Squares:1800s

1800

1800² = 80³ + 100³ + 120³ = 30⁴ + 30⁴ + 30⁴ + 30⁴.

Komachi equation: 1800² = 1² / 2² * 3456² + 7² * 8² * 9².

1800² = (1)(2)(3)(4)(5)(6 + 7 + 8 + 9)(10 + 11 + 12 + 13 + 14)(15),
1800² = (1)(2)(3)(4 + 5 + 6 + ... + 12)(13 + 14 + 15 + 16 + 17)(18 + 19 + 20 + 21 + 22),
1800² = (1)(2)(3)(4 + 5 + 6 + 7 + 8)(9 + 10 + 11 + ... + 23)(24 + 25 + 26),
1800² = (1)(2)(3)(4 + 5 + 6 + 7 + 8)(9 + 10 + 11)(12 + 13 + 14 + ... + 36),
1800² = (1)(2)(3 + 4 + 5 + 6)(7 + 8 + 9 + 10 + 11)(12 + 13)(14 + 15 + 16 + 17 + 18),
1800² = (1)(2)(3 + 4 + 5 + 6 + 7)(8)(9)(10 + 11 + 12 + 13 + 14)(15),
1800² = (1)(2)(3 + 4 + 5 + 6 + 7)(8)(9 + 10 + 11 + ... + 18)(19 + 20 + 21),
1800² = (1)(2)(3 + 4 + 5 + 6 + 7)(8 + 9 + 10 + 11 + 12)(13 + 14)(15 + 16 + 17),
1800² = (1)(2)(3 + 4 + 5 + 6 + 7)(8 + 9 + 10)(11 + 12 + 13 + 14)(15 + 16 + 17),
1800² = (1)(2 + 3)(4)(5)(6)(7 + 8)(9 + 10 + 11)(12),
1800² = (1)(2 + 3)(4)(5)(6 + 7 + 8 + ... + 14)(15 + 16 + 17 + ... + 30),
1800² = (1)(2 + 3)(4)(5 + 6 + 7 + ... + 10)(11 + 12 + 13 + ... + 85),
1800² = (1)(2 + 3)(4 + 5)(6)(7 + 8 + 9)(10)(11 + 12 + 13 + 14),
1800² = (1)(2 + 3 + 4 + 5 + 6)(7 + 8)(9)(10 + 11 + 12 + ... + 15)(16),
1800² = (1)(2 + 3 + 4 + 5 + 6)(7 + 8 + 9 + 10 + 11)(12)(13 + 14 + 15 + ... + 27),
1800² = (1)(2 + 3 + 4 + 5 + 6)(7 + 8 + 9 + 33)(34 + 35 + 36 + ... + 41),
1800² = (1)(2 + 3 + 4 + 5 + 6)(7 + 8 + 9)(10 + 11 + 12 + ... + 15)(16 + 17 + 18 + ... + 20),
1800² = (1)(2 + 3 + 4)(5)(6)(7 + 8 + 9)(10)(11 + 12 + 13 + 14),
1800² = (1 + 2)(3 + 4 + 5 + ... + 17)(18 + 19 + 20 + 21 + 22)(23 + 24 + 25),
1800² = (1 + 2)(3 + 4 + 5 + 6 + 7)(8)(9 + 10 + 11 + ... + 16)(17 + 18 + 19),
1800² = (1 + 2 + 3 + 4)(5)(6 + 7 + 8 + 9 + 10)(11 + 12 + 13)(14 + 15 + 16),
1800² = (1 + 2 + 3 + 4)(5 + 6 + 7 + ... + 19)(20)(21 + 22 + 23 + 24),
1800² = (1 + 2 + 3 + 4)(5 + 6 + 7 + ... + 20)(21 + 22 + 23 + ... + 60),
1800² = (1 + 2 + 3)(4)(5)(6 + 7 + 8 + 9)(10 + 11 + 12 + 13 + 14)(15),
1800² = (1 + 2 + 3)(4 + 5 + 6 + ... + 12)(13 + 14 + 15 + 16 + ... + 17)(18 + 19 + 20 + 21 + 22),
1800² = (1 + 2 + 3)(4 + 5 + 6 + 7 + 8)(9 + 10 + 11 + ... + 23)(24 + 25 + 26),
1800² = (1 + 2 + 3)(4 + 5 + 6 + 7 + 8)(9 + 10 + 11)(12 + 13 + 14 + ... + 36).

