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4000

4000² = 200³ + 200³ = 20⁵ + 20⁵ + 20⁵ + 20⁵ + 20⁵.

400040001 = 20001².

Page of Squares : First Upload August 6, 2007 ; Last Revised August 26, 2008
by Yoshio Mimura, Kobe, Japan

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4002

4002²± 5 are primes.

4002² = 7⁴ + 19⁴ + 19⁴ + 63⁴.

Page of Squares : First Upload August 26, 2008 ; Last Revised January 18, 2014
by Yoshio Mimura, Kobe, Japan

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4004

4004⁵ = 1029130250245121024 : 10² + 2² + 9² + 1² + 30² + 25² + 0² + 2² + 45² + 12² + 10² + 2² + 4² = 4004.

2080^k + 3497^k + 4004^k + 4108^k are squares for k = 1,2,3 (117², 7033², 430443²).

Page of Squares : First Upload December 8, 2008 ; Last Revised June 7, 2011
by Yoshio Mimura, Kobe, Japan

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4007

4007² = 34⁴ + 46⁴ + 46⁴ + 49⁴.

4007⁵ = 1032991414928036807 :
  10² + 32² + 9² + 9² + 1² + 41² + 4² + 9² + 28² + 0² + 3² + 6² + 8² + 0² + 7² = 4007.

Page of Squares : First Upload August 26, 2008 ; Last Revised December 8, 2008
by Yoshio Mimura, Kobe, Japan

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4011

4011² = 41³ + 196³ + 204³.

The square root of 4011 is 63.33245..., and 63 = 3² + 3² + 2² + 4² + 5².

Page of Squares : First Upload August 6, 2007 ; Last Revised August 26, 2008
by Yoshio Mimura, Kobe, Japan

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4016

4016³ = 64771076096, and 6² + 4² + 7² + 7² + 10² + 7² + 60² + 9² + 6² = 4016.

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

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4017

S₂(4017) = S₂(3076) + S₂(3293), where S₂(n) = 1² + 2² + 3² + 4² + 5² + ... + n².

4017² = 27² + 28² + 29² + 30² + 31² + ... + 364².

Page of Squares : First Upload August 6, 2007 ; Last Revised August 6, 2007
by Yoshio Mimura, Kobe, Japan

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4018

4018² = 7⁴ + 21⁴ + 21⁴ + 63⁴ = 17⁴ + 23⁴ + 53⁴ + 53⁴.

Page of Squares : First Upload August 26, 2008 ; Last Revised August 26, 2008
by Yoshio Mimura, Kobe, Japan

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4020

Komachi equations:
4020² = 1² * 2345² / 6² / 7² * 8² * 9² = 1² * 2² / 3² / 4² * 5² * 67² * 8² * 9²
 = 1² / 2² * 3² * 45² * 67² * 8² / 9².

Page of Squares : First Upload August 26, 2008 ; Last Revised October 8, 2010
by Yoshio Mimura, Kobe, Japan

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4022

4022²± 3 are primes.

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

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4023

4023² = 16184529, a zigzag square.

Page of Squares : First Upload August 6, 2007 ; Last Revised August 6, 2007
by Yoshio Mimura, Kobe, Japan

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4024

4024² = 100³ + 162³ + 222³.

Page of Squares : First Upload August 26, 2008 ; Last Revised August 26, 2008
by Yoshio Mimura, Kobe, Japan

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4026

330^k + 1353^k + 4026^k + 4092^k are squares for k = 1,2,3 (99², 5907², 369171²).

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

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4027

4027² = 16216729, 1 - 6 + 2 * 16 * 7 * 2 * 9 = 4027.

Page of Squares : First Upload August 6, 2007 ; Last Revised August 6, 2007
by Yoshio Mimura, Kobe, Japan

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4028

4028² = 16224784, 1 * 6 / 2 * 24 * 7 * 8 - 4 = 4028.

Page of Squares : First Upload August 6, 2007 ; Last Revised August 6, 2007
by Yoshio Mimura, Kobe, Japan

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4031

4031² = 81³ + 94³ + 246³.

4031² = 16248961, 1 * 624 * 8 - 961 = 4031.

