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53000

53000² = (1² + 4)(6² + 4)(7² + 4)(11² + 4)(46² + 4) = (7² + 4)(11² + 4)(14² + 4)(46² + 4).

Page of Squares : First Upload March 15, 2014 ; Last Revised March 15, 2014
by Yoshio Mimura, Kobe, Japan

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53016

235² + 53016 = 329², 235² - 53016 = 47².

Page of Squares : First Upload September 1, 2012 ; Last Revised September 1, 2012
by Yoshio Mimura, Kobe, Japan

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53064

53064²± 5 are primes.

53064² = (1² + 8)(16² + 8)(27² + 8)(40² + 8) = (5² + 8)(8² + 8)(27² + 8)(40² + 8).

Page of Squares : First Upload March 15, 2014 ; Last Revised March 15, 2014
by Yoshio Mimura, Kobe, Japan

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53084

53084²± 3 are primes.

Page of Squares : First Upload March 15, 2014 ; Last Revised March 15, 2014
by Yoshio Mimura, Kobe, Japan

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53108

53108²± 3 are primes.

Page of Squares : First Upload March 15, 2014 ; Last Revised March 15, 2014
by Yoshio Mimura, Kobe, Japan

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53126

53126²± 3 are primes.

Page of Squares : First Upload March 15, 2014 ; Last Revised March 15, 2014
by Yoshio Mimura, Kobe, Japan

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53130

53130² + 52051² = 5532103501, a mosaic.

Page of Squares : First Upload September 1, 2012 ; Last Revised September 1, 2012
by Yoshio Mimura, Kobe, Japan

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53134

53134²± 3 are primes.

Page of Squares : First Upload March 15, 2014 ; Last Revised March 15, 2014
by Yoshio Mimura, Kobe, Japan

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53136

53136² = (1² + 8)(4² + 8)(8² + 8)(19² + 8)(22² + 8) = (8² + 8)(44² + 8)(142² + 8).

Page of Squares : First Upload March 15, 2014 ; Last Revised March 15, 2014
by Yoshio Mimura, Kobe, Japan

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53144

53144² = (2² + 3)(7² + 3)(17² + 3)(163² + 3) = (17² + 3)(19² + 3)(163² + 3).

Page of Squares : First Upload March 15, 2014 ; Last Revised March 15, 2014
by Yoshio Mimura, Kobe, Japan

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53152

53152² = (1² + 7)(9² + 7)(12² + 7)(163² + 7) = (2² + 7)(9² + 7)(12² + 7)(139² + 7)
= (12² + 7)(31² + 7)(139² + 7).

Page of Squares : First Upload March 15, 2014 ; Last Revised March 15, 2014
by Yoshio Mimura, Kobe, Japan

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53194

53194²± 3 are primes.

Page of Squares : First Upload March 15, 2014 ; Last Revised March 15, 2014
by Yoshio Mimura, Kobe, Japan

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53200

53200² = 295³ + 296³ + 297³ + 298² + 299² + ... + 370³.

Page of Squares : First Upload September 1, 2012 ; Last Revised September 1, 2012
by Yoshio Mimura, Kobe, Japan

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53231

53231 is the 8th prime for which the Legendre Symbol (a/53231) = 1 for a = 1,2,3,...,22.

Page of Squares : First Upload September 1, 2012 ; Last Revised September 1, 2012
by Yoshio Mimura, Kobe, Japan

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53265

The quadratic polynomial 53265X² - 235200X + 278656 takes the values 311², 146², 229², 436², 659², 886² at X = 1, 2,..., 6.

Page of Squares : First Upload September 1, 2012 ; Last Revised September 1, 2012
by Yoshio Mimura, Kobe, Japan

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53280

53280² = (2² - 1)(3² - 1)(73² - 1)(149² - 1) = (5² - 1)(73² - 1)(149² - 1).

Page of Squares : First Upload March 15, 2014 ; Last Revised March 15, 2014
by Yoshio Mimura, Kobe, Japan

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53298

53298² = (18² + 5)(29² + 5)(101² + 5) = (2² + 5)(4² + 5)(7² + 5)(18² + 5)(29² + 5).

