## 1600

The square of 40.

1600^{2} = 02560000, 5600^{2} = 31360000, 3600^{2} = 12960000, 9600^{2} = 92160000,

Other examples: 2916 - 5030 - 3009 - 0540 - 2916, and 2100 - 4100 - 8100 - 6100 - 2100.

by Yoshio Mimura, Kobe, Japan

## 1602

S_{2}(1602) = S_{2}(742) + S_{2}(1547), where S_{2}(n) = 1^{2} + 2^{2} + 3^{2} + ... + n^{2}.

1602_{2} = 3^{4} + 21^{4} + 33^{4} + 33^{4}.

1602_{2} = (1^{2} + 2)(4^{2} + 2)(218^{2} + 2).

Komachi equation: 1602^{2} = 12^{2} * 3^{2} * 4^{2} / 56^{2} * 7^{2} * 89^{2}.

by Yoshio Mimura, Kobe, Japan

## 1603

1 / 1603 = 0.000623830318, 6^{2} + 23^{2} + 8^{2} + 30^{2} + 3^{2} + 1^{2} + 8^{2} = 1603.

by Yoshio Mimura, Kobe, Japan

## 1604

1604^{2} = 2572816, 2 + 57 * 28 + 1 * 6 = 2 + 57 * 28 * 1 + 6 = 1604.

by Yoshio Mimura, Kobe, Japan

## 1605

The square root of 1605 is 40.062..., and 40 = 0^{2} + 6^{2} + 2^{2}.

by Yoshio Mimura, Kobe, Japan

## 1607

1607^{2} = 9^{3} + 56^{3} + 134^{3}.

by Yoshio Mimura, Kobe, Japan

## 1608

1608^{2} = 48^{3} + 61^{3} + 131^{3}.

by Yoshio Mimura, Kobe, Japan

## 1611

1611^{2} = 2595321, 2 - 5 + 9 + 5 * 321 = 1611.

1611^{2}± 2 are primes.

1611_{2} = 32_{3} + 105_{3} + 112_{3}.

by Yoshio Mimura, Kobe, Japan

## 1612

1612^{2} = (7^{2} + 3)(11^{2} + 3)(20^{2} + 3).

1612^{2} = 30^{2} + 31^{2} + 32^{2} + 33^{2} + 34^{2} + 35^{2} + 36^{2} + ... + 198^{2}.

A cubic polynomial:

(X + 1440^{2})(X + 1612^{2})(X + 5859^{2}) = X^{3} + 6245^{2}X^{2} + 12875292^{2}X + 13600379520^{2}.

by Yoshio Mimura, Kobe, Japan

## 1614

786^{k} + 1614^{k} + 3882^{k} + 4122^{k} are squares for k = 1,2,3 (102^{2}, 5940^{2}, 365004^{2}).

48^{k} + 177^{k} + 186^{k} + 1614^{k} are squares for k = 1,2,3 (45^{2}, 1635^{2}, 64935^{2}).

by Yoshio Mimura, Kobe, Japan

## 1615

293930^{k} + 529720^{k} + 687990^{k} + 1096585^{k} are squares for k = 1,2,3 (1615^{2}, 1429275^{2}, 1348452325^{2}).

by Yoshio Mimura, Kobe, Japan

## 1616

1 / 1616 = 0.000618811881188118811, 618^{2} + 81^{2} + 1^{2} + 881^{2} + 188^{2} + 1188^{2} + 11^{2} = 1616^{2},

1 / 1616 = 0.000618811881188118811, 618^{2} + 81^{2} + 188^{2} + 1188^{2} + 11^{2} + 881^{2} + 1^{2} = 1616^{2}.

by Yoshio Mimura, Kobe, Japan

## 1617

1617^{2}± 2 are primes.

1617^{2} = 50^{3} + 70^{3} + 129^{3}.

1617^{k} + 8106^{k} + 10290^{k} + 15708^{k} are squares for k = 1,2,3 (189^{2}, 20517^{2}, 2345679^{2}).

294^{k} + 1288^{k} + 1617^{k} + 2730^{k} are squares for k = 1,2,3 (77^{2}, 3437^{2}, 163513^{2}).