Page of Squares : First Upload February 26, 2007 ; Last Revised September 7, 2010
by Yoshio Mimura, Kobe, Japan

1801

1801² = 3243601, a zigzag square.

1801² = 6⁶ + 6⁶ + 6⁶ + 7⁶ + 12⁶.

Page of Squares : First Upload February 26, 2007 ; Last Revised July 14, 2008
by Yoshio Mimura, Kobe, Japan

1803

1803² = 3250809, a zigzag square.

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

1804

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

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

1805

1805² = (48 + 49 + 50 + ... + 66)² + (67 + 68 + 69 + ... + 85)² + (86 + 87 + 88 + ... + 104)² + ... + (67 + 68 + 69 + ... + 85)².

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

1806

1806² = 3261636, a zigzag square.

1806²± 5 are primes.

1806² = (3² + 5)(9² + 5)(52² + 5).

1806² = 19³ + 105³ + 128³.

(1806² - 6) = (3² - 6)(8² - 6)(11² - 6)(13² - 6).

Page of Squares : First Upload February 26, 2007 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

1807

404768^k + 722800^k + 851097^k + 1286584^k are squares for k = 1,2,3 (1807², 1750983², 1786091203²).

Page of Squares : First Upload May 10, 2011 ; Last Revised May 10, 2011
by Yoshio Mimura, Kobe, Japan

1808

1808² = 68³ + 76³ + 136³.

Page of Squares : First Upload July 14, 2008 ; Last Revised July 14, 2008
by Yoshio Mimura, Kobe, Japan

1809

1809² = 3272481, 3 * 2 * 72 * 4 + 81 = 1809.

Loop of length 56 by the function f(N) = ... + c² + b² + a² where N = ... + 100²c + 100b + a:
1809 - 405 - 41 - ... - 5072 - 7684 - 12832 - 1809
(Note f(1809) = 18² + 09² = 405, f(405) = 4² + 05² = 41, etc. See 41)

Page of Squares : First Upload February 26, 2007 ; Last Revised October 9, 2008
by Yoshio Mimura, Kobe, Japan

1810

1810² = 1⁵ + 3⁵ + 7⁵ + 9⁵ + 20⁵.

Page of Squares : First Upload July 14, 2008 ; Last Revised July 14, 2008
by Yoshio Mimura, Kobe, Japan

1812

1812²± 5 are primes.

1812² = 36³ + 94³ + 134³.

1812⁵ = 19533990254072832 : 19² + 5² + 33² + 9² + 9² + 0² + 2² + 5² + 4² + 0² + 7² + 2² + 8² + 3² + 2² = 1812.

226^k + 610^k + 822^k + 1478^k are squares for k = 1,2,3 (56², 1812², 63424²).

Page of Squares : First Upload July 14, 2008 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

1813

1813² = 3286969, a zigzag square.

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

1814

1² + 2² + 3² + 4² + ... + 1814² = 1991359315, the first 10-digit sum consisting of odd digits (there are such 4 sums in all, cf. 2526, 2773, 3014).

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

1815

1815² = (1)(2 + 3)(4 + 5)(6 + 7 + 8 + ... + 16)(17 + 18 + 19 + ... + 38),
1815² = (1)(2 + 3 + 4)(5)(6 + 7 + 8 + ... + 16)(17 + 18 + 19 + ... + 38).

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

1816

1816² = 3297856, a zigzag square with different digits.

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

1817

1817² = 50³ + 112³ + 121³ = 6⁴ + 17⁴ + 18⁴ + 42⁴.

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

Page of Squares : First Upload February 26, 2007 ; Last Revised July 14, 2008
by Yoshio Mimura, Kobe, Japan

1818

1818² = 3305124, 3² + 3² + 0² + 5² + 1² + 2² + 4² = 8².

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

1819

S₂(1819) = S₂(77) + S₂(211), where S₂(n) = 1² + 2² + 3² + ... + n².

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

1821

1821² = 25³ + 68³ + 144³.

Page of Squares : First Upload July 14, 2008 ; Last Revised July 14, 2008
by Yoshio Mimura, Kobe, Japan

1822

1822² + 1823² + 1824² + ... + 7489² = 7490² + 7491² + 7492² + ... + 9413².

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

1823

1823² = 3323329, a square with just 3 kinds of digits.

