Squares:1600s

1600

The square of 40.

1600² = 02560000, 5600² = 31360000, 3600² = 12960000, 9600² = 92160000,
Other examples: 2916 - 5030 - 3009 - 0540 - 2916, and 2100 - 4100 - 8100 - 6100 - 2100.

Page of Squares : First Upload February 13, 2007 ; Last Revised February 13, 2007
by Yoshio Mimura, Kobe, Japan

1602

S₂(1602) = S₂(742) + S₂(1547), where S₂(n) = 1² + 2² + 3² + ... + n².

1602₂ = 3⁴ + 21⁴ + 33⁴ + 33⁴.

1602₂ = (1² + 2)(4² + 2)(218² + 2).

Komachi equation: 1602² = 12² * 3² * 4² / 56² * 7² * 89².

Page of Squares : First Upload February 13, 2007 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1603

1 / 1603 = 0.000623830318, 6² + 23² + 8² + 30² + 3² + 1² + 8² = 1603.

Page of Squares : First Upload February 13, 2007 ; Last Revised February 13, 2007
by Yoshio Mimura, Kobe, Japan

1604

1604² = 2572816, 2 + 57 * 28 + 1 * 6 = 2 + 57 * 28 * 1 + 6 = 1604.

Page of Squares : First Upload February 13, 2007 ; Last Revised February 13, 2007
by Yoshio Mimura, Kobe, Japan

1605

The square root of 1605 is 40.062..., and 40 = 0² + 6² + 2².

Page of Squares : First Upload February 13, 2007 ; Last Revised February 13, 2007
by Yoshio Mimura, Kobe, Japan

1607

1607² = 9³ + 56³ + 134³.

Page of Squares : First Upload July 10, 2008 ; Last Revised July 10, 2008
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1608

1608² = 48³ + 61³ + 131³.

Page of Squares : First Upload July 10, 2008 ; Last Revised July 10, 2008
by Yoshio Mimura, Kobe, Japan

1611

1611² = 2595321, 2 - 5 + 9 + 5 * 321 = 1611.

1611²± 2 are primes.

1611₂ = 32₃ + 105₃ + 112₃.

Page of Squares : First Upload February 13, 2007 ; Last Revised December 29, 2013
by Yoshio Mimura, Kobe, Japan

1612

1612² = (7² + 3)(11² + 3)(20² + 3).

1612² = 30² + 31² + 32² + 33² + 34² + 35² + 36² + ... + 198².

A cubic polynomial:
(X + 1440²)(X + 1612²)(X + 5859²) = X³ + 6245²X² + 12875292²X + 13600379520².

Page of Squares : First Upload February 13, 2007 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1614

786^k + 1614^k + 3882^k + 4122^k are squares for k = 1,2,3 (102², 5940², 365004²).
48^k + 177^k + 186^k + 1614^k are squares for k = 1,2,3 (45², 1635², 64935²).

Page of Squares : First Upload April 28, 2011 ; Last Revised April 28, 2011
by Yoshio Mimura, Kobe, Japan

1615

293930^k + 529720^k + 687990^k + 1096585^k are squares for k = 1,2,3 (1615², 1429275², 1348452325²).

Page of Squares : First Upload April 28, 2011 ; Last Revised April 28, 2011
by Yoshio Mimura, Kobe, Japan

1616

1 / 1616 = 0.000618811881188118811, 618² + 81² + 1² + 881² + 188² + 1188² + 11² = 1616²,
1 / 1616 = 0.000618811881188118811, 618² + 81² + 188² + 1188² + 11² + 881² + 1² = 1616².

Page of Squares : First Upload February 13, 2007 ; Last Revised February 13, 2007
by Yoshio Mimura, Kobe, Japan

1617

1617²± 2 are primes.

1617² = 50³ + 70³ + 129³.

1617^k + 8106^k + 10290^k + 15708^k are squares for k = 1,2,3 (189², 20517², 2345679²).
294^k + 1288^k + 1617^k + 2730^k are squares for k = 1,2,3 (77², 3437², 163513²).
924^k + 1078^k + 1617^k + 2310^k are squares for k = 1,2,3 (77², 3157², 136367²).