Page of Squares : First Upload August 6, 2007 ; Last Revised August 26, 2008
by Yoshio Mimura, Kobe, Japan

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4032

4032² = 120³ + 174³ + 210³ = 16⁴ + 32⁴ + 48⁴ + 56⁴ = 24⁴ + 48⁴ + 48⁴ + 48⁴ = 10⁵ + 14⁵ + 14⁵ + 20⁵ + 26⁵.

4032² = 16257024, 1 * 6 * 2 / 5 * 70 * 24 = 4032.

4032² = (2² - 1)(3² - 1)(15² - 1)(55² - 1) = (2² - 1)(3² - 1)(7² - 1)(8² - 1)(15² - 1)
= (5² - 1)(15² - 1)(55² - 1) = (5² - 1)(7² - 1)(8² - 1)(15² - 1) = (4³ + 8)(6³ + 8)(10³ + 8).

Komachi equations:
4032² = 12² * 3456² * 7² / 8² / 9² = 98² / 7² / 6² * 54² * 32² */ 1²
 = 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.

Page of Squares : First Upload August 6, 2007 ; Last Revised December 28, 2013
by Yoshio Mimura, Kobe, Japan

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4039

S₂(4039) = S₂(580) + S₂(4035), where S₂(n) = 1² + 2² + 3² + 4² + 5² + ... + n².

4039² = S₂(265) + S₂(311).

Page of Squares : First Upload August 6, 2007 ; Last Revised August 6, 2007
by Yoshio Mimura, Kobe, Japan

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4040

4040² = 72³ + 160³ + 228³ = 74³ + 169³ + 223³ = 124³ + 177³ + 207³.

Page of Squares : First Upload August 26, 2008 ; Last Revised August 26, 2008
by Yoshio Mimura, Kobe, Japan

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4042

1505^k + 4042^k + 4988^k + 6106^k are squares for k = 1,2,3 (129², 8987², 648999²).

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

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4044

4044² = 16353936, a zigzag square.

Page of Squares : First Upload August 6, 2007 ; Last Revised August 6, 2007
by Yoshio Mimura, Kobe, Japan

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4045

4045⁴ = 267715862100625, and 2² + 6² + 7² + 7² + 1² + 58² + 6² + 21² + 0² + 0² + 6² + 2² + 5² = 4045.

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

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4047

4047² = 16378209, a square with different digits.

Page of Squares : First Upload August 6, 2007 ; Last Revised August 6, 2007
by Yoshio Mimura, Kobe, Japan

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4048

4048² = (19² + 7)(211² + 7) = (2² + 7)(4² + 7)(13² + 7)(19² + 7) = (2² + 7)(4² + 7)(9² + 7)(27² + 7)
= (3² + 7)(4² + 7)(211² + 7) = (4² + 7)(27² + 7)(31² + 7).

A cubic polynomial :
(X + 756²)(X + 4048²)(X + 28497²) = X³ + 28793²X² + 117390252²X + 87209027136².

Page of Squares : First Upload August 6, 2007 ; Last Revised December 28, 2013
by Yoshio Mimura, Kobe, Japan

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4049

4049² = 64³ + 168³ + 225³.

Page of Squares : First Upload August 26, 2008 ; Last Revised August 26, 2008
by Yoshio Mimura, Kobe, Japan

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4050

4050² = 45⁴ + 45⁴ + 45⁴ + 45⁴.

Page of Squares : First Upload August 26, 2008 ; Last Revised August 26, 2008
by Yoshio Mimura, Kobe, Japan

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4054

4054² = 16434916, and 1² + 6² + 4² + 3² + 4² + 9² + 1² + 6² = 14².

Page of Squares : First Upload August 6, 2007 ; Last Revised August 6, 2007
by Yoshio Mimura, Kobe, Japan

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4056

4056² = (1² + 3)(6² + 3)(7² + 3)(45² + 3).

65² + 4056 = 91², 65² - 4056 = 13².

Page of Squares : First Upload August 2, 2011 ; Last Revised December 28, 2013
by Yoshio Mimura, Kobe, Japan

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4058

4058² = 16467364, 1 * 6 * 4 + 673 * 6 - 4 = 4058.

