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2000 - 2099

2000

20002 = 106 + 106 + 106 + 106.

20002 = (12 + 4)(22 + 4)(42 + 4)(62 + 4)(112 + 4) = (22 + 4)(42 + 4)(112 + 4)(142 + 4).

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

2001

20012 = 4004001, a square with just 3 kinds of digits.

20012 = 4004001, a reversible square (1004004 = 10022).

Page of Squares : First Upload March 12, 2007 ; Last Revised March 12, 2007
by Yoshio Mimura, Kobe, Japan

2002

2002 = (12 + 22 + 32 + ... + 2202) / (12 + 22 + 32 + ... + 172).

20022 = 4008004, a palindromic square with just 3 kinds of even digits.

20022 = 2252 + 2272 + 2292 + 2312 + 2322 + 2332 + ... + 3272.

Page of Squares : First Upload March 12, 2007 ; Last Revised November 25, 2008
by Yoshio Mimura, Kobe, Japan

2005

20052 = 4020025, 42 + 02 + 22 + 02 + 02 + 22 + 52 = 72.

20052 = 4020025, and 4020 / 02 - 5 = 2005.

Page of Squares : First Upload March 12, 2007 ; Last Revised March 12, 2007
by Yoshio Mimura, Kobe, Japan

2006

20062 = 4024036, 42 + 02 + 22 + 42 + 02 + 32 + 62 = 92.

Page of Squares : First Upload March 12, 2007 ; Last Revised March 12, 2007
by Yoshio Mimura, Kobe, Japan

2007

20072 = 623 + 1183 + 1293 = 74 + 264 + 264 + 424.

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

2008

20082 = 163 + 1203 + 1323.

20082 = 4032064, 42 + 02 + 32 + 22 + 02 + 62 + 42 = 92.

Page of Squares : First Upload March 12, 2007 ; Last Revised July 17, 2008
by Yoshio Mimura, Kobe, Japan

2010

830k + 2010k + 4570k + 4690k are squares for k = 1,2,3 (1102, 69002, 4553002).

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

2011

20112 = 4044121, a reversible square (1214404 = 11022).

Page of Squares : First Upload March 12, 2007 ; Last Revised March 12, 2007
by Yoshio Mimura, Kobe, Japan

2012

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

20122 = 44 + 164 + 224 + 444 = 284 + 324 + 324 + 344.

Page of Squares : First Upload March 12, 2007 ; Last Revised July 17, 2008
by Yoshio Mimura, Kobe, Japan

2013

20132 = 4052169, a square with different digits.

Page of Squares : First Upload March 12, 2007 ; Last Revised March 12, 2007
by Yoshio Mimura, Kobe, Japan

2014

2014 = (12 + 22 + 32 + ... + 1322) / (12 + 22 + 32 + ... + 102).

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

2015

20152 = 163 + 783 + 1533.

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

2016

20162 = (32 - 1)(132 - 1)(552 - 1) = (32 - 1)(72 - 1)(82 - 1)(132 - 1) = (82 - 1)(152 - 1)(172 - 1).

652 + 2016 = 792, 652 - 2016 = 472.

Komachi equations:
20162 = 12 * 3 * 4 * 56 * 7 * 8 * 9
20162 = 982 / 72 / 62 / 52 * 4322 * 102.

A cubic polynomial :
(X + 3632)(X + 13642)(X + 20162) = X3 + 24612X2 + 28882922X + 9981861122.

Page of Squares : First Upload March 12, 2007 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

2017

20172 = 4068289, and 4068 / 2 - 8 - 9 = 2017.

20172 = 124 + 184 + 214 + 444.

Page of Squares : First Upload March 12, 2007 ; Last Revised July 17, 2008
by Yoshio Mimura, Kobe, Japan

2018

20182 = 4072324, a zigzag square.

20182 = 1! + 1! + 1! + 1! + 8! + 8! + 9! + 10! = 1! + 1! + 2! + 8! + 8! + 9! + 10! = 2! + 2! + 8! + 8! + 9! + 10!.