Page of Squares : First Upload July 10, 2008 ; Last Revised December 29, 2013
by Yoshio Mimura, Kobe, Japan

1618

1618² = 23⁴ + 29⁴ + 29⁴ + 31⁴.

Page of Squares : First Upload July 10, 2008 ; Last Revised July 10, 2008
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1619

1619² = 2621161, a square with just 3 kinds of digits 1, 2 and 6.

Page of Squares : First Upload February 13, 2007 ; Last Revised February 13, 2007
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1620

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

1620² = (1)(2)(3)(4)(5 + 6 + 7 + ... + 10)(11 + 12 + 13 + ... + 70),
1620² = (1)(2)(3)(4)(5 + 6 + 7 + ... + 22)(23 + 24 + 25 + ... + 37),
1620² = (1)(2)(3)(4 + 5)(6)(7 + 8)(9)(10 + 11 + 12 + 13 + 14),
1620² = (1)(2 + 3)(4)(5 + 6 + 7 + ... + 13)(14 + 15 + 16 + ... + 58),
1620² = (1)(2 + 3)(4)(5 + 6 + 7)(8 + 9 + 10)(11 + 12 + 13 + ... + 25),
1620² = (1)(2 + 3)(4 + 5)(6 + 7 + 8 + 9)(10 + 11 + 12 + ... + 17)(18),
1620² = (1)(2 + 3)(4 + 5 + 6 + 7 + 8)(9)(10 + 11 + 12 + ... + 17)(18),
1620² = (1)(2 + 3 + 4 + ... + 7)(8)(9)(10)(11 + 12 + 13 + ... + 19),
1620² = (1)(2 + 3 + 4 + ... + 7)(8 + 9 + 10)(11 + 12 + 13 + ... + 85),
1620² = (1)(2 + 3 + 4)(5)(6 + 7 + ... + 9)(10 + 11 + 12 + ... + 17)(18),
1620² = (1)(2 + 3 + 4)(5 + 6 + 7)(8 + 9 + 10 + ... + 22)(23 + 24 + 25),
1620² = (1 + 2)(3)(4)(5)(6)(7 + 8 + 9 + 10 + 11)(12 + 13 + 14 + 15),
1620² = (1 + 2)(3)(4)(5 + 6 + 7 + ... + 10)(11 + 12 + 13)(14 + 15 + 16),
1620² = (1 + 2)(3)(4)(5 + 6 + 7 + ... + 19)(20 + 21 + 22 + ... + 34),
1620² = (1 + 2)(3)(4 + 5)(6)(7 + 8)(9 + 10 + 11)(12),
1620² = (1 + 2)(3)(4 + 5)(6 + 7 + 8 + ... + 14)(15 + 16 + 17 + ... + 30),
1620² = (1 + 2)(3)(4 + 5 + 6 + ... + 11)(12)(13 + 14)(15),
1620² = (1 + 2)(3)(4 + 5 + 6 + ... + 11)(12 + 13 + 14 + 15)(16 + 17 + 18 + 19 + 20),
1620² = (1 + 2)(3)(4 + 5 + 6 + 7 + 8)(9 + 10 + 11)(12)(13 + 14),
1620² = (1 + 2)(3)(4 + 5 + 6)(7 + 8 + 9)(10)(11 + 12 + 13 + ... + 16),
1620² = (1 + 2)(3 + 4 + 5 + 6)(7 + 8)(9)(10)(11 + 12 + 13),
1620² = (1 + 2)(3 + 4 + 5 + 6 + 7)(8 + 9 + 10 + ... + 19)(20 + 21 + 22 + ... + 28),
1620² = (1 + 2 + 3 + 4)(5 + 6 + 7 + ... + 10)(11 + 12 + 13)(14 + 15 + 16 + ... + 22),
1620² = (1 + 2 + 3 + 4)(5 + 6 + 7 + ... + 724),
1620² = (1 + 2 + 3 + 4)(5 + 6 + 7)(8 + 9 + 10)(11 + 12 + 13 + ... + 34),
1620² = (1 + 2 + 3 + 4 + 5)(6 + 7 + 8 + 9 + 10)(11 + 12 + 13 + ... + 16)(17 + 18 + 19),
1620² = (1 + 2 + 3 + ... + 9)(10)(11 + 12 + 13)(14 + 15 + 16 + ... + 22),
1620² = (1 + 2 + 3)(4)(5 + 6 + 7 + ... + 10)(11 + 12 + 13 + ... + 70),
1620² = (1 + 2 + 3)(4)(5 + 6 + 7 + ... + 22)(23 + 24 + 25 + ... + 37),
1620² = (1 + 2 + 3)(4 + 5)(6)(7 + 8)(9)(10 + 11 + 12 + 13 + 14).