924^{k} + 1078^{k} + 1617^{k} + 2310^{k} are squares for k = 1,2,3 (77^{2}, 3157^{2}, 136367^{2}).

by Yoshio Mimura, Kobe, Japan

## 1618

1618^{2} = 23^{4} + 29^{4} + 29^{4} + 31^{4}.

by Yoshio Mimura, Kobe, Japan

## 1619

1619^{2} = 2621161, a square with just 3 kinds of digits 1, 2 and 6.

by Yoshio Mimura, Kobe, Japan

## 1620

A cubic polynomial :

(X + 1620^{2})(X + 2079^{2})(X + 4400^{2}) = X^{3} + 5129^{2}X^{2} + 12076020^{2}X + 14819112000^{2}.

1620^{2} = (1)(2)(3)(4)(5 + 6 + 7 + ... + 10)(11 + 12 + 13 + ... + 70),

1620^{2} = (1)(2)(3)(4)(5 + 6 + 7 + ... + 22)(23 + 24 + 25 + ... + 37),

1620^{2} = (1)(2)(3)(4 + 5)(6)(7 + 8)(9)(10 + 11 + 12 + 13 + 14),

1620^{2} = (1)(2 + 3)(4)(5 + 6 + 7 + ... + 13)(14 + 15 + 16 + ... + 58),

1620^{2} = (1)(2 + 3)(4)(5 + 6 + 7)(8 + 9 + 10)(11 + 12 + 13 + ... + 25),

1620^{2} = (1)(2 + 3)(4 + 5)(6 + 7 + 8 + 9)(10 + 11 + 12 + ... + 17)(18),

1620^{2} = (1)(2 + 3)(4 + 5 + 6 + 7 + 8)(9)(10 + 11 + 12 + ... + 17)(18),

1620^{2} = (1)(2 + 3 + 4 + ... + 7)(8)(9)(10)(11 + 12 + 13 + ... + 19),

1620^{2} = (1)(2 + 3 + 4 + ... + 7)(8 + 9 + 10)(11 + 12 + 13 + ... + 85),

1620^{2} = (1)(2 + 3 + 4)(5)(6 + 7 + ... + 9)(10 + 11 + 12 + ... + 17)(18),

1620^{2} = (1)(2 + 3 + 4)(5 + 6 + 7)(8 + 9 + 10 + ... + 22)(23 + 24 + 25),

1620^{2} = (1 + 2)(3)(4)(5)(6)(7 + 8 + 9 + 10 + 11)(12 + 13 + 14 + 15),

1620^{2} = (1 + 2)(3)(4)(5 + 6 + 7 + ... + 10)(11 + 12 + 13)(14 + 15 + 16),

1620^{2} = (1 + 2)(3)(4)(5 + 6 + 7 + ... + 19)(20 + 21 + 22 + ... + 34),

1620^{2} = (1 + 2)(3)(4 + 5)(6)(7 + 8)(9 + 10 + 11)(12),

1620^{2} = (1 + 2)(3)(4 + 5)(6 + 7 + 8 + ... + 14)(15 + 16 + 17 + ... + 30),

1620^{2} = (1 + 2)(3)(4 + 5 + 6 + ... + 11)(12)(13 + 14)(15),

1620^{2} = (1 + 2)(3)(4 + 5 + 6 + ... + 11)(12 + 13 + 14 + 15)(16 + 17 + 18 + 19 + 20),

1620^{2} = (1 + 2)(3)(4 + 5 + 6 + 7 + 8)(9 + 10 + 11)(12)(13 + 14),

1620^{2} = (1 + 2)(3)(4 + 5 + 6)(7 + 8 + 9)(10)(11 + 12 + 13 + ... + 16),

1620^{2} = (1 + 2)(3 + 4 + 5 + 6)(7 + 8)(9)(10)(11 + 12 + 13),

1620^{2} = (1 + 2)(3 + 4 + 5 + 6 + 7)(8 + 9 + 10 + ... + 19)(20 + 21 + 22 + ... + 28),

1620^{2} = (1 + 2 + 3 + 4)(5 + 6 + 7 + ... + 10)(11 + 12 + 13)(14 + 15 + 16 + ... + 22),

