 + 9 = 46 * 9 * 5 + 8 + 89 = 2167.

Page of Squares : First Upload March 19, 2007 ; Last Revised March 19, 2007
by Yoshio Mimura, Kobe, Japan

2169

1 / 2169 = 0.000461041...., and 461041 = 6792.

21692 = (93 + 94 + 95 + 96 + 97 + 98)2 + (99 + 100 + 101 + 102 + 103 + 104)2 + (105 + 106 + 107 + 108 + 109 + 110)2 + ... + (141 + 142 + 143 + 144 + 145 + 146)2.

Page of Squares : First Upload March 19, 2007 ; Last Revised March 19, 2007
by Yoshio Mimura, Kobe, Japan

2170

21702 = 963 + 973 + 983 + 993 + 1003.

21702 = (12 + 6)(82 + 6)(982 + 6) = (222 + 6)(982 + 6).

Page of Squares : First Upload November 25, 2008 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

2171

21712 = 4713241, a zigzag square.

21712 = 4713241 appears in the decimal expressions of π:
  π = 3.14159•••4713241••• (from the 26172nd digit)
  (4713241 is the seventh 7-digit square in the expression of π.)

Page of Squares : First Upload March 19, 2007 ; Last Revised November 4, 2008
by Yoshio Mimura, Kobe, Japan

2172

21722 = 4717584, a zigzag square.

21722 = 4! + 5! + 9! + 9! + 9! + 10!.

21722 = (32 + 3)(6272 + 3).

Komachi equations:
21722 = 92 * 82 * 72 / 62 * 5432 / 212 = 982 / 72 * 62 * 5432 / 212.

21722 = 803 + 943 + 1503.

21722 = (66 + 67 + 68 + ... + 77)2 + (78 + 79 + 80 + ... + 89)2 + (90 + 91 + 92 + ... + 101)2 + ... + (102 + 103 + 104 + ... + 113)2.

Page of Squares : First Upload March 19, 2007 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

2174

21742 = 4726276, a zigzag square.

Page of Squares : First Upload March 19, 2007 ; Last Revised March 19, 2007
by Yoshio Mimura, Kobe, Japan

2175

21752 = 4730625, a square with different digits.

Page of Squares : First Upload March 19, 2007 ; Last Revised March 19, 2007
by Yoshio Mimura, Kobe, Japan

2176

21762 = 4734976, 42 + 72 + 32 + 42 + 92 + 72 + 62 = 162.

21762 = 4734976, and 4 * 7 + 34 * 9 * 7 + 6 = 2176.

Page of Squares : First Upload March 19, 2007 ; Last Revised March 19, 2007
by Yoshio Mimura, Kobe, Japan

2177

A cubic polynomial :
(X + 4682)(X + 11042)(X + 18172) = X3 + 21772X2 + 22391882X + 9387930242.

Page of Squares : First Upload March 19, 2007 ; Last Revised March 19, 2007
by Yoshio Mimura, Kobe, Japan

2178

21782 = 443 + 553 + 1653 = 334 + 334 + 334 + 334.

21782 = (12 + 2)(22 + 2)(32 + 2)(82 + 2)(192 + 2) = (22 + 2)(32 + 2)(142 + 2)(192 + 2)
= (32 + 2)(42 + 2)(82 + 2)(192 + 2) = (82 + 2)(142 + 2)(192 + 2).

21784 = 22502537891856, and 222 + 52 + 02 + 22 + 52 + 372 + 82 + 92 + 12 + 82 + 52 + 62 = 2178.

Page of Squares : First Upload July 17, 2008 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

2179

21792 = 4748041, a zigzag square.

Page of Squares : First Upload March 19, 2007 ; Last Revised March 19, 2007
by Yoshio Mimura, Kobe, Japan

2183

21832 = 4765489, and 4 + 7 * 6 * 54 - 89 = 2183.

21832 = 743 + 923 + 1533 = 84 + 84 + 234 + 464.

Page of Squares : First Upload March 19, 2007 ; Last Revised July 17, 2008
by Yoshio Mimura, Kobe, Japan

2184

852 + 2184 = 972, 852 - 2184 = 712.

The integral triangle of sides 10656, 10985, 21623 has square area 21842.

21842 = (12 + 3)(22 + 3)(32 + 3)(62 + 3)(192 + 3) = (12 + 3)(22 + 3)(62 + 3)(72 + 3)(92 + 3)
= (12 + 3)(22 + 3)(92 + 3)(452 + 3) = (12 + 3)(52 + 3)(62 + 3)(332 + 3)
= (12 + 3)(62 + 3)(92 + 3)(192 + 3) = (22 + 3)(32 + 3)(52 + 3)(452 + 3)
= (22 + 3)(32 + 3)(52 + 3)(62 + 3)(72 + 3) = (22 + 3)(32 + 3)(72 + 3)(332 + 3)
= (32 + 3)(192 + 3)(332 + 3) = (32 + 3)(52 + 3)(62 + 3)(192 + 3)
= (52 + 3)(62 + 3)(72 + 3)(92 + 3) = (52 + 3)(92 + 3)(452 + 3) = (72 + 3)(92 + 3)(332 + 3).

Komachi equations:
21842 = 92 * 82 * 72 * 652 * 42 / 32 / 22 / 102 = 982 / 72 * 652 * 42 * 32 * 22 / 102.

21842 = 223 + 263 + 1683 = 423 + 1213 + 1433 = 493 + 1203 + 1433 = 913 + 1013 + 1443.

Page of Squares : First Upload July 17, 2008 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

2186

21862 = 4778596, and 4 * 7 * 78 + 5 - 9 + 6 = 2186.

Page of Squares : First Upload March 19, 2007 ; Last Revised March 19, 2007
by Yoshio Mimura, Kobe, Japan

2187

21872 = 813 + 1623.

12 + 22 + 32 + 42 + 52 + ... + 21872 = 3489176250, the unique 10-digit sum which consists of distinct 10 digits.

Komachi equations:
21872 = 127 - 37 * 47 + 567 - 77 * 87 + 97 = 127 - 37 * 47 + 567 / 77 - 87 + 97
 = 127 - 37 * 47 + 567 / 77 / 87 * 97 = 127 - 37 * 47 - 567 + 77 * 87 + 97
 = 127 - 37 * 47 - 567 / 77 + 87 + 97 = 127 - 37 * 47 * 567 / 77 / 87 + 97
 = 127 - 37 * 47 / 567 * 77 * 87 + 97 = 127 / 37 - 47 + 567 - 77 * 87 + 97
 = 127 / 37 - 47 + 567 / 77 - 87 + 97 = 127 / 37 - 47 + 567 / 77 / 87 * 97
 = 127 / 37 - 47 - 567 + 77 * 87 + 97 = 127 / 37 - 47 - 567 / 77 + 87 + 97
 = 127 / 37 - 47 * 567 / 77 / 87 + 97 = 127 / 37 - 47 / 567 * 77 * 87 + 97
 = 127 / 37 / 47 - 567 / 77 / 87 + 97 = 127 / 37 / 47 * 567 - 77 * 87 + 97
 = 127 / 37 / 47 * 567 / 77 - 87 + 97 = 127 / 37 / 47 * 567 / 77 / 87 * 97
 = 127 / 37 / 47 / 567 * 77 * 87 * 97 = - 127 + 37 * 47 - 567 + 77 * 87 + 97
 = - 127 + 37 * 47 - 567 / 77 + 87 + 97 = - 127 + 37 * 47 + 567 - 77 * 87 + 97
 = - 127 + 37 * 47 + 567 / 77 - 87 + 97 = - 127 + 37 * 47 + 567 / 77 / 87 * 97
 = - 127 + 37 * 47 * 567 / 77 / 87 + 97 = - 127 + 37 * 47 / 567 * 77 * 87 + 97
 = - 127 / 37 + 47 - 567 + 77 * 87 + 97 = - 127 / 37 + 47 - 567 / 77 + 87 + 97
 = - 127 / 37 + 47 + 567 - 77 * 87 + 97 = - 127 / 37 + 47 + 567 / 77 - 87 + 97
 = - 127 / 37 + 47 + 567 / 77 / 87 * 97 = - 127 / 37 + 47 * 567 / 77 / 87 + 97
 = - 127 / 37 + 47 / 567 * 77 * 87 + 97 = - 127 / 37 / 47 + 567 / 77 / 87 + 97
 = - 127 / 37 / 47 * 567 + 77 * 87 + 97 = - 127 / 37 / 47 * 567 / 77 + 87 + 97.

Page of Squares : First Upload March 19, 2007 ; Last Revised September 14, 2010
by Yoshio Mimura, Kobe, Japan

2188

21882 = 4787344, and 4 * 78 * 7 + 3 + 4 / 4 = 2188.

21882 = 43 + 773 + 1633 = 15 + 125 + 155 + 185 + 185 = 17 + 37 + 37 + 97.

Page of Squares : First Upload March 19, 2007 ; Last Revised July 17, 2008
by Yoshio Mimura, Kobe, Japan

2189

21892 = 30 + 31 + 37 + 38 + 314.

21892 = 4791721 appears in the decimal expressions of e:
  e = 2.71828•••4791721••• (from the 116036th digit)

Page of Squares : First Upload November 4, 2008 ; Last Revised August 29, 2011
by Yoshio Mimura, Kobe, Japan

2190

21902 = (12 + 9)(82 + 9)(812 + 9).

Komachi equation: 21902 = 92 * 8762 / 542 * 32 / 22 * 102.

62 + 192 + 322 + 452 + ... + (13x + 6)2 + ... + 21902 = 164842.

Page of Squares : First Upload March 19, 2007 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

2192

21922 = 4804864, a square with even digits.

21922 = 84 + 84 + 324 + 444.

72 + 302 + 532 + 762 + ... + (23x + 7)2 + ... + 21922 = 124522.

Page of Squares : First Upload March 19, 2007 ; Last Revised July 17, 2008
by Yoshio Mimura, Kobe, Japan

2193

21932 = 14 + 24 + 244 + 464.

Page of Squares : First Upload July 17, 2008 ; Last Revised July 17, 2008
by Yoshio Mimura, Kobe, Japan

2197

A cubic polynomial :
(X + 6002)(X + 14402)(X + 15472) = X3 + 21972X2 + 25633202X + 13366080002.

Komachi equation: 21972 = 96 * 86 * 76 * 656 / 46 / 36 / 2106.

Page of Squares : First Upload March 19, 2007 ; Last Revised September 14, 2010
by Yoshio Mimura, Kobe, Japan

2199

21992 = 4835601, a square with different digits.

Page of Squares : First Upload March 19, 2007 ; Last Revised March 19, 2007
by Yoshio Mimura, Kobe, Japan