Page of Squares : First Upload March 12, 2007 ; Last Revised March 12, 2007
by Yoshio Mimura, Kobe, Japan

2021

20212 = 4084441, a reversible square (1444804 = 12022).

Page of Squares : First Upload March 12, 2007 ; Last Revised March 12, 2007
by Yoshio Mimura, Kobe, Japan

2022

20222 = 4088484, a square with just 3 kinds of even digits.

20222 = 4088484, a reversible square (4848804 = 22022).

20222 = 663 + 793 + 1493.

Page of Squares : First Upload March 12, 2007 ; Last Revised July 17, 2008
by Yoshio Mimura, Kobe, Japan

2023

20232 = 4092529, a zigzag square.

20232 = 174 + 344 + 344 + 344.

Page of Squares : First Upload March 12, 2007 ; Last Revised July 17, 2008
by Yoshio Mimura, Kobe, Japan

2024

20242 = 43 + 83 + 1603 = 483 + 1003 + 1443.

20242 = (22 + 7)(92 + 7)(652 + 7) = (312 + 7)(652 + 7).

20242 = 4096576, and 4096 = 642, 576 = 242.

Page of Squares : First Upload March 12, 2007 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

2025

The square of 45.

2025 = (20 + 25)2.

2025 = 452, and 3136 = 562 (3136 = 2025 + 1111).

Page of Squares : First Upload March 12, 2007 ; Last Revised March 12, 2007
by Yoshio Mimura, Kobe, Japan

2027

20272 = 4108729, a square with different digits.

Page of Squares : First Upload March 12, 2007 ; Last Revised March 12, 2007
by Yoshio Mimura, Kobe, Japan

2028

20282 = 4! + 5! + 8! + 8! + 8! + 9! + 10!.

20282 = (62 + 3)(72 + 3)(452 + 3).

133 + 2028 = 652, 133 - 2028 = 132.

Page of Squares : First Upload March 12, 2007 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

2030

20302 = (12 + 6)(202 + 6)(382 + 6).

2030 = 212 + 222 + 232 + 242 = 252 + 262 + 272.

Page of Squares : First Upload September 1, 2011 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

2033

A cubic polynomial :
(X + 5672)(X + 12002)(X + 15402) = X3 + 20332X2 + 21541802X + 10478160002.

Page of Squares : First Upload March 12, 2007 ; Last Revised March 12, 2007
by Yoshio Mimura, Kobe, Japan

2034

20342 = 35 + 145 + 155 + 175 + 175.

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

2036

20362± 3 are primes.

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

2037

20372 = (122 + 3)(1682 + 3).

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

2038

20382 = 4153444 appears in the decimal expressions of e:
  e = 2.71828•••4153444••• (from the 120398th digit)

Page of Squares : First Upload July 17, 2008 ; Last Revised November 4, 2008
by Yoshio Mimura, Kobe, Japan

2039

20392 = 413 + 413 + 1593.

20392 = 4157521, 42 + 12 + 52 + 72 + 52 + 22 + 12 = 112.

Page of Squares : First Upload March 12, 2007 ; Last Revised July 17, 2008
by Yoshio Mimura, Kobe, Japan

2040

20402 = (42 - 1)(162 - 1)(332 - 1).

20402 = 563 + 1003 + 1443.

The integral triangle of sides 1825, 5689, 6936 has square area 20402.

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

2043

20432 = 4173849, a zigzag square.

20435 = 35591134615018443 :
32 + 52 + 52 + 92 + 112 + 342 + 62 + 152 + 02 + 182 + 42 + 42 + 32 = 2043.

Page of Squares : First Upload March 12, 2007 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

2045

The square root of 2045 is 45.2216..., and 45 = 22 + 22 + 12 + 62.

Page of Squares : First Upload March 12, 2007 ; Last Revised March 12, 2007
by Yoshio Mimura, Kobe, Japan

2046

2046 = (12 + 22 + 32 + ... + 2472) / (12 + 22 + 32 + ... + 192).

20462 = 133 + 1083 + 1433.