1620^{2} = (1 + 2 + 3 + 4)(5 + 6 + 7 + ... + 724),

1620^{2} = (1 + 2 + 3 + 4)(5 + 6 + 7)(8 + 9 + 10)(11 + 12 + 13 + ... + 34),

1620^{2} = (1 + 2 + 3 + 4 + 5)(6 + 7 + 8 + 9 + 10)(11 + 12 + 13 + ... + 16)(17 + 18 + 19),

1620^{2} = (1 + 2 + 3 + ... + 9)(10)(11 + 12 + 13)(14 + 15 + 16 + ... + 22),

1620^{2} = (1 + 2 + 3)(4)(5 + 6 + 7 + ... + 10)(11 + 12 + 13 + ... + 70),

1620^{2} = (1 + 2 + 3)(4)(5 + 6 + 7 + ... + 22)(23 + 24 + 25 + ... + 37),

1620^{2} = (1 + 2 + 3)(4 + 5)(6)(7 + 8)(9)(10 + 11 + 12 + 13 + 14).

by Yoshio Mimura, Kobe, Japan

## 1621

The square root of 1621 is 40.26..., and 40 = 2^{2} + 6^{2}.

1621^{2} = 2627641 appears in the decimal expressions of e:

e = 2.71828•••2627641••• (from the 48086th digit)

by Yoshio Mimura, Kobe, Japan

## 1623

1623^{2} = 6^{3} + 106^{3} + 113^{3}.

by Yoshio Mimura, Kobe, Japan

## 1624

1624^{2} = 2637376, a zigzag square.

1624^{2}± 3 are primes.

1624^{2} = (7^{2} + 7)(14^{2} + 7)(15^{2} + 7) = (7^{2} + 7)(217^{2} + 7).

S_{2}(1624) = S_{2}(9) * S_{2}(12) * S_{2}(28), where S_{2}(n) = 1^{2} + 2^{2} + 3^{2} + ... + n^{2}.

1624_{2} = 50_{3} + 80_{3} + 126_{3}.

by Yoshio Mimura, Kobe, Japan

## 1625

1625^{2} = 2640625, 2^{2} + 6^{2} + 4^{2} + 0^{2} + 6^{2} + 2^{2} + 5^{2} = 11^{2}.

1625_{2} = 41_{3} + 78_{3} + 128_{3}.

by Yoshio Mimura, Kobe, Japan

## 1626

1626^{2} = 2643876, 26 + 4 + 38 * 7 * 6 = 1626.

by Yoshio Mimura, Kobe, Japan

## 1628

1628^{2} = 2650384, a square with different digits.

1628^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 1632

1632_{2} = 92_{3} + 96_{3} + 100_{3}.

1632^{2} = 2663424, 2^{2} + 6^{2} + 6^{2} + 3^{2} + 4^{2} + 2^{2} + 4^{2} = 11^{2}.

1632^{2} = 2663424, 2 * 6 / 6 * 34 * 24 = 2 / 6 * 6 * 34 * 24 = 1632,

1632^{2} = 2663424, 26 * 63 - 4 + 2 - 4 = 26 * 63 - 4 / 2 - 4 = 1632.

1632^{2} = (3^{2} - 1)(577^{2} - 1).

by Yoshio Mimura, Kobe, Japan

## 1633

1633^{2} = 2666689, a square the digits of which are non-decreasing.

1633^{2} = 2666689, 26 * 66 + 6 - 89 = 1633.

1633^{2} = 20^{3} + 80^{3} + 129^{3}.

by Yoshio Mimura, Kobe, Japan

## 1634

1634^{2} = 2669956 appears in the decimal expressions of π:

π = 3.14159•••2669956••• (from the 80656st digit)

by Yoshio Mimura, Kobe, Japan

## 1635

1635^{2} + 1636^{2} + 1637^{2} + ... + 2470^{2} = 2471^{2} + 2472^{2} + 2473^{2} + ... + 2954^{2}.

48^{k} + 177^{k} + 186^{k} + 1614^{k} are squares for k = 1,2,3 (45^{2}, 1635^{2}, 64935^{2}).

by Yoshio Mimura, Kobe, Japan

## 1638

The integral triangle of sides 1869, 2873, 3370 has square area 1638^{2}.

Komachi eqaution: 1638^{2} = 1^{2} / 2^{2} / 3^{2} / 4^{2} * 56^{2} * 78^{2} * 9^{2}.