Page of Squares : First Upload February 13, 2007 ; Last Revised February 13, 2007
by Yoshio Mimura, Kobe, Japan

1621

The square root of 1621 is 40.26..., and 40 = 2² + 6².

1621² = 2627641 appears in the decimal expressions of e:
e = 2.71828•••2627641••• (from the 48086th digit)

Page of Squares : First Upload February 13, 2007 ; Last Revised November 4, 2008
by Yoshio Mimura, Kobe, Japan

1623

1623² = 6³ + 106³ + 113³.

Page of Squares : First Upload July 10, 2008 ; Last Revised July 10, 2008
by Yoshio Mimura, Kobe, Japan

1624

1624² = 2637376, a zigzag square.

1624²± 3 are primes.

1624² = (7² + 7)(14² + 7)(15² + 7) = (7² + 7)(217² + 7).

S₂(1624) = S₂(9) * S₂(12) * S₂(28), where S₂(n) = 1² + 2² + 3² + ... + n².

1624₂ = 50₃ + 80₃ + 126₃.

Page of Squares : First Upload February 13, 2007 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

1625

1625² = 2640625, 2² + 6² + 4² + 0² + 6² + 2² + 5² = 11².

1625₂ = 41₃ + 78₃ + 128₃.

Page of Squares : First Upload February 13, 2007 ; Last Revised July 10, 2008
by Yoshio Mimura, Kobe, Japan

1626

1626² = 2643876, 26 + 4 + 38 * 7 * 6 = 1626.

Page of Squares : First Upload February 13, 2007 ; Last Revised February 13, 2007
by Yoshio Mimura, Kobe, Japan

1628

1628² = 2650384, a square with different digits.

1628²± 3 are primes.

Page of Squares : First Upload February 13, 2007 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

1632

1632₂ = 92₃ + 96₃ + 100₃.

1632² = 2663424, 2² + 6² + 6² + 3² + 4² + 2² + 4² = 11².

1632² = 2663424, 2 * 6 / 6 * 34 * 24 = 2 / 6 * 6 * 34 * 24 = 1632,
1632² = 2663424, 26 * 63 - 4 + 2 - 4 = 26 * 63 - 4 / 2 - 4 = 1632.

1632² = (3² - 1)(577² - 1).

Page of Squares : First Upload February 13, 2007 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1633

1633² = 2666689, a square the digits of which are non-decreasing.

1633² = 2666689, 26 * 66 + 6 - 89 = 1633.

1633² = 20³ + 80³ + 129³.

Page of Squares : First Upload February 13, 2007 ; Last Revised July 10, 2008
by Yoshio Mimura, Kobe, Japan

1634

1634² = 2669956 appears in the decimal expressions of π:
π = 3.14159•••2669956••• (from the 80656st digit)

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

1635

1635² + 1636² + 1637² + ... + 2470² = 2471² + 2472² + 2473² + ... + 2954².

48^k + 177^k + 186^k + 1614^k are squares for k = 1,2,3 (45², 1635², 64935²).

Page of Squares : First Upload April 28, 2011 ; Last Revised September 9, 2011
by Yoshio Mimura, Kobe, Japan

1638

The integral triangle of sides 1869, 2873, 3370 has square area 1638².

Komachi eqaution: 1638² = 1² / 2² / 3² / 4² * 56² * 78² * 9².

1638² = (74 + 75 + 76)² + (77 + 78 + 79)² + (80 + 81 + 82)² + ... + (143 + 144 + 145)².