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

1824

1824² = 44³ + 148³.

1824² = 48³ + 20⁵ + 4⁷.

Komachi equations:
1824² = 9² * 8² * 76² * 5² * 4² / 3² / 2² / 10² = 9² * 8² * 76² / 5² / 4² / 3² * 2² * 10².

Page of Squares : First Upload July 14, 2008 ; Last Revised January 6, 2011
by Yoshio Mimura, Kobe, Japan

1826

1 / 1826 = 0.0005476...., 5476 = 74².

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

1827

1827²± 2 are primes.

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

1828

1828² = 137² + 138² + 139² + 140² + 141² + 142² + ... + 232².

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

1830

1830² = 1³ + 2³ + 3³ + 4³ + 5³ + 6³ + ... + 60³.

1830² + 1831² + 1832² + ... + 1860² = 1861² + 1862² + 1863² + ... + 1890².

990^k + 1020^k + 1785^k + 1830^k are squares for k = 1,2,3 (75², 2925², 117675²).

Page of Squares : First Upload February 26, 2007 ; Last Revised September 9, 2011
by Yoshio Mimura, Kobe, Japan

1832

1832² = 3356224, 33 * 56 - 2 * 2 * 4 = 1832.

1832² = 163² + 165² + 167² + 169² + 170² + 171² + ... +289².

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

1833

S₂(1833) = S₂(148) + S₂(189), where S₂(n) = 1² + 2² + 3² + ... + n².

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

1834

1834² = 3363556, a square with just 3 kinds of digits.

1834² = 3363556, 3 * 3 * 6 * 35 - 56 = 1834.

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

1836

1836² = 24³ + 90³ + 138³.

1836² = (1² + 8)(3² + 8)(10² + 8)(14² + 8) = (2² + 8)(14² + 8)(37² + 8) = (2² + 8)(530² + 8).

A cubic polynomial :
(X + 912²)(X + 1211²)(X + 1836²) = X³ + 2381²X² + 2994492²X + 2027737152².

1836² = (1)(2)(3 + 4 + 5)(6 + 7 + 8 + 9 + 10 + 11)(12 + 13 + 14 + 15)(16 + 17 + 18),
1836² = (1 + 2)(3)(4)(5 + 6 + 7)(8 + 9)(10 + 11 + 12 + ... + 26),
1836² = (1 + 2)(3 + 4 + 5 + 6)(7 + 8 + 9 + 10)(11 + 12 + 13 + ... + 61),
1836² = (1 + 2)(3 + 4 + 5)(6 + 7 + 8 + 9 + 10 + 11)(12)(13 + 14 + 15 + ... + 21).

Page of Squares : First Upload February 26, 2007 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1837

1837² = 3374569, 3² + 3² + 7² + 4² + 5² + 6² + 9² = 15².

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

1838

1838² = 3378244, 33 * 7 * 8 - 2 - 4 - 4 = 1838.

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

1839

1839²± 2 are primes.

1839² = 3381921, 3² + 3² + 8² + 1² + 9² + 2² + 1² = 13².

Page of Squares : First Upload February 26, 2007 ; Last Revised December 29, 2013
by Yoshio Mimura, Kobe, Japan

1840

1840² = 44³ + 68³ + 144³.

Page of Squares : First Upload July 14, 2008 ; Last Revised July 14, 2008
by Yoshio Mimura, Kobe, Japan

1841

Komachi equation: 1841² = 1² / 2² / 3² / 4² * 56² * 789².

Page of Squares : First Upload September 7, 2010 ; Last Revised September 7, 2010
by Yoshio Mimura, Kobe, Japan

1843

1843⁴ = 11537224429201, and 1² + 1² + 5² + 3² + 7² + 22² + 4² + 4² + 29² + 20² + 1² = 1843.

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

1844

1844² = 64³ + 98³ + 130³.

Page of Squares : First Upload July 14, 2008 ; Last Revised July 14, 2008
by Yoshio Mimura, Kobe, Japan

1845

1845⁵ = 21378727540153125 : 2² + 1² + 37² + 8² + 7² + 2² + 7² + 5² + 4² + 0² + 15² + 3² + 1² + 2² + 5² = 1845.

Komachi equations:
1845² = 1⁶ * 2⁶ + 3⁶ * 4⁶ - 5⁶ + 6⁶ + 7⁶ - 8⁶ + 9⁶ = 9⁶ - 8⁶ + 7⁶ + 6⁶ - 5⁶ + 4⁶ * 3⁶ + 2⁶ */ 1⁶.