1638^{2} = (74 + 75 + 76)^{2} + (77 + 78 + 79)^{2} + (80 + 81 + 82)^{2} + ... + (143 + 144 + 145)^{2}.

1638^{2} = (1)(2)(3)(4 + 5)(6 + 7)(8 + 9 + ... + 20)(21),

1638^{2} = (1)(2)(3 + 4)(5 + 6 + 7 + 8)(9)(10 + 11)(12 + 13 + 14),

1638^{2} = (1)(2 + 3 + 4 + 5)(6 + 7)(8 + 9 + 10 + ... + 13)(14 + 15 + 16 + ... + 25),

1638^{2} = (1)(2 + 3 + 4 + 5)(6 + 7 + 8 + ... + 12)(13)(14 + 15 + 16 + ... + 25),

1638^{2} = (1)(2 + 3 + 4 + 5)(6 + 7 + 8 + ... + 57)(58 + 59),

1638^{2} = (1)(2 + 3 + 4 + 5)(6 + 7 + 8)(9 + 10 + 11 + ... + 17)(18 + 19 + 20 + 21),

1638^{2} = (1)(2 + 3 + 4)(5 + 6 + 7 + 8)(9 + 10 + 11 + 12)(13 + 14 + 15 + ... + 26),

1638^{2} = (1 + 2)(3 + 4)(5 + 6 + 7 + 8)(9 + 10 + 11 + ... + 99),

1638^{2} = (1 + 2)(3 + 4 + 5 + ... + 15)(16 + 17 + 18 + ... + 23)(24 + 25),

1638^{2} = (1 + 2 + 3 + ... + 13)(14)(15 + 16 + ... + 66),

1638^{2} = (1 + 2 + 3 + ... + 13)(14 + 15 + ... + 22)(23 + 24 + 25 + ... + 29),

1638^{2} = (1 + 2 + 3 + ... + 6)(7 + 8 + 9 + ... + 19)(20 + 21 + 22 + ... + 43),

1638^{2} = (1 + 2 + 3 + ... + 6)(7 + 8 + 9 + ... + 32)(33 + 34 + 35 + ... + 39),

1638^{2} = (1 + 2 + 3)(4 + 5)(6 + 7)(8 + 9 + 10 + ... + 20)(21).

by Yoshio Mimura, Kobe, Japan

## 1639

1639^{2} = 2686321, 268 * 6 + 32 - 1 = 1639 .

by Yoshio Mimura, Kobe, Japan

## 1640

1640^{2} = 2689600 appears in the decimal expressions of π:

π = 3.14159•••2689600••• (from the 80129th digit)

by Yoshio Mimura, Kobe, Japan

## 1642

1642^{2} = 2696164, a square pegged by 6.

by Yoshio Mimura, Kobe, Japan

## 1643

S_{2}(1643) = S_{2}(1239) + S_{2}(1363).

1643^{2} = 62^{3} + 68^{3} + 129^{3}.

by Yoshio Mimura, Kobe, Japan

## 1644

1644^{4} = 7304781885696, and 7^{2} + 30^{2} + 4^{2} + 7^{2} + 8^{2} + 18^{2} + 8^{2} + 5^{2} + 6^{2} + 9^{2} + 6^{2} = 1644.

by Yoshio Mimura, Kobe, Japan

## 1645

1645^{2} = 2706025, 270 * 6 + 0 + 25 = 1645.

1645^{k} + 54285^{k} + 56165^{k} + 108805^{k} are squares for k = 1,2,3 (470^{2}, 133950^{2}, 40314250^{2}).

141^{k} + 1645^{k} + 2773^{k} + 4277^{k} are squares for k = 1,2,3 (94^{2}, 5358^{2}, 322514^{2}).

Komachi equation: 1645^{2} = 987^{2} * 6^{2} / 54^{2} * 3^{2} / 2^{2} * 10^{2}.

by Yoshio Mimura, Kobe, Japan

## 1646

1646^{2} = 2709316, a square with different digits.

1646^{2}± 3 are primes.

1646^{2} = 36^{3} + 109^{3} + 111^{3}.