1638² = (1)(2)(3)(4 + 5)(6 + 7)(8 + 9 + ... + 20)(21),
1638² = (1)(2)(3 + 4)(5 + 6 + 7 + 8)(9)(10 + 11)(12 + 13 + 14),
1638² = (1)(2 + 3 + 4 + 5)(6 + 7)(8 + 9 + 10 + ... + 13)(14 + 15 + 16 + ... + 25),
1638² = (1)(2 + 3 + 4 + 5)(6 + 7 + 8 + ... + 12)(13)(14 + 15 + 16 + ... + 25),
1638² = (1)(2 + 3 + 4 + 5)(6 + 7 + 8 + ... + 57)(58 + 59),
1638² = (1)(2 + 3 + 4 + 5)(6 + 7 + 8)(9 + 10 + 11 + ... + 17)(18 + 19 + 20 + 21),
1638² = (1)(2 + 3 + 4)(5 + 6 + 7 + 8)(9 + 10 + 11 + 12)(13 + 14 + 15 + ... + 26),
1638² = (1 + 2)(3 + 4)(5 + 6 + 7 + 8)(9 + 10 + 11 + ... + 99),
1638² = (1 + 2)(3 + 4 + 5 + ... + 15)(16 + 17 + 18 + ... + 23)(24 + 25),
1638² = (1 + 2 + 3 + ... + 13)(14)(15 + 16 + ... + 66),
1638² = (1 + 2 + 3 + ... + 13)(14 + 15 + ... + 22)(23 + 24 + 25 + ... + 29),
1638² = (1 + 2 + 3 + ... + 6)(7 + 8 + 9 + ... + 19)(20 + 21 + 22 + ... + 43),
1638² = (1 + 2 + 3 + ... + 6)(7 + 8 + 9 + ... + 32)(33 + 34 + 35 + ... + 39),
1638² = (1 + 2 + 3)(4 + 5)(6 + 7)(8 + 9 + 10 + ... + 20)(21).

Page of Squares : First Upload February 13, 2007 ; Last Revised October 7, 2011
by Yoshio Mimura, Kobe, Japan

1639

1639² = 2686321, 268 * 6 + 32 - 1 = 1639 .

Page of Squares : First Upload February 13, 2007 ; Last Revised February 13, 2007
by Yoshio Mimura, Kobe, Japan

1640

1640² = 2689600 appears in the decimal expressions of π:
π = 3.14159•••2689600••• (from the 80129th digit)

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

1642

1642² = 2696164, a square pegged by 6.

Page of Squares : First Upload February 13, 2007 ; Last Revised February 13, 2007
by Yoshio Mimura, Kobe, Japan

1643

S₂(1643) = S₂(1239) + S₂(1363).

1643² = 62³ + 68³ + 129³.

Page of Squares : First Upload February 13, 2007 ; Last Revised July 10, 2008
by Yoshio Mimura, Kobe, Japan

1644

1644⁴ = 7304781885696, and 7² + 30² + 4² + 7² + 8² + 18² + 8² + 5² + 6² + 9² + 6² = 1644.

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

1645

1645² = 2706025, 270 * 6 + 0 + 25 = 1645.

1645^k + 54285^k + 56165^k + 108805^k are squares for k = 1,2,3 (470², 133950², 40314250²).
141^k + 1645^k + 2773^k + 4277^k are squares for k = 1,2,3 (94², 5358², 322514²).

Komachi equation: 1645² = 987² * 6² / 54² * 3² / 2² * 10².

Page of Squares : First Upload February 13, 2007 ; Last Revised April 28, 2011
by Yoshio Mimura, Kobe, Japan

1646

1646² = 2709316, a square with different digits.

1646²± 3 are primes.

1646² = 36³ + 109³ + 111³.

1646⁵ = 12082287187210976 : 1² + 2² + 0² + 8² + 22² + 8² + 7² + 18² + 7² + 21² + 0² + 9² + 7² + 6² = 1646.

Page of Squares : First Upload February 13, 2007 ; Last Revised January 16, 2014
by Yoshio Mimura, Kobe, Japan

1648

1648² = 2715904, a square with different digits.

1648² = 8⁴ + 28⁴ + 32⁴ + 32⁴.