Page of Squares : First Upload March 15, 2014 ; Last Revised March 15, 2014
by Yoshio Mimura, Kobe, Japan

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53300

53300² = (1² + 1)(2² + 1)(3² + 1)(8² + 1)(9² + 1)(73² + 1)
= (1² + 1)(3² + 1)(5² + 1)(32² + 1)(73² + 1) = (1² + 1)(3² + 1)(5² + 1)(8² + 1)(9² + 1)(32² + 1)
= (1² + 1)(3² + 1)(9² + 1)(18² + 1)(73² + 1) = (1² + 1)(7² + 1)(8² + 1)(9² + 1)(73² + 1)
= (1² + 1)(9² + 1)(57² + 1)(73² + 1) = (2² + 1)(5² + 1)(7² + 1)(9² + 1)(73² + 1)
= (1² + 4)(3² + 4)(4² + 4)(23² + 4)(64² + 4) = (1² + 4)(16² + 4)(23² + 4)(64² + 4)
= (23² + 4)(36² + 4)(64² + 4).

Page of Squares : First Upload November 16, 2013 ; Last Revised March 15, 2014
by Yoshio Mimura, Kobe, Japan

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53307

53307²± 2 are primes.

Page of Squares : First Upload March 15, 2014 ; Last Revised March 15, 2014
by Yoshio Mimura, Kobe, Japan

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53340

The quadratic polynomial 53340X² - 337260X + 561649 takes the values 527², 317², 173², 257², 457², 677² at X = 1, 2,..., 6.

Page of Squares : First Upload September 1, 2012 ; Last Revised September 1, 2012
by Yoshio Mimura, Kobe, Japan

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53352

53352² = (2² - 1)(5² - 1)(37² - 1)(170² - 1).

Page of Squares : First Upload March 15, 2014 ; Last Revised March 15, 2014
by Yoshio Mimura, Kobe, Japan

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53380

53380² = (4² + 4)(56² + 4)(213² + 4).

Page of Squares : First Upload March 15, 2014 ; Last Revised March 15, 2014
by Yoshio Mimura, Kobe, Japan

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53406

53406² = (1² + 5)(8² + 5)(34² + 5)(77² + 5) = (9² + 5)(34² + 5)(169² + 5).

Page of Squares : First Upload March 15, 2014 ; Last Revised March 15, 2014
by Yoshio Mimura, Kobe, Japan

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53416

53416²± 3 are primes.

Page of Squares : First Upload March 15, 2014 ; Last Revised March 15, 2014
by Yoshio Mimura, Kobe, Japan

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53417

53417² = 287² + 288² + 289² + 290² + ... + 2047².

Page of Squares : First Upload September 1, 2012 ; Last Revised September 1, 2012
by Yoshio Mimura, Kobe, Japan

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53438

53438²± 3 are primes.

Page of Squares : First Upload March 15, 2014 ; Last Revised March 15, 2014
by Yoshio Mimura, Kobe, Japan

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53494

53494²± 3 are primes.

Page of Squares : First Upload March 15, 2014 ; Last Revised March 15, 2014
by Yoshio Mimura, Kobe, Japan

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53502

53502² = 2862464004 is a square with even digits.

Page of Squares : First Upload September 1, 2012 ; Last Revised September 1, 2012
by Yoshio Mimura, Kobe, Japan

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53508

53508² = (1³ + 2³ + 3³ + ... + 13³)(14³ + 15³ + 16³ + ... + 34³).

53508² = (1² + 3)(2² + 3)(5² + 3)(6² + 3)(306² + 3) = (1² + 3)(2² + 3)(33² + 3)(306² + 3)
= (1² + 3)(7² + 3)(12² + 3)(306² + 3) = (2² + 3)(3² + 3)(19² + 3)(306² + 3)
= (2² + 3)(6² + 3)(7² + 3)(12² + 3)(37² + 3) = (2² + 3)(7² + 3)(9² + 3)(306² + 3)
= (2² + 3)(12² + 3)(37² + 3)(45² + 3) = (5² + 3)(33² + 3)(306² + 3)
= (6² + 3)(12² + 3)(19² + 3)(37² + 3) = (9² + 3)(19² + 3)(306² + 3).

Page of Squares : First Upload February 9, 2013 ; Last Revised March 15, 2014
by Yoshio Mimura, Kobe, Japan

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53522

53522² = 2864604484 is a square with even digits.

Page of Squares : First Upload September 1, 2012 ; Last Revised September 1, 2012
by Yoshio Mimura, Kobe, Japan

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53546

53546²= 372 x 373 + 373 x 374 + 374 x 375 + ... + 2052 x 2053.