1646^{5} = 12082287187210976 : 1^{2} + 2^{2} + 0^{2} + 8^{2} + 22^{2} + 8^{2} + 7^{2} + 18^{2} + 7^{2} + 21^{2} + 0^{2} + 9^{2} + 7^{2} + 6^{2} = 1646.

by Yoshio Mimura, Kobe, Japan

## 1648

1648^{2} = 2715904, a square with different digits.

1648^{2} = 8^{4} + 28^{4} + 32^{4} + 32^{4}.

by Yoshio Mimura, Kobe, Japan

## 1649

1649^{2} = 10^{4} + 20^{4} + 32^{4} + 35^{4} = 15^{4} + 30^{4} + 30^{4} + 32^{4}.

85^{k} + 1649^{k} + 4097^{k} + 4573^{k} are squares for k = 1,2,3 (102^{2}, 6358^{2}, 410958^{2}).

336396^{k} + 463369^{k} + 728858^{k} + 1190578^{k} are squares for k = 1,2,3 (1649^{2}, 1508835^{2}, 1487402947^{2}).

474912^{k} + 534276^{k} + 745348^{k} + 964665^{k} are squares for k = 1,2,3 (1649^{2}, 1413193^{2}, 1253551661^{2}).

1649^{5} = 12192795175283249 :

12^{2} + 1^{2} + 9^{2} + 2^{2} + 7^{2} + 9^{2} + 5^{2} + 1^{2} + 7^{2} + 5^{2} + 2^{2} + 8^{2} + 32^{2} + 4^{2} + 9^{2} = 1649.

by Yoshio Mimura, Kobe, Japan

## 1650

The square root of 1650 is 40.62..., and 40 = 6^{2} + 2^{2}.

The integral triangle of sides 521, 12104, 12375 has square area 1650^{2}.

1650^{2} = (1)(2 + 3)(4)(5 + 6)(7 + 8)(9 + 10 + 11 + ... + 41),

1650^{2} = (1)(2 + 3)(4 + 5)(6 + 7 + 8 + ... + 49)(50),

1650^{2} = (1)(2 + 3)(4 + 5 + 6 + 7)(8 + 9 + 10 + ... + 17)(18 + 19 + 20 + ... + 26),

1650^{2} = (1)(2 + 3 + 4 + ... + 9)(10 + 11 + 12)(13 + 14 + 15 + ... + 62),

1650^{2} = (1)(2 + 3 + 4)(5)(6 + 7 + ... + 49)(50),

1650^{2} = (1 + 2 + 3 + 4)(5)(6 + 7 + 8 + ... + 27)(28 + 29 + 30 + 31 + 32),

1650^{2} = (1 + 2 + 3 + 4)(5 + 6 + 7 + ... + 10)(11 + 12 + 13 + ... + 110),

1650^{2} = (1 + 2 + 3 + ... + 9)(10)(11 + 12 + 13 + ... + 110).

1650^{2} = (2^{2} + 6)(3^{2} + 6)(7^{2} + 6)(18^{2} + 6) = (3^{2} + 6)(4^{2} + 6)(7^{2} + 6)(12^{2} + 6)

= (7^{2} + 6)(12^{2} + 6)(18^{2} + 6).

by Yoshio Mimura, Kobe, Japan

## 1652

1652^{2}± 3 are primes.

by Yoshio Mimura, Kobe, Japan

## 1653

1653^{2} = 1^{3} + 2^{3} + 3^{3} + 4^{3} + 5^{3} + ... + 57^{3}.

281010^{k} + 608304^{k} + 629793^{k} + 1213302^{k} are squares for k = 1,2,3 (1653^{2}, 1522413^{2}, 1511022177^{2}).

by Yoshio Mimura, Kobe, Japan

## 1656

1656^{2} = 42^{3} + 64^{3} + 134^{3}.