Page of Squares : First Upload February 13, 2007 ; Last Revised July 10, 2008
by Yoshio Mimura, Kobe, Japan

1649

1649² = 10⁴ + 20⁴ + 32⁴ + 35⁴ = 15⁴ + 30⁴ + 30⁴ + 32⁴.

85^k + 1649^k + 4097^k + 4573^k are squares for k = 1,2,3 (102², 6358², 410958²).
336396^k + 463369^k + 728858^k + 1190578^k are squares for k = 1,2,3 (1649², 1508835², 1487402947²).
474912^k + 534276^k + 745348^k + 964665^k are squares for k = 1,2,3 (1649², 1413193², 1253551661²).

1649⁵ = 12192795175283249 :
12² + 1² + 9² + 2² + 7² + 9² + 5² + 1² + 7² + 5² + 2² + 8² + 32² + 4² + 9² = 1649.

Page of Squares : First Upload July 10, 2008 ; Last Revised April 28, 2011
by Yoshio Mimura, Kobe, Japan

1650

The square root of 1650 is 40.62..., and 40 = 6² + 2².

The integral triangle of sides 521, 12104, 12375 has square area 1650².

1650² = (1)(2 + 3)(4)(5 + 6)(7 + 8)(9 + 10 + 11 + ... + 41),
1650² = (1)(2 + 3)(4 + 5)(6 + 7 + 8 + ... + 49)(50),
1650² = (1)(2 + 3)(4 + 5 + 6 + 7)(8 + 9 + 10 + ... + 17)(18 + 19 + 20 + ... + 26),
1650² = (1)(2 + 3 + 4 + ... + 9)(10 + 11 + 12)(13 + 14 + 15 + ... + 62),
1650² = (1)(2 + 3 + 4)(5)(6 + 7 + ... + 49)(50),
1650² = (1 + 2 + 3 + 4)(5)(6 + 7 + 8 + ... + 27)(28 + 29 + 30 + 31 + 32),
1650² = (1 + 2 + 3 + 4)(5 + 6 + 7 + ... + 10)(11 + 12 + 13 + ... + 110),
1650² = (1 + 2 + 3 + ... + 9)(10)(11 + 12 + 13 + ... + 110).

1650² = (2² + 6)(3² + 6)(7² + 6)(18² + 6) = (3² + 6)(4² + 6)(7² + 6)(12² + 6)
= (7² + 6)(12² + 6)(18² + 6).

Page of Squares : First Upload February 13, 2007 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1652

1652²± 3 are primes.

Page of Squares : First Upload January 16, 2014 ; Last Revised January 16, 2014
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1653

1653² = 1³ + 2³ + 3³ + 4³ + 5³ + ... + 57³.

281010^k + 608304^k + 629793^k + 1213302^k are squares for k = 1,2,3 (1653², 1522413², 1511022177²).

Page of Squares : First Upload February 13, 2007 ; Last Revised April 28, 2011
by Yoshio Mimura, Kobe, Japan

1656

1656² = 42³ + 64³ + 134³.

1656⁴ = 7520406736896, and 7² + 5² + 2² + 0² + 4² + 0² + 6² + 7² + 36² + 8² + 9² + 6² = 1656.

Page of Squares : First Upload July 10, 2008 ; Last Revised December 1, 2008
by Yoshio Mimura, Kobe, Japan

1659

1659² = 2752281, 2 + 75 * 22 + 8 - 1 = 1659.

Page of Squares : First Upload February 13, 2007 ; Last Revised February 13, 2007
by Yoshio Mimura, Kobe, Japan

1662

1662² = 2762244, 27 * 62 - 2 * 4 - 4 = 1662.

Page of Squares : First Upload February 13, 2007 ; Last Revised February 13, 2007
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1663

1663² = 2765569, 2² + 7² + 6² + 5² + 5² + 6² + 9² = 16².

Page of Squares : First Upload February 13, 2007 ; Last Revised February 13, 2007
by Yoshio Mimura, Kobe, Japan

1664

1664² = 6³ + 52³ + 138³ = 10³ + 104³ + 118³.

Komachi equation: 1664² = 1² * 234² * 56² / 7² * 8² / 9².

