Squares:21

The smallest squares containing k 21's :
121 = 11², 212521 = 461², 121242121 = 11011²,
21319212121 = 146011², 52121021421121 = 7219489².

The squares which begin with 21 and end in 21 are
212521 = 461², 2134521 = 1461², 21058921 = 4589², 21261321 = 4611²,
21520321 = 4639²,...

The first integer which is the sum of 6 squares in just two ways (see 20).

The 2nd integer which is the sum of 3 distinct squares: 1² + 2² + 4².

1 / 21 = 0.04761..., 4761 = 69².

21² = 441, 4 + 4 + 1 = 3².

21² = 1³ + 2³ + 6³ + 6³ = 1³ + 2³ + 3³ + 4³ + 5³ + 6³.

Every integer greater than 21 is the sum of 8 nonzero squares.

21² = 441 is a reversible square (144 = 12²).

21² = 441, every digit of which is a square.

21⁸ = 37822859361, 3 + 78 + 2 + 285 + 9 + 3 + 61 = 378 + 2 + 2 + 8 + 5 + 9 + 36 + 1 = 21²,
21⁹ = 794280046581, 7 + 94 + 280 + 046 + 5 + 8 + 1 = 79 + 4 + 280 + 04 + 65 + 8 + 1 = 21².

21¹⁰ = 16679880978201, 16 + 6 + 7 + 98 + 8 + 097 + 8 + 201 = 21² and more 7 equations,
21¹¹ = 350277500542221, 3 + 50 + 277 + 50 + 054 + 2 + 2 + 2 + 1 = 21² and more 9 equations,
21¹² = 7355827511386641, 7 + 3 + 5