Page of Squares : First Upload December 8, 2008 ; Last Revised September 7, 2010
by Yoshio Mimura, Kobe, Japan

1847

1847² = 4⁴ + 18⁴ + 21⁴ + 42⁴.

Page of Squares : First Upload July 14, 2008 ; Last Revised July 14, 2008
by Yoshio Mimura, Kobe, Japan

1848

1848² = 49³ + 72³ + 143³.

1848² = (1)(2)(3 + 4 + 5 + ... + 13)(14 + 15 + 16 + ... + 35)(36),
1848² = (1)(2)(3 + 4 + 5 + ... + 8)(9 + 10 + 11 + ... + 24)(25 + 26 + 27 + ... + 31),
1848² = (1)(2)(3 + 4 + 5)(6 + 7 + 8 + ... + 533),
1848² = (1)(2 + 3 + 4 + ... + 12)(13 + 14 + 15)(16 + 17 + 18 + ... + 48),
1848² = (1)(2 + 3 + 4 + 5)(6 + 7 + 8 + ... + 698),
1848² = (1 + 2)(3)(4)(5 + 6)(7)(8)(9 + 10 + 11 + ... + 19),
1848² = (1 + 2)(3 + 4)(5 + 6)(7)(8)(9 + 10 + 11 + ... + 24).

Page of Squares : First Upload February 26, 2007 ; Last Revised July 14, 2008
by Yoshio Mimura, Kobe, Japan

1849

The square of 43.

1849² = 86³ + 86³ + 129³.

A cubic polynomial :
(X + 504²)(X + 999²)(X + 1472²) = X³ + 1849²X² + 1722312² + 741146112².

Page of Squares : First Upload February 26, 2007 ; Last Revised July 14, 2008
by Yoshio Mimura, Kobe, Japan

1850

1850² = (1² + 1)(3² + 1)(6² + 1)(68² + 1) = (2² + 1)(3² + 1)(6² + 1)(43² + 1)
= (2² + 1)(7² + 1)(117² + 1) = (6² + 1)(7² + 1)(43² + 1).

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

1851

1851² = 3426201, 342 * 6 - 201 = 1851.

1851² = 9³ + 88³ + 140³ = 64³ + 84³ + 137³.

Page of Squares : First Upload February 26, 2007 ; Last Revised July 14, 2008
by Yoshio Mimura, Kobe, Japan

1852

126^k + 538^k + 1050^k + 1422^k are squares for k = 1,2,3 (56², 1852², 64736²).

Page of Squares : First Upload May 10, 2011 ; Last Revised May 10, 2011
by Yoshio Mimura, Kobe, Japan

1856

1856² = (1² + 7)(15² + 7)(43² + 7).

Page of Squares : First Upload May 10, 2011 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1858

1858² = 7⁴ + 7⁴ + 13⁴ + 43⁴.

Page of Squares : First Upload July 14, 2008 ; Last Revised July 14, 2008
by Yoshio Mimura, Kobe, Japan

1859

The square root of 1859 is 43.11612..., and 43 = 1² + 1² + 6² + 1² + 2².

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

1862

Komachi equation: 1862² = 98² * 76² / 5² * 4² / 32² * 10².

Page of Squares : First Upload September 7, 2010 ; Last Revised September 7, 2010
by Yoshio Mimura, Kobe, Japan

1863

1863² = 9⁴ + 12⁴ + 24⁴ + 42⁴ = 9⁴ + 18⁴ + 36⁴ + 36⁴ = 12⁴ + 13⁴ + 34⁴ + 38⁴.

A cubic polynomial :
(X + 560²)(X + 1863²)(X + 4284²) = X³ + 4705²X² + 8398908²X + 4469411520².

The second square which is the sums of four 4th powers in 3 ways :
1863² = 9⁴ + 12⁴ + 24⁴ + 42⁴ = 9⁴ + 18⁴ + 36⁴ + 36⁴ = 12⁴ + 13⁴ + 34⁴ + 38⁴.

Page of Squares : First Upload February 26, 2007 ; Last Revised July 14, 2008
by Yoshio Mimura, Kobe, Japan

1865

1865² = 3478225, 3 * 4 * 78 * 2 - 2 - 5 = 1865.