Page of Squares : First Upload July 10, 2008 ; Last Revised September 7, 2010
by Yoshio Mimura, Kobe, Japan

1665

1665² = 2772225, a square with just 3 kinds of digits.

Page of Squares : First Upload February 13, 2007 ; Last Revised February 13, 2007
by Yoshio Mimura, Kobe, Japan

1666

1666² = 2775556, 2 + 7 + 7 + 5 * 55 * 6 = 2 + 7 + 7 + 55 * 5 * 6 = 1666.

1666² = 63³ + 69³ + 130³.

1666^k + 2210^k + 5270^k + 9350^k are squares for k = 1,2,3 (136², 11084², 989536²).

Page of Squares : First Upload February 13, 2007 ; Last Revised April 28, 2011
by Yoshio Mimura, Kobe, Japan

1667

1667² = 2778889, the digits of which are non-decreasing.

1667² = 2778889 appears in the decimal expressions of π:
π = 3.14159•••2778889••• (from the 47042nd digit)

Page of Squares : First Upload February 13, 2007 ; Last Revised November 4, 2008
by Yoshio Mimura, Kobe, Japan

1668

1668² = 2782224, 278 / 2 / 2 * 24 = 1668.

1668² = 16⁴ + 24⁴ + 32⁴ + 34⁴.

138^k + 570^k + 1086^k + 1122^k are squares for k = 1,2,3 (54², 1668², 53676²).
210^k + 606^k + 762^k + 1338^k are squares for k = 1,2,3 (54², 1668², 55404²).

Page of Squares : First Upload February 13, 2007 ; Last Revised April 28, 2011
by Yoshio Mimura, Kobe, Japan

1669

1669² = 2785561, 278 * 5 / 5 * 6 + 1 = 278 / 5 * 5 * 6 + 1 = 1669.

Page of Squares : First Upload February 13, 2007 ; Last Revised February 13, 2007
by Yoshio Mimura, Kobe, Japan

1671

(1671² + 9) = (4² + 9)(5² + 9)(6² + 9)(8² + 9) = (1² + 9)(4² + 9)(8² + 9)(12² + 9).

1671² = 2792241 appears in the decimal expressions of π:
π = 3.14159•••2792241••• (from the 16310th digit)
(2792241 is the third 7-digit square in the expression of π.)

Page of Squares : First Upload February 13, 2007 ; Last Revised November 4, 2008
by Yoshio Mimura, Kobe, Japan

1673

1673² = 25³ + 34³ + 140³ = 1⁵ + 3⁵ + 13⁵ + 14⁵ + 18⁵.

The sum of the squares of divisors of 1673 is a square, 1690².

Page of Squares : First Upload July 10, 2008 ; Last Revised November 1, 2011
by Yoshio Mimura, Kobe, Japan

1674

1674² = 2802276, 2 + 8 * 0 + 22 * 76 = 2 - 8 * 0 + 22 * 76 = 1674 .

Page of Squares : First Upload February 13, 2007 ; Last Revised February 13, 2007
by Yoshio Mimura, Kobe, Japan

1679

1679² = 2819041, a zigzag square.

1679² = 2819041, 2 * 819 + 0 + 41 = 1679.

1679^2 = 83^3 + 93^3 + 113^3.

Page of Squares : First Upload February 13, 2007 ; Last Revised July 10, 2008
by Yoshio Mimura, Kobe, Japan

1680

1680² = 10³ + 63³ + 137³.

1680, 1681, 1682, 1683 and 1684 are five consecutive integers having square factors (the second case).

The integral triangle of sides 1521, 3712, 3983 (or 1904, 4285, 5811) has square area 1680².

Komachi equations:
1680² = 12² * 3² / 4² + 5² * 6² * 7² * 8² - 9² = 12² * 3² / 4² * 5² * 6² * 7² * 8² / 9²
= 12² * 3² * 45² / 6² * 7² * 8² / 9² = - 12² * 3² / 4² + 5² * 6² * 7² * 8² + 9².

A quartic polynomial: (See 420)
(X + 420)(X + 525)(X + 1344)(X + 1680) = X⁴ + 63²X³ + 2310²X² + 52920²X + 705600².