Page of Squares : First Upload September 1, 2012 ; Last Revised September 1, 2012
by Yoshio Mimura, Kobe, Japan

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53578

53578²± 3 are primes.

Page of Squares : First Upload March 15, 2014 ; Last Revised March 15, 2014
by Yoshio Mimura, Kobe, Japan

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53612

53612²± 3 are primes.

Page of Squares : First Upload March 15, 2014 ; Last Revised March 15, 2014
by Yoshio Mimura, Kobe, Japan

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53650

53650² = (1² + 1)(2² + 1)(3² + 1)(6² + 1)(882² + 1)
= (1² + 1)(2² + 1)(6² + 1)(41² + 1)(68² + 1) = (1² + 1)(43² + 1)(882² + 1)
= (1² + 1)(6² + 1)(7² + 1)(882² + 1) = (2² + 1)(12² + 1)(17² + 1)(117² + 1)
= (3² + 1)(6² + 1)(41² + 1)(68² + 1) = (6² + 1)(12² + 1)(17² + 1)(43² + 1).

Page of Squares : First Upload November 16, 2013 ; Last Revised November 16, 2013
by Yoshio Mimura, Kobe, Japan

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53664

53664²± 5 are primes.

Page of Squares : First Upload March 15, 2014 ; Last Revised March 15, 2014
by Yoshio Mimura, Kobe, Japan

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53676

53676² = 96*97*98 + 98*99*100 + 100*101*102 + 102*103*104 + ... + 388*389*390.

Page of Squares : First Upload October 26, 2013 ; Last Revised October 26, 2013
by Yoshio Mimura, Kobe, Japan

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53680

53680² = 4759 x 4760 + 4761 x 4762 + 4763 x 4764 + 4765 x 4766 + ... + 4999 x 5000.

Page of Squares : First Upload September 1, 2012 ; Last Revised September 1, 2012
by Yoshio Mimura, Kobe, Japan

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53698

(53698² - 5) = (4² - 5)(6² - 5)(8² - 5)(16² - 5)(24² - 5).

Page of Squares : First Upload September 1, 2012 ; Last Revised September 1, 2012
by Yoshio Mimura, Kobe, Japan

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53739

53739²± 2 are primes.

Page of Squares : First Upload March 15, 2014 ; Last Revised March 15, 2014
by Yoshio Mimura, Kobe, Japan

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53756

The quadratic polynomial -53756X² + 364412X - 250631 takes the values 245², 513², 599², 589², 477², 25² at X = 1, 2,..., 6.

Page of Squares : First Upload September 1, 2012 ; Last Revised September 1, 2012
by Yoshio Mimura, Kobe, Japan

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53760

53760² = (3² - 1)(5² - 1)(9² - 1)(15² - 1)(29² - 1) = (3² - 1)(6² - 1)(7² - 1)(15² - 1)(31² - 1)
= (3² - 1)(9² - 1)(11² - 1)(13² - 1)(15² - 1) = (3² - 1)(15² - 1)(31² - 1)(41² - 1)
= (9² - 1)(13² - 1)(15² - 1)(31² - 1).

452² + 53760 = 508², 452² - 53760 = 388²,
296² + 53760 = 376², 296² - 53760 = 184²,
232² + 53760 = 328², 232² - 53760 = 8².

Page of Squares : First Upload September 1, 2012 ; Last Revised arch 15, 2014
by Yoshio Mimura, Kobe, Japan

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53770

53770²± 3 are primes.

Page of Squares : First Upload March 15, 2014 ; Last Revised March 15, 2014
by Yoshio Mimura, Kobe, Japan

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53802

53802² = (17² + 5)(19² + 5)(164² + 5) = (3² + 5)(4² + 5)(19² + 5)(164² + 5).

Page of Squares : First Upload March 15, 2014 ; Last Revised March 15, 2014
by Yoshio Mimura, Kobe, Japan

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53824

53824 = 232² is a zigzag square with different digits.

Page of Squares : First Upload September 1, 2012 ; Last Revised September 1, 2012
by Yoshio Mimura, Kobe, Japan

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53830

53830²± 3 are primes.

Page of Squares : First Upload March 15, 2014 ; Last Revised March 15, 2014
by Yoshio Mimura, Kobe, Japan

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53848

53848²± 3 are primes.