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

1866

1866² = 3481956, a square with different digits.

1866² = S₂(99) + S₂(211), where S₂(n) = 1² + 2² + 3² + ... + n².

464634^k + 696018^k + 1065486^k + 1255818^k are squares for k = 1,2,3 (1866², 1847340², 1904629932²).

Page of Squares : First Upload February 26, 2007 ; Last Revised May 10, 2011
by Yoshio Mimura, Kobe, Japan

1867

1867² = 3485689, 3 + 4 * 8 * 56 + 8 * 9 = 1867.

1867² = 20³ + 110³ + 129³.

Page of Squares : First Upload February 26, 2007 ; Last Revised July 14, 2008
by Yoshio Mimura, Kobe, Japan

1869

1869² = 8³ + 49³ + 150³.

Page of Squares : First Upload July 14, 2008 ; Last Revised July 14, 2008
by Yoshio Mimura, Kobe, Japan

1870

1870² = (1 + 2 + 3 + 4)(5 + 6)(7 + 8 + 9 + 10)(11 + 12 + 13 + ... + 44).

704^k + 1034^k + 1870^k + 6193^k are squares for k = 1,2,3 (99², 6589², 495495²).

Page of Squares : First Upload February 26, 2007 ; Last Revised May 10, 2011
by Yoshio Mimura, Kobe, Japan

1872

1872² = 3504384, a zigzag square.

Komachi equation: 1872² = 9² * 8² * 7² * 65² * 4² * 3² / 210².

1872² = (1)(2)(3)(4 + 5 + 6 + ... + 9)(10 + 11 + 12 + ... + 22)(23 + 24 + 25),
1872² = (1)(2 + 3 + 4 + ... + 14)(15 + 16 + 17)(18)(19 + 20),
1872² = (1 + 2 + 3)(4 + 5 + 6 + ... + 9)(10 + 11 + 12 + ... + 22)(23 + 24 + 25).

Page of Squares : First Upload February 26, 2007 ; Last Revised September 7, 2010
by Yoshio Mimura, Kobe, Japan

1873

1873² = 3508129, a square with different digits.

1873² = 57³ + 99³ + 133³.

Page of Squares : First Upload February 26, 2007 ; Last Revised July 14, 2008
by Yoshio Mimura, Kobe, Japan

1874

1 / 1874 = 0.00053361..., and 53361 = 231².

1874² = 3511876, 3 - 5 + 1 * 1876 = 3 - 5 * 1 + 1876 = 1874.

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

1875

1875² = 3515625, 3 * 5 * 1 / 5 * 625 = 3 / 5 * 1 * 5 * 625 = 1875.

1875² = 25³ + 50³ + 150³.

1875² = (1 + 2)(3 + 4 + 5 + 6 + 7)(8 + 9 + 10 + ... + 17)(18 + 19 + 20 + ... + 32).

Page of Squares : First Upload February 26, 2007 ; Last Revised July 14, 2008
by Yoshio Mimura, Kobe, Japan

1876

1876² = 3519376, a zigzag square with odd digits except the last digit 6.

Page of Squares : First Upload February 26, 2007 ; Last Revised August 24, 2013
by Yoshio Mimura, Kobe, Japan

1877

1877² = 59³ + 95³ + 135³.

Page of Squares : First Upload July 14, 2008 ; Last Revised July 14, 2008
by Yoshio Mimura, Kobe, Japan

1878

The square root of 1878 is 43.335..., and 43 = 3² + 3² + 5².

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

1880

1880²= 145 x 146 + 146 x 147 + 147 x 148 + 148 x 149 + 149 x 150 + ...+ 238 x 239.

1880² = 71³ + 112³ + 121³.

The sum of the squares of divisors of 1880 is a square, 2210².

Komachi equation: 1880² = 987² * 6² * 5² * 4² / 3² / 21².

Page of Squares : First Upload February 26, 2007 ; Last Revised November 1, 2011
by Yoshio Mimura, Kobe, Japan

1881

1881² = 3538161, a zigzag square.

1881² = (1² + 2)(3² + 2)(13² + 2)(25² + 2) = (1² + 2)(3² + 2)(5² + 2)(63² + 2) = (1² + 8)(7² + 8)(83² + 8).

Page of Squares : First Upload February 26, 2007 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1883

1883² = 3545689, 3² + 5² + 4² + 5² + 6² + 8² + 9² = 16².