Page of Squares : First Upload August 6, 2007 ; Last Revised August 6, 2007
by Yoshio Mimura, Kobe, Japan

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4060

4060 = (1² + 2² + 3² + ... + 87²) / (1² + 2² + 3² + 4² + 5²).

4060² = 2⁵ + 8⁵ + 12⁵ + 23⁵ + 25⁵.

S₂(4060) = S₂(2650) + S₂(3642), where S₂(n) = 1² + 2² + 3² + 4² + 5² + ... + n².

Page of Squares : First Upload August 6, 2007 ; Last Revised November 25, 2008
by Yoshio Mimura, Kobe, Japan

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4062

4062²± 5 are primes.

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

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4063

4063 = (1² + 2² + 3² + ... + 119²) / (1² + 2² + 3² + ... + 7²).

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

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4065

4065 = (1² + 2² + 3² + ... + 135²) / (1² + 2² + 3² + ... + 8²).

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

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4068

4068² = 16³ + 54³ + 254³.

1 / 4068 = 0.0002458210422, 24² + 58² + 2² + 10² + 4² + 2² + 2² = 4068.

Page of Squares : First Upload August 6, 2007 ; Last Revised August 26, 2008
by Yoshio Mimura, Kobe, Japan

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4069

4069² = 16556761, 1 + 6 - 5 * 5 + 67 * 61 = 4069.

Page of Squares : First Upload August 6, 2007 ; Last Revised August 6, 2007
by Yoshio Mimura, Kobe, Japan

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4070

4070 = (1² + 2² + 3² + ... + 55²) / (1² + 2² + 3²).

4070² = S₂(39) + S₂(367), where S₂(n) = 1² + 2² + 3² + 4² + 5² + ... + n².

4070² = 16564900, 1 * 65 * 64 - 90 + 0 = 4070.

Page of Squares : First Upload August 6, 2007 ; Last Revised November 25, 2008
by Yoshio Mimura, Kobe, Japan

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4072

4072² = 22⁴ + 22⁴ + 34⁴ + 62⁴.

Page of Squares : First Upload August 26, 2008 ; Last Revised August 26, 2008
by Yoshio Mimura, Kobe, Japan

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4073

4073² = 27⁴ + 42⁴ + 42⁴ + 56⁴.

Page of Squares : First Upload August 26, 2008 ; Last Revised August 26, 2008
by Yoshio Mimura, Kobe, Japan

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4074

4074² = 16597476, 1 + 65 * 9 * 7 - 4 * 7 + 6 = 4074.

4074² = (1² + 3)(12² + 3)(168² + 3) = 4074,(2² + 3)(9² + 3)(168² + 3).

Page of Squares : First Upload August 6, 2007 ; Last Revised December 28, 2013
by Yoshio Mimura, Kobe, Japan

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4076

1 / 4076 = 0.00024533856722, 2² + 45² + 3² + 38² + 5² + 6² + 7² + 22² = 4076.

Page of Squares : First Upload August 6, 2007 ; Last Revised August 6, 2007
by Yoshio Mimura, Kobe, Japan

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4077

4077⁵ = 1126428311047604157 : 11² + 26² + 42² + 8² + 31² + 10² + 4² + 7² + 6² + 0² + 4² + 15² + 7² = 4077.

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

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4078

6948² + 4078² = 64904788, a mosaic square.

Page of Squares : First Upload August 6, 2007 ; Last Revised August 6, 2007
by Yoshio Mimura, Kobe, Japan

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4082

4082² = 16662724, 1 * 66 * 62 - 7 * 2 + 4 = 4082.

Page of Squares : First Upload August 6, 2007 ; Last Revised August 6, 2007
by Yoshio Mimura, Kobe, Japan

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4086

4086 = (1² + 2² + 3² + ... + 567²) / (1² + 2² + 3² + ... + 35²).

4086² = 87³ + 157³ + 230³.

4086² = 16662724, 16 * 6 * 9 * 5 - 39 * 6 = 4086.

Page of Squares : First Upload August 6, 2007 ; Last Revised November 25, 2008
by Yoshio Mimura, Kobe, Japan

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4087

4087² = 25³ + 136³ + 242³ = 74³ + 121³ + 244³.