Page of Squares : First Upload July 17, 2008 ; Last Revised November 25, 2008
by Yoshio Mimura, Kobe, Japan

2047

20472 = 4190209, a zigzag square.

Page of Squares : First Upload March 12, 2007 ; Last Revised March 12, 2007
by Yoshio Mimura, Kobe, Japan

2048

The square root of 2048 is 45.254..., 45 = 22 + 52 + 42.

20482 = (12 + 7)(32 + 7)(1812 + 7) = (112 + 7)(1812 + 7).

Komachi equations:
20482 = 127 * 37 / 47 + 567 / 77 + 87 - 97 = -127 * 37 / 47 + 567 / 77 + 87 + 97.

20482 = 1283 + 1283 = 324 + 324 + 324 + 324 = 165 + 165 + 165 + 165 = 87 + 87.

Page of Squares : First Upload March 12, 2007 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

2049

20492 = 40 + 46 + 411.

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

2050

20502 = 54 + 154 + 154 + 454.

20502 = 4202500, and 4 = 22, 202500 = 4502.

20502 = (22 + 1)(32 + 1)(92 + 1)(322 + 1) = (72 + 1)(92 + 1)(322 + 1).

Page of Squares : First Upload March 12, 2007 ; Last Revised November 2, 2013
by Yoshio Mimura, Kobe, Japan

2051

20512 + 20522 + 20532 + ... + 42902 = 42912 + 42922 + 42932 + ... + 53052.

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

2052

20522 = 183 + 573 + 1593 = 843 + 1053 + 1353.

20522 = (12 + 8)(22 + 8)(72 + 8)(262 + 8) = (22 + 2)(42 + 2)(62 + 2)(322 + 2)
= (72 + 8)(102 + 8)(262 + 8).

10k + 74k + 98k + 142k are squares for k = 1,2,3 (182, 1882, 20522).
14k + 58k + 122k + 130k are squares for k = 1,2,3 (182, 1882, 20522).

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

2057

20572 = 114 + 224 + 224 + 444.

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

2058

20582 = 493 + 983 + 1473.

20582 = (12 + 6)(62 + 6)(1202 + 6) = (32 + 5)(172 + 5)(322 + 5).

The square root of 2058 is 45.36...., 45 = 32 + 62.

Page of Squares : First Upload March 12, 2007 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

2060

20602 = 613 + 1103 + 1393.

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

2062

1 / 2062 = 0.000484...., and 484 = 222.

20622 = 4251844, and 42 * 51 - 84 + 4 = 2062.

Page of Squares : First Upload March 12, 2007 ; Last Revised March 12, 2007
by Yoshio Mimura, Kobe, Japan

2063

1 / 2063 = 0.000484...., and 484 = 222.

Page of Squares : First Upload March 12, 2007 ; Last Revised March 12, 2007
by Yoshio Mimura, Kobe, Japan

2064

1 / 2064 = 0.000484...., and 484 = 222.

Page of Squares : First Upload March 12, 2007 ; Last Revised March 12, 2007
by Yoshio Mimura, Kobe, Japan

2065

1 / 2065 = 0.000484...., and 484 = 222.

Page of Squares : First Upload March 12, 2007 ; Last Revised March 12, 2007
by Yoshio Mimura, Kobe, Japan

2066

1 / 2066 = 0.000484...., and 484 = 222.

Page of Squares : First Upload March 12, 2007 ; Last Revised March 12, 2007
by Yoshio Mimura, Kobe, Japan

2068

20682± 3 are primes.

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

2069

20692 = 4280761, a square with different digits.

Page of Squares : First Upload March 12, 2007 ; Last Revised March 12, 2007
by Yoshio Mimura, Kobe, Japan

2070

20702 = (22 + 5)(52 + 5)(82 + 5)(152 + 5).

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

2073

20732 = 573 + 1003 + 1463.

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

2074

578k + 2074k + 2890k + 4862k are squares for k = 1,2,3 (1022, 60522, 3849482).

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

2075

20752 = 93 + 1163 + 1403.

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

2078

20782± 3 are primes.

20782 = 513 + 733 + 1563.