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

1884

1884² = 3549456, 3 + 5 * 4 * 94 - 5 + 6 = 3 - 5 + 4 * 94 * 5 + 6 = 1884.

1884² = 22³ + 30³ + 152³.

66^k + 210^k + 1230^k + 1410^k are squares for k = 1,2,3 (54², 1884², 68364²).

Page of Squares : First Upload February 26, 2007 ; Last Revised May 10, 2011
by Yoshio Mimura, Kobe, Japan

1885

1885² = 3553225, a square with just 3 kinds of digits.

1885² = (2² + 1)(12² + 1)(70² + 1).

1885² = S₂(137) + S₂(200), where S₂(n) = 1² + 2² + 3² + ... + n².

Page of Squares : First Upload February 26, 2007 ; Last Revised November 2, 2013
by Yoshio Mimura, Kobe, Japan

1887

1 / 1887 = 0.000529...., and 529 = 23².

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

1888

1 / 1888 = 0.000529...., and 529 = 23².

The square root of 1888 is 43.4511..., and 43 = 4² + 5² + 1² + 1².

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

1889

1 / 1889 = 0.000529...., and 529 = 23².

1889² = 4³ + 89³ + 142³.

Page of Squares : First Upload February 26, 2007 ; Last Revised July 14, 2008
by Yoshio Mimura, Kobe, Japan

1890

Komachi equations:
1890² = 9² * 8² * 7² * 6² * 5² * 4² / 32² */ 1² = 9² / 8² * 7² * 6² * 5² / 4² * 32² */ 1²
= 98² / 7² / 6² * 54² * 3² / 2² * 10².

1890² = (1² + 5)(2² + 5)(10² + 5)(25² + 5) = (1² + 5)(2² + 5)(4² + 5)(5² + 5)(10² + 5)
= (1² + 5)(3² + 5)(10² + 5)(20² + 5) = (2² + 5)(4² + 5)(5² + 5)(25² + 5)
= (2² + 5)(5² + 5)(10² + 5)(11² + 5) = (2² + 5)(5² + 5)(115² + 5) = (3² + 5)(20² + 5)(25² + 5)
= (3² + 5)(4² + 5)(5² + 5)(20² + 5) = (4² + 5)(5² + 5)(7² + 5)(10² + 5) = (5² + 5)(17² + 5)(20² + 5)
= (7² + 5)(10² + 5)(25² + 5).