Page of Squares : First Upload August 26, 2008 ; Last Revised August 26, 2008
by Yoshio Mimura, Kobe, Japan

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4088

4088² = 16711744, 1² + 6² + 7² + 1² + 1² + 7² + 4² + 4² = 13².

4088² = 442² + 443² + 444² + 445² + 446² + ... + 514².

Page of Squares : First Upload August 6, 2007 ; Last Revised August 6, 2007
by Yoshio Mimura, Kobe, Japan

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4089

4089^k + 9870^k + 15792^k + 25474^k are squares for k = 1,2,3 (235², 31819², 4636691²).

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

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4092

4092² = 4⁴ + 36⁴ + 44⁴ + 58⁴.

330^k + 1353^k + 4026^k + 4092^k are squares for k = 1,2,3 (99², 5907², 369171²).

The integral triangle of sides 3330, 10057, 10571 has square area 4092².

Page of Squares : First Upload August 26, 2008 ; Last Revised October 11, 2011
by Yoshio Mimura, Kobe, Japan

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4094

4094²± 3 are primes.

4094² = 18³ + 151³ + 237³ = 107³ + 145³ + 232³.

Komachi equation: 4094² = 1² * 23² / 4² * 56² / 7² * 89².

4094³ = 68618862584, and 6² + 8² + 61² + 8² + 8² + 6² + 2² + 5² + 8² + 4² = 4084.

4094⁵ = 1150109502276960224, and 1² + 1² + 5² + 0² + 10² + 9² + 50² + 2² + 27² + 6² + 9² + 6² + 0² + 22² + 4² = 1² + 1² + 50² + 10² + 9² + 5² + 0² + 2² + 27² + 6² + 9² + 6² + 0² + 22² + 4² = 4094.

Page of Squares : First Upload August 26, 2008 ; Last Revised January 18, 2014
by Yoshio Mimura, Kobe, Japan

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4095

4095² = 16769025, 1 + 6 * 76 * 9 + 0 - 2 * 5 = 4095.

4095² + 4096² + 4097² + ... + 4140² = 4141² + 4142² + 4143² + ... + 4185².

Komachi equations
4095² = 9² * 8² * 7² * 65² * 4² / 32² */ 1² = 9² / 8² * 7² * 65² / 4² * 32² */ 1².

Page of Squares : First Upload August 6, 2007 ; Last Revised September 13, 2011
by Yoshio Mimura, Kobe, Japan

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4096

The square of 64.

4096² = 9³ + 58³ + 255³ = 9³ + 183³ + 220³ = 22³ + 57³ + 255³.

1 / 4096 = 0.000244140625, 244140625 = 15625².

4096² = (3² + 7)(5² + 7)(181² + 7).

Komachi equations
4096² = 12⁴ * 3⁴ / 4⁴ + 56⁴ / 7⁴ * 8⁴ - 9⁴ = 12⁴ * 3⁴ / 4⁴ * 56⁴ / 7⁴ * 8⁴ / 9⁴
 = - 12⁴ * 3⁴ / 4⁴ + 56⁴ / 7⁴ * 8⁴ + 9⁴,
4096² = 12⁶ * 3⁶ * 4⁶ * 56⁶ / 7⁶ / 8⁶ / 9⁶ = 12⁶ * 3⁶ * 4⁶ / 56⁶ * 7⁶ * 8⁶ / 9⁶.

4096² = 16777216, 1² + 6² + 7² + 7² + 7² + 2² + 1² + 6² = 15².

Page of Squares : First Upload August 6, 2007 ; Last Revised December 28, 2013
by Yoshio Mimura, Kobe, Japan

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4097

4097² = 1⁴ + 8⁴ + 8⁴ + 64⁴ = 1⁶ + 4⁶ + 4⁶ + 16⁶.

4097² = 16785409, a square with different digits.

85^k + 1649^k + 4097^k + 4573^k are squares for k = 1,2,3 (102², 6358², 410958²).

Page of Squares : First Upload August 6, 2007 ; Last Revised June 7, 2011
by Yoshio Mimura, Kobe, Japan

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4098

4098² = (1² + 5)(1673² + 5).

Page of Squares : First Upload December 28, 2013 ; Last Revised December 28, 2013
by Yoshio Mimura, Kobe, Japan

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