Page of Squares : First Upload March 15, 2014 ; Last Revised March 15, 2014
by Yoshio Mimura, Kobe, Japan

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53856

53856² = (1² + 8)(2² + 8)(6² + 8)(16² + 8)(48² + 8)
= (2² + 8)(3² + 8)(4² + 8)(5² + 8)(6² + 8)(20² + 8) = (2² + 8)(3² + 8)(4² + 8)(6² + 8)(116² + 8)
= (2² + 8)(3² + 8)(4² + 8)(20² + 8)(38² + 8) = (2² + 8)(3² + 8)(6² + 8)(20² + 8)(28² + 8)
= (2² + 8)(5² + 8)(6² + 8)(8² + 8)(48² + 8) = (2² + 8)(6² + 8)(20² + 8)(116² + 8)
= (2² + 8)(8² + 8)(38² + 8)(48² + 8) = (23² - 1)(35² - 1)(67² - 1)
= (3² + 8)(4² + 8)(6² + 8)(14² + 8)(28² + 8) = (3² + 8)(6² + 8)(8² + 8)(14² + 8)(16² + 8)
= (4² + 8)(5² + 8)(6² + 8)(14² + 8)(20² + 8) = (4² + 8)(6² + 8)(14² + 8)(116² + 8)
= (4² + 8)(6² + 8)(20² + 8)(82² + 8) = (4² + 8)(14² + 8)(20² + 8)(38² + 8)
= (6² + 8)(10² + 8)(16² + 8)(48² + 8) = (6² + 8)(14² + 8)(20² + 8)(28² + 8).

Page of Squares : First Upload March 15, 2014 ; Last Revised March 15, 2014
by Yoshio Mimura, Kobe, Japan

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53867

53867² = 1433² + 1435² + 1437² + 1439² + 1441² + ... + 2729².

Page of Squares : First Upload September 1, 2012 ; Last Revised September 1, 2012
by Yoshio Mimura, Kobe, Japan

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53811

53881 is the 9th Legendre Symbol (a/53881) = 1 for a = 1,2,3,...,22.

Page of Squares : First Upload September 1, 2012 ; Last Revised September 1, 2012
by Yoshio Mimura, Kobe, Japan

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53892

53892²± 5 are primes.

Page of Squares : First Upload March 15, 2014 ; Last Revised March 15, 2014
by Yoshio Mimura, Kobe, Japan

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53900

53900² = (1² + 6)(2² + 6)(4² + 6)(7² + 6)(8² + 6)(22² + 6)
= (1² + 6)(2² + 6)(4² + 6)(22² + 6)(62² + 6) = (2² + 6)(8² + 6)(22² + 6)(92² + 6)
= (4² + 6)(8² + 6)(22² + 6)(62² + 6).

Page of Squares : First Upload March 15, 2014 ; Last Revised March 15, 2014
by Yoshio Mimura, Kobe, Japan

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53914

53914²± 3 are primes.

Page of Squares : First Upload March 15, 2014 ; Last Revised March 15, 2014
by Yoshio Mimura, Kobe, Japan

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53937

53937² = 2909199969 is a square pegged by 9.

Page of Squares : First Upload September 1, 2012 ; Last Revised September 1, 2012
by Yoshio Mimura, Kobe, Japan

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53940

53940² = (30² - 1)(1799² - 1).

Page of Squares : First Upload March 15, 2014 ; Last Revised March 15, 2014
by Yoshio Mimura, Kobe, Japan

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53943

53943²± 2 are primes.

Page of Squares : First Upload March 15, 2014 ; Last Revised March 15, 2014
by Yoshio Mimura, Kobe, Japan

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53966

53966²± 3 are primes.

Page of Squares : First Upload March 15, 2014 ; Last Revised March 15, 2014
by Yoshio Mimura, Kobe, Japan

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53976

53976² = 2913408576 is a square with different digits.

Page of Squares : First Upload September 1, 2012 ; Last Revised September 1, 2012
by Yoshio Mimura, Kobe, Japan

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53984

53984²± 3 are primes.

Page of Squares : First Upload March 15, 2014 ; Last Revised March 15, 2014
by Yoshio Mimura, Kobe, Japan

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53992

53992²± 3 are primes.

Page of Squares : First Upload March 15, 2014 ; Last Revised March 15, 2014
by Yoshio Mimura, Kobe, Japan

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