1656^{4} = 7520406736896, and 7^{2} + 5^{2} + 2^{2} + 0^{2} + 4^{2} + 0^{2} + 6^{2} + 7^{2} + 36^{2} + 8^{2} + 9^{2} + 6^{2} = 1656.

by Yoshio Mimura, Kobe, Japan

## 1659

1659^{2} = 2752281, 2 + 75 * 22 + 8 - 1 = 1659.

by Yoshio Mimura, Kobe, Japan

## 1662

1662^{2} = 2762244, 27 * 62 - 2 * 4 - 4 = 1662.

by Yoshio Mimura, Kobe, Japan

## 1663

1663^{2} = 2765569, 2^{2} + 7^{2} + 6^{2} + 5^{2} + 5^{2} + 6^{2} + 9^{2} = 16^{2}.

by Yoshio Mimura, Kobe, Japan

## 1664

1664^{2} = 6^{3} + 52^{3} + 138^{3} = 10^{3} + 104^{3} + 118^{3}.

Komachi equation: 1664^{2} = 1^{2} * 234^{2} * 56^{2} / 7^{2} * 8^{2} / 9^{2}.

by Yoshio Mimura, Kobe, Japan

## 1665

1665^{2} = 2772225, a square with just 3 kinds of digits.

by Yoshio Mimura, Kobe, Japan

## 1666

1666^{2} = 2775556, 2 + 7 + 7 + 5 * 55 * 6 = 2 + 7 + 7 + 55 * 5 * 6 = 1666.

1666^{2} = 63^{3} + 69^{3} + 130^{3}.

1666^{k} + 2210^{k} + 5270^{k} + 9350^{k} are squares for k = 1,2,3 (136^{2}, 11084^{2}, 989536^{2}).

by Yoshio Mimura, Kobe, Japan

## 1667

1667^{2} = 2778889, the digits of which are non-decreasing.

1667^{2} = 2778889 appears in the decimal expressions of π:

π = 3.14159•••2778889••• (from the 47042nd digit)

by Yoshio Mimura, Kobe, Japan

## 1668

1668^{2} = 2782224, 278 / 2 / 2 * 24 = 1668.

1668^{2} = 16^{4} + 24^{4} + 32^{4} + 34^{4}.

138^{k} + 570^{k} + 1086^{k} + 1122^{k} are squares for k = 1,2,3 (54^{2}, 1668^{2}, 53676^{2}).

210^{k} + 606^{k} + 762^{k} + 1338^{k} are squares for k = 1,2,3 (54^{2}, 1668^{2}, 55404^{2}).

by Yoshio Mimura, Kobe, Japan

## 1669

1669^{2} = 2785561, 278 * 5 / 5 * 6 + 1 = 278 / 5 * 5 * 6 + 1 = 1669.

by Yoshio Mimura, Kobe, Japan

## 1671

(1671^{2} + 9) = (4^{2} + 9)(5^{2} + 9)(6^{2} + 9)(8^{2} + 9) = (1^{2} + 9)(4^{2} + 9)(8^{2} + 9)(12^{2} + 9).

1671^{2} = 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.)

by Yoshio Mimura, Kobe, Japan

## 1673

1673^{2} = 25^{3} + 34^{3} + 140^{3} = 1^{5} + 3^{5} + 13^{5} + 14^{5} + 18^{5}.

The sum of the squares of divisors of 1673 is a square, 1690^{2}.

by Yoshio Mimura, Kobe, Japan

## 1674

1674^{2} = 2802276, 2 + 8 * 0 + 22 * 76 = 2 - 8 * 0 + 22 * 76 = 1674 .

by Yoshio Mimura, Kobe, Japan

## 1679

1679^{2} = 2819041, a zigzag square.

1679^{2} = 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, 2008by Yoshio Mimura, Kobe, Japan

## 1680

1680^{2} = 10^{3} + 63^{3} + 137^{3}.

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 1680^{2}.

Komachi equations:

1680^{2} = 12^{2} * 3^{2} / 4^{2} + 5^{2} * 6^{2} * 7^{2} * 8^{2} - 9^{2} = 12^{2} * 3^{2} / 4^{2} * 5^{2} * 6^{2} * 7^{2} * 8^{2} / 9^{2}

= 12^{2} * 3^{2} * 45^{2} / 6^{2} * 7^{2} * 8^{2} / 9^{2} = - 12^{2} * 3^{2} / 4^{2} + 5^{2} * 6^{2} * 7^{2} * 8^{2} + 9^{2}.

A quartic polynomial: (See 420)

(X + 420)(X + 525)(X + 1344)(X + 1680) = X^{4} + 63^{2}X^{3} + 2310^{2}X^{2} + 52920^{2}X + 705600^{2}.