1680² = (17² - 1)(99² - 1) = (2² - 1)(3² - 1)(4² - 1)(6² - 1)(15² - 1)
= (2² - 1)(6² - 1)(11² - 1)(15² - 1) = (4² - 1)(15² - 1)(29² - 1)
= (4² - 1)(5² - 1)(6² - 1)(15² - 1) = (6² - 1)(15² - 1)(19² - 1) = (6² - 1)(7² - 1)(41² - 1).

1680² = (1)(2)(3)(4)(5 + 6 + 7 + ... + 11)(12)(13 + 14 + 15 + ... + 22),
1680² = (1)(2)(3 + 4)(5)(6)(7 + 8 + 9)(10 + 11 + 12 + ... + 25),
1680² = (1)(2)(3 + 4)(5 + 6 + 7 + ... + 11)(12)(13 + 14 + 15 + ... + 27),
1680² = (1)(2 + 3)(4)(5)(6)(7)(8)(9 + 10 + 11 + ... + 15),
1680² = (1)(2 + 3)(4)(5)(6 + 7 + 8 + ... + 26)(27 + 28 + 29),
1680² = (1)(2 + 3)(4)(5)(6 + 7 + 8)(9 + 10 + 11 + ... + 15)(16),
1680² = (1)(2 + 3 + 4 + 5)(6 + 7 + ... + 10)(11 + 12 + 13)(14 + 15 + ... + 21),
1680² = (1)(2 + 3 + 5 + 6)(7)(8)(9)(10 + 11 + 12 + ... + 25),
1680² = (1)(2 + 3 + 4 + 5 + 6)(7)(8)(9 + 10 + 11 + ... + 71),
1680² = (1)(2 + 3 + 4 + 5 + 6)(7 + 8 + 9 + ... + 21)(22 + 23 + 24 + ... + 42),
1680² = (1 + 2)(3 + 4 + 5 + 6 + 7)(8)(9 + 10 + 11 + 12)(13 + 14 + 15 + ... + 19),
1680² = (1 + 2 + 4)(5 + 6 + 7 + ... + 11)(12 + 13 + 14 + ... + 23)(24),
1680² = (1 + 2 + 3 + ... + 6)(7 + 8 + 9 + ... + 518),
1680² = (1 + 2 + 3 + ... + 7)(8)(9 + 10 + 11 + ... + 12)(13 + 14 + 15 + ... + 27),
1680² = (1 + 2 + 3 + ... + 7)(8 + 9 + 10 + ... + 27)(28 + 29 + 30 + ... + 36),
1680² = (1 + 2 + 3)(4)(5 + 6 + 7 + ... + 11)(12)(13 + 14 + 15 + ... + 22).

Page of Squares : First Upload February 13, 2007 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1681

The square of 41.

1681 = 41², 16 = 4², 81 = 9².

1 + 2 + 3 + 4 + 5 + 6 + ... + 1681 = 1189².

1681² - 1680² - 1679² + 1678² - 1677² + ... + 1² = 1189².

A cubic polynomial :
(X + 576²)(X + 744²)(X + 1393²) = X³ + 1681²X² + 1378968²X + 596961792².

1681² = 2825761, 2 * 8 / 2 * 5 * 7 * 6 + 1 = 1681.

Loop of length 56 by the function f(N) = ... + c² + b² + a² where N = ... + 100²c + 100b + a:
1681 - 6817 - 4913 - 2570 - ... - 1809 - 405 - 41 - 1681
(Note f(1681) = 16² + 81² = 6817, f(6817) = 68² + 17² = 4913, etc. See 41)

1681² = 2825761 appears in the decimal expressions of e:
e = 2.71828•••2825761••• (from the 37236th digit)
(2825761 is the tenth 7-digit square in the expression of e.)

Page of Squares : First Upload February 13, 2007 ; Last Revised November 4, 2008
by Yoshio Mimura, Kobe, Japan

1682

1682² = 2829124, 2 * 829 + 1 * 24 = 2 * 829 * 1 + 24 = 1682.

1682² = 29⁴ + 29⁴ + 29⁴ + 29⁴.