1890² = (1)(2)(3)(4 + 5)(6 + 7 + 8 + ... + 12)(13 + 14 + 15 + ... + 47),
1890² = (1)(2)(3)(4 + 5)(6 + 7 + 8 + 9)(10 + 11)(12 + 13 + 14 + ... + 18),
1890² = (1)(2)(3)(4 + 5)(6 + 7 + 8)(9 + 10 + 11 + 12)(13 + 14 + 15 + 16 + 17),
1890² = (1)(2)(3)(4 + 5)(6 + 7 + 8)(9 + 10 + 11)(12 + 13 + 14 + 15 + 16 + 17 + 18),
1890² = (1)(2)(3)(4 + 5 + 6 + ... + 24)(25)(26 + 27 + 28),
1890² = (1)(2)(3)(4 + 5 + 6 + ... + 31)(32 + 33 + 34 + ... + 58),
1890² = (1)(2)(3)(4 + 5 + 6 + 7 + 8)(9)(10 + 11)(12 + 13 + 14 + ... + 18),
1890² = (1)(2)(3 + 4 + 5 + ... + 11)(12 + 13)(14 + 15 + 15 + ... + 49),
1890² = (1)(2)(3 + 4 + 5 + 6)(7 + 8)(9)(10 + 11 + 12 + ... + 39),
1890² = (1)(2)(3 + 4 + 5 + 6 + 7)(8 + 9 + 10 + ... + 13)(14 + 15 + 15 + ... + 49),
1890² = (1)(2)(3 + 4 + 5 + 6 + 7)(8 + 9 + 10 + ... + 28)(29 + 30 + 31 + ... + 34),
1890² = (1)(2)(3 + 4 + 5 + 6 + 7)(8 + 9 + 10)(11 + 12 + 13 + ... + 73),
1890² = (1)(2)(3 + 4 + 5 + ... + 9)(10 + 11 + 12 + ... + 44)(45),
1890² = (1)(2 + 3)(4 + 5)(6 + 7 + 8)(9)(10 + 11 + 12 + ... + 30),
1890² = (1)(2 + 3)(4 + 5 + 6 + ... + 10)(11 + 12 + 13)(14 + 15 + 16 + ... + 31),
1890² = (1)(2 + 3 + 4 + ... + 16)(17 + 18 + 19 + ... + 23)(24 + 25 + 26 + ... + 30),
1890² = (1)(2 + 3 + 4 + ... + 16)(17 + 18 + ... + 25)(26 + 27 + 28 + 29 + 30),
1890² = (1)(2 + 3 + 4 + ... + 19)(20)(21)(22 + 23),
1890² = (1)(2 + 3 + 4 + ... + 19)(20 + 21 + 22 + ... + 25)(26 + 27 + 28 + 29 + 30),
1890² = (1)(2 + 3 + 4 + 5)(6)(7 + 8 + 9 + ... + 20)(21 + 22 + 23 + ... + 29),
1890² = (1)(2 + 3 + 4 + 5)(6 + 7 + 8)(9)(10)(11 + 12 + 13 + ... + 19),
1890² = (1)(2 + 3 + 4 + 5 + 6)(7)(8 + 9 + 10 + ... + 13)(14 + 15 + 16 + ... + 31),
1890² = (1)(2 + 3 + 4 + 5 + 6)(7 + 8 + 9 + ... + 20)(21)(22 + 23),
1890² = (1)(2 + 3 + 4)(5)(6 + 7 + 8)(9)(10 + 11 + 12 + ... + 30),
1890² = (1)(2 + 3 + 4)(5 + 6 + 7)(8 + 9 + 10 + ... + 22)(23 + 24 + 25 + ... + 26),
1890² = (1 + 2)(3)(4)(5 + 6 + 7 + ... + 445),
1890² = (1 + 2)(3)(4 + 5)(6 + 7 + 8 + ... + 14)(15 + 16 + 17 + ... + 34),
1890² = (1 + 2)(3)(4 + 5)(6 + 7 + 8 + 9)(10 + 11)(12 + 13 + 14 + 15 + 16),
1890² = (1 + 2)(3)(4 + 5)(6 + 7 + 8)(9 + 10 + 11)(12 + 13 + 14 + 15 + 16),
1890² = (1 + 2)(3)(4 + 5 + 6 + 7 + 8)(9)(10 + 11)(12 + 13 + 14 + 15 + 16),
1890² = (1 + 2)(3)(4 + 5 + 6)(7)(8 + 9 + 10 + ... + 16)(17 + 18),
1890² = (1 + 2)(3)(4 + 5 + 6)(7 + 8)(9 + 10 + 11 + 12)(13 + 14 + 15),
1890² = (1 + 2)(3 + 4)(5)(6 + 7 + 8 + 9)(10 + 11)(12 + 13 + 14 + 15),
1890² = (1 + 2)(3 + 4)(5)(6 + 7 + 8)(9 + 10 + 11)(12 + 13 + 14 + 15),
1890² = (1 + 2)(3 + 4 + 5 + ... + 11)(12)(13 + 14 + 15 + ... + 57),
1890² = (1 + 2)(3 + 4 + 5 + ... + 11)(12 + 13 + 14 + 15 + 16)(17 + 18 + 19 + ... + 28),
1890² = (1 + 2)(3 + 4 + 5 + ... + 11)(12 + 13 + 14 + ... + 18)(19 + 20 + 21 + ... + 26),
1890² = (1 + 2)(3 + 4 + 5 + 6 + 7)(8 + 9 + 10)(11 + 12 + 13 + ... + 17)(18),
1890² = (1 + 2)(3 + 4 + 5 + ... + 9)(10)(11 + 12 + 13 + ... + 16)(17 + 18),
1890² = (1 + 2)(3 + 4 + 5)(6 + 7 + 8 + ... + 12)(13 + 14 + 15 + ... + 57),
1890² = (1 + 2)(3 + 4 + 5)(6 + 7 + 8)(9)(10 + 11)(12 + 13),
1890² = (1 + 2 + 3 + 4)(5 + 6 + 7)(8 + 9 + 10 + ... + 13)(14 + 15 + 16 + ... + 28),
1890² = (1 + 2 + 3 + 4)(5 + 6 + 7)(8 + 9 + 10 + ... + 34)(35),
1890² = (1 + 2 + 3 + 4 + 5)(6 + 7 + 8 + ... + 12)(13 + 14 + 15)(16 + 17 + 18 + 19 + 20),
1890² = (1 + 2 + 3 + ... + 6)(7 + 8 + 9 + 10 + 11)(12 + 13 + 14 + 15)(16 + 17 + 18 + 19),
1890² = (1 + 2 + 3 + ... + 6)(7 + 8 + 9 + 10 + 11)(12 + 13 + 14 + 15 + 16)(17 + 18 + 19),
1890² = (1 + 2 + 3 + ... + 6)(7 + 8 + 9 + ... + 13)(14 + 15 + 16)(17 + 18 + 19),
1890² = (1 + 2 + 3 + ... + 7)(8 + 9 + 10 + ... + 22)(23 + 24 + 25 + ... + 40),
1890² = (1 + 2 + 3 + ... + 7)(8 + 9 + 10 + ... + 34)(35 + 36 + 37 + ... + 40),
1890² = (1 + 2 + 3 + ... + 8)(9 + 10 + 11 + ... + 26)(27 + 28 + 29 + ... + 36),
1890² = (1 + 2 + 3 + ... + 9)(10 + 11)(12 + 13 + 14 + 15)(16 + 17 + 18 + 19),
1890² = (1 + 2 + 3 + ... + 9)(10 + 11)(12 + 13 + 14 + 15 + 16)(17 + 18 + 19),
1890² = (1 + 2 + 3)(4 + 5)(6 + 7 + 8 + ... + 12)(13 + 14 + 15 + ... + 47),
1890² = (1 + 2 + 3)(4 + 5)(6 + 7 + 8 + 9)(10 + 11)(12 + 13 + 14 + ... + 18),
1890² = (1 + 2 + 3)(4 + 5)(6 + 7 + 8)(9 + 10 + 11 + 12)(13 + 14 + 15 + 16 + 17),
1890² = (1 + 2 + 3)(4 + 5)(6 + 7 + 8)(9 + 10 + 11)(12 + 13 + 14 + ... + 18),
1890² = (1 + 2 + 3)(4 + 5 + 6 + ... + 24)(25)(26 + 27 + 28),
1890² = (1 + 2 + 3)(4 + 5 + 6 + ... + 31)(32 + 33 + 34 + ... + 58),
1890² = (1 + 2 + 3)(4 + 5 + 6 + 7 + 8)(9)(10 + 11)(12 + 13 + 14 + ... + 18).