1680^{2} = (17^{2} - 1)(99^{2} - 1) = (2^{2} - 1)(3^{2} - 1)(4^{2} - 1)(6^{2} - 1)(15^{2} - 1)

= (2^{2} - 1)(6^{2} - 1)(11^{2} - 1)(15^{2} - 1) = (4^{2} - 1)(15^{2} - 1)(29^{2} - 1)

= (4^{2} - 1)(5^{2} - 1)(6^{2} - 1)(15^{2} - 1) = (6^{2} - 1)(15^{2} - 1)(19^{2} - 1) = (6^{2} - 1)(7^{2} - 1)(41^{2} - 1).

1680^{2} = (1)(2)(3)(4)(5 + 6 + 7 + ... + 11)(12)(13 + 14 + 15 + ... + 22),

1680^{2} = (1)(2)(3 + 4)(5)(6)(7 + 8 + 9)(10 + 11 + 12 + ... + 25),

1680^{2} = (1)(2)(3 + 4)(5 + 6 + 7 + ... + 11)(12)(13 + 14 + 15 + ... + 27),

1680^{2} = (1)(2 + 3)(4)(5)(6)(7)(8)(9 + 10 + 11 + ... + 15),

1680^{2} = (1)(2 + 3)(4)(5)(6 + 7 + 8 + ... + 26)(27 + 28 + 29),

1680^{2} = (1)(2 + 3)(4)(5)(6 + 7 + 8)(9 + 10 + 11 + ... + 15)(16),

1680^{2} = (1)(2 + 3 + 4 + 5)(6 + 7 + ... + 10)(11 + 12 + 13)(14 + 15 + ... + 21),

1680^{2} = (1)(2 + 3 + 5 + 6)(7)(8)(9)(10 + 11 + 12 + ... + 25),

1680^{2} = (1)(2 + 3 + 4 + 5 + 6)(7)(8)(9 + 10 + 11 + ... + 71),

1680^{2} = (1)(2 + 3 + 4 + 5 + 6)(7 + 8 + 9 + ... + 21)(22 + 23 + 24 + ... + 42),

1680^{2} = (1 + 2)(3 + 4 + 5 + 6 + 7)(8)(9 + 10 + 11 + 12)(13 + 14 + 15 + ... + 19),

1680^{2} = (1 + 2 + 4)(5 + 6 + 7 + ... + 11)(12 + 13 + 14 + ... + 23)(24),

1680^{2} = (1 + 2 + 3 + ... + 6)(7 + 8 + 9 + ... + 518),

1680^{2} = (1 + 2 + 3 + ... + 7)(8)(9 + 10 + 11 + ... + 12)(13 + 14 + 15 + ... + 27),

1680^{2} = (1 + 2 + 3 + ... + 7)(8 + 9 + 10 + ... + 27)(28 + 29 + 30 + ... + 36),

1680^{2} = (1 + 2 + 3)(4)(5 + 6 + 7 + ... + 11)(12)(13 + 14 + 15 + ... + 22).

by Yoshio Mimura, Kobe, Japan

## 1681

The square of 41.

1681 = 41^{2}, 16 = 4^{2}, 81 = 9^{2}.

1 + 2 + 3 + 4 + 5 + 6 + ... + 1681 = 1189^{2}.

1681^{2} - 1680^{2} - 1679^{2} + 1678^{2} - 1677^{2} + ... + 1^{2} = 1189^{2}.

A cubic polynomial :

(X + 576^{2})(X + 744^{2})(X + 1393^{2}) = X^{3} + 1681^{2}X^{2} + 1378968^{2}X + 596961792^{2}.

1681^{2} = 2825761, 2 * 8 / 2 * 5 * 7 * 6 + 1 = 1681.

Loop of length 56 by the function f(N) = ... + c^{2} + b^{2} + a^{2} where N = ... + 100^{2}c + 100b + a:

1681 - 6817 - 4913 - 2570 - ... - 1809 - 405 - 41 - 1681

(Note f(1681) = 16^{2} + 81^{2} = 6817, f(6817) = 68^{2} + 17^{2} = 4913, etc. See 41)

1681^{2} = 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.)

by Yoshio Mimura, Kobe, Japan

## 1682

1682^{2} = 2829124, 2 * 829 + 1 * 24 = 2 * 829 * 1 + 24 = 1682.