Page of Squares : First Upload February 13, 2007 ; Last Revised July 10, 2008
by Yoshio Mimura, Kobe, Japan

1683

1683² = (1)(2 + 3 + 4 + 5 + 6 + 7)(8 + 9)(10 + 11 + 12 + ... + 111).

1683² = (1² + 2)(3² + 2)(7² + 2)(41² + 2) = (3² + 8)(5² + 8)(71² + 8).

51^k + 1683^k + 6069^k + 10693^k are squares for k = 1,2,3 (136², 12410², 1204552²).
18^k + 282^k + 921^k + 1380^k are squares for k = 1,2,3 (51², 1683², 58581²).

Page of Squares : First Upload February 13, 2007 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1684

A cubic polynomial :
(X + 336²)(X + 1684²)(X + 10773²) = X³ + 10909²X² + 18507972²X + 6095621952².

Page of Squares : First Upload February 13, 2007 ; Last Revised February 13, 2007
by Yoshio Mimura, Kobe, Japan

1686

1686² = 2842596, 2 * 842 + 5 - 9 + 6 = 1686.

Page of Squares : First Upload February 13, 2007 ; Last Revised February 13, 2007
by Yoshio Mimura, Kobe, Japan

1687

452116^k + 705166^k + 767585^k + 921102^k are squares for k = 1,2,3 (1687², 1462629², 1294915895²).

Page of Squares : First Upload April 28, 2011 ; Last Revised April 28, 2011
by Yoshio Mimura, Kobe, Japan

1688

(1688² - 8) = (5² - 8)(7² - 8)(8² - 8)(9² - 8) = (3² - 8)(5² - 8)(7² - 8)(8² - 8)(9² - 8).

Page of Squares : First Upload February 13, 2007 ; Last Revised February 13, 2007
by Yoshio Mimura, Kobe, Japan

1689

1689² = 2852721, 2 * 852 - 7 * 2 - 1 = 1689.

Page of Squares : First Upload February 13, 2007 ; Last Revised February 13, 2007
by Yoshio Mimura, Kobe, Japan

1690

1690² = (1)(2 + 3 + 4 + 5 + 6)(7 + 8 + 9 + ... + 19)(20 + 21 + 22 + ... + 45).

1690²= 36 x 37 + 37 x 38 + 38 x 39 + 39 x 40 + 40 x 41 + ... + 204 x 205.

1690² = (2² + 1)(3² + 1)(239² + 1) = (7² + 1)(239² + 1) = (3² + 4)(16² + 4)(29² + 4)
= (2² + 9)(11² + 9)(41² + 9).

1010^k + 1690^k + 2570^k + 2830^k are squares for k = 1,2,3 (90², 4300², 213300²).

Komachi equation: 1690² = 1² * 234² * 5² / 6² * 78² / 9².

Page of Squares : First Upload February 13, 2007 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan

1691

1691² = 2859481, a zigzag square.

Page of Squares : First Upload February 13, 2007 ; Last Revised February 13, 2007
by Yoshio Mimura, Kobe, Japan

1692

1692² = 2862864, a square with even digits.

1692² = 2862864, 2 * 862 - 8 - 6 * 4 = 28 * 62 - 8 * 6 + 4 = 1692.

34^k + 82^k + 98^k + 110^k are squares for k = 1,2,3 (18², 172², 1692²).
146^k + 154^k + 654^k + 1546^k are squares for k = 1,2,3 (50², 1692², 63100²).

Komachi equation: 1692² = 987² * 6² * 5² * 4² * 3² / 210².

Page of Squares : First Upload February 13, 2007 ; Last Revised April 28, 2011
by Yoshio Mimura, Kobe, Japan

1694

1694² = (1)(2 + 3 + 4 + 5)(6 + 7 + 8 + ... + 16)(17 + 18 + 19 + ... + 60).

Page of Squares : First Upload February 13, 2007 ; Last Revised February 13, 2007
by Yoshio Mimura, Kobe, Japan

1698

436386^k + 524682^k + 660522^k + 1261614^k are squares for k = 1,2,3 (1698², 1579140², 1588645404²).

Page of Squares : First Upload April 28, 2011 ; Last Revised April 28, 2011
by Yoshio Mimura, Kobe, Japan