Page of Squares : First Upload February 26, 2007 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1891

1891 = (1² + 2² + 3² + ... + 30²) / (1² + 2²).

1891² = 1³ + 2³ + 3³ + 4³ + 5³ + 6³ + ... + 61³.

Page of Squares : First Upload February 26, 2007 ; Last Revised November 25, 2008
by Yoshio Mimura, Kobe, Japan

1893

1893² = 3583449, 3 * 583 + 4 * 4 * 9 = 1893.

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

1894

1894² = 3587236, 3² + 5² + 8² + 7² + 2² + 3² + 6² = 14².

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

1895

1895² = 8⁵ + 12⁵ + 17⁵ + 18⁵.

Page of Squares : First Upload July 14, 2008 ; Last Revised July 14, 2008
by Yoshio Mimura, Kobe, Japan

1896

1896² = 3594816, a square with different digits.

1 / 1896 = 0.00052742616033, 5² + 2² + 7² + 4² + 26² + 1² + 6² + 033² = 1896.

1896² = 3594816, 3 * 5 * 94 + 81 * 6 = 3 + 59 * 4 * 8 - 1 + 6 = 1896.

1896² = 83³ + 109³ + 120³.

Page of Squares : First Upload February 26, 2007 ; Last Revised July 14, 2008
by Yoshio Mimura, Kobe, Japan

1898

1898² = 3602404, 3² + 6² + 0² + 2² + 4² + 0² + 4² = 9².

1898² = 3602404, and 324 = 18², 64 = 8², 00 = 0².

1898² = 9⁵ + 9⁵ + 12⁵ + 15⁵ + 19⁵.

1898^k + 4862^k + 5954^k + 20410^k are squares for k = 1,2,3 (182², 21892², 2972372²).

Page of Squares : First Upload February 26, 2007 ; Last Revised May 10, 2011
by Yoshio Mimura, Kobe, Japan