1682^{2} = 29^{4} + 29^{4} + 29^{4} + 29^{4}.

by Yoshio Mimura, Kobe, Japan

## 1683

1683^{2} = (1)(2 + 3 + 4 + 5 + 6 + 7)(8 + 9)(10 + 11 + 12 + ... + 111).

1683^{2} = (1^{2} + 2)(3^{2} + 2)(7^{2} + 2)(41^{2} + 2) = (3^{2} + 8)(5^{2} + 8)(71^{2} + 8).

51^{k} + 1683^{k} + 6069^{k} + 10693^{k} are squares for k = 1,2,3 (136^{2}, 12410^{2}, 1204552^{2}).

18^{k} + 282^{k} + 921^{k} + 1380^{k} are squares for k = 1,2,3 (51^{2}, 1683^{2}, 58581^{2}).

by Yoshio Mimura, Kobe, Japan

## 1684

A cubic polynomial :

(X + 336^{2})(X + 1684^{2})(X + 10773^{2}) = X^{3} + 10909^{2}X^{2} + 18507972^{2}X + 6095621952^{2}.

by Yoshio Mimura, Kobe, Japan

## 1686

1686^{2} = 2842596, 2 * 842 + 5 - 9 + 6 = 1686.

by Yoshio Mimura, Kobe, Japan

## 1687

452116^{k} + 705166^{k} + 767585^{k} + 921102^{k} are squares for k = 1,2,3 (1687^{2}, 1462629^{2}, 1294915895^{2}).

by Yoshio Mimura, Kobe, Japan

## 1688

(1688^{2} - 8) = (5^{2} - 8)(7^{2} - 8)(8^{2} - 8)(9^{2} - 8) = (3^{2} - 8)(5^{2} - 8)(7^{2} - 8)(8^{2} - 8)(9^{2} - 8).

by Yoshio Mimura, Kobe, Japan

## 1689

1689^{2} = 2852721, 2 * 852 - 7 * 2 - 1 = 1689.

by Yoshio Mimura, Kobe, Japan

## 1690

1690^{2} = (1)(2 + 3 + 4 + 5 + 6)(7 + 8 + 9 + ... + 19)(20 + 21 + 22 + ... + 45).

1690^{2}= 36 x 37 + 37 x 38 + 38 x 39 + 39 x 40 + 40 x 41 + ... + 204 x 205.

1690^{2} = (2^{2} + 1)(3^{2} + 1)(239^{2} + 1) = (7^{2} + 1)(239^{2} + 1) = (3^{2} + 4)(16^{2} + 4)(29^{2} + 4)

= (2^{2} + 9)(11^{2} + 9)(41^{2} + 9).

1010^{k} + 1690^{k} + 2570^{k} + 2830^{k} are squares for k = 1,2,3 (90^{2}, 4300^{2}, 213300^{2}).

Komachi equation: 1690^{2} = 1^{2} * 234^{2} * 5^{2} / 6^{2} * 78^{2} / 9^{2}.

by Yoshio Mimura, Kobe, Japan

## 1691

1691^{2} = 2859481, a zigzag square.

by Yoshio Mimura, Kobe, Japan

## 1692

1692^{2} = 2862864, a square with even digits.

1692^{2} = 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 (18^{2}, 172^{2}, 1692^{2}).

146^{k} + 154^{k} + 654^{k} + 1546^{k} are squares for k = 1,2,3 (50^{2}, 1692^{2}, 63100^{2}).

Komachi equation: 1692^{2} = 987^{2} * 6^{2} * 5^{2} * 4^{2} * 3^{2} / 210^{2}.

by Yoshio Mimura, Kobe, Japan

## 1694

1694^{2} = (1)(2 + 3 + 4 + 5)(6 + 7 + 8 + ... + 16)(17 + 18 + 19 + ... + 60).

by Yoshio Mimura, Kobe, Japan

## 1698

436386^{k} + 524682^{k} + 660522^{k} + 1261614^{k} are squares for k = 1,2,3 (1698^{2}, 1579140^{2}, 1588645404^{2}).

by Yoshio Mimura, Kobe, Japan