Page of Squares : First Upload July 17, 2008 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

2079

20792 = 4322241, a square every digit of which is non-zero and smaller than 5.

20792 = 333 + 343 + 353 + 363 + 373 + 383 + ... + 653.

A cubic polynomial :
(X + 16202)(X + 20792)(X + 44002) = X3 + 51292X2 + 120760202X + 148191120002.

Page of Squares : First Upload March 12, 2007 ; Last Revised September 7, 2013
by Yoshio Mimura, Kobe, Japan

2080

20802 = (22 + 4)(42 + 4)(102 + 4)(162 + 4).

Komachi equation: 20802 = 92 * 82 * 72 * 652 * 42 / 32 / 212.

20802 + 20812 + 20822 + ... + 21122 = 21132 + 21142 + 21152 + ... + 21442.

2080k + 3497k + 4004k + 4108k are squares for k = 1,2,3 (1172, 70332, 4304432).
960k + 2080k + 2745k + 3240k are squares for k = 1,2,3 (952, 48252, 2541252).

Page of Squares : First Upload September 14, 2010 ; Last Revised December 21, 2013
by Yoshio Mimura, Kobe, Japan

2082

20822 = 4334724, and 4 + 3 * 347 * 2 - 4 = 4 * 3 * 347 * 2 / 4 = 2082.

Page of Squares : First Upload March 12, 2007 ; Last Revised March 12, 2007
by Yoshio Mimura, Kobe, Japan

2083

The square root of 2083 is 45.63...., 45 = 62 + 32.

20832 = 4338889 appears in the decimal expressions of e:
  e = 2.71828•••4338889••• (from the 146953rd digit)

Page of Squares : First Upload March 12, 2007 ; Last Revised November 4, 2008
by Yoshio Mimura, Kobe, Japan

2084

(20842 + 2) = (52 + 2)(62 + 2)(72 + 2)(92 + 2).

Page of Squares : First Upload March 12, 2007 ; Last Revised March 12, 2007
by Yoshio Mimura, Kobe, Japan

2085

1 / 2085 = 0.00047961...., and 47961 = 2192.

Page of Squares : First Upload March 12, 2007 ; Last Revised March 12, 2007
by Yoshio Mimura, Kobe, Japan

2087

12 + 22 + 32 + 42 + 52 + ... + 20872 = 3032202300, which consists of 3 kinds of digits.

Page of Squares : First Upload March 12, 2007 ; Last Revised March 12, 2007
by Yoshio Mimura, Kobe, Japan

2088

20882 = 4! + 5! + 7! + 9! + 9! + 10!.

20882± 5 are primes.

Komachi equation: 20882 = 92 * 872 / 62 / 52 / 42 * 322 * 102.

Page of Squares : First Upload March 12, 2007 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

2090

2090k + 8074k + 8558k + 12254k are squares for k = 1,2,3 (1762, 171162, 17327202).
154k + 814k + 1298k + 2090k are squares for k = 1,2,3 (662, 25962, 1089002).

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

2092

20922± 3 are primes.

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

2093

20932 = 4380649. a zigzag square.

20932 = (12 + 22 + 32 + 42 + 52 + 62)(72 + 82 + 92 + ... + 522).

Page of Squares : First Upload March 12, 2007 ; Last Revised February 2, 2013
by Yoshio Mimura, Kobe, Japan

2094

20942 = 4384836, a zigzag square.

20942 = 473 + 1093 + 1443.

Page of Squares : First Upload March 12, 2007 ; Last Revised July 17, 2008
by Yoshio Mimura, Kobe, Japan

2095

20952 = 4389025, a square with different digits.

Page of Squares : First Upload March 12, 2007 ; Last Revised March 12, 2007
by Yoshio Mimura, Kobe, Japan

2096

20962 = 163 + 843 + 1563 = 503 + 1123 + 1423.

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

2099

20995 = 40743862065910499 : 402 + 72 + 42 + 32 + 82 + 62 + 22 + 02 + 62 + 52 + 92 + 12 + 02 + 42 + 92 + 92 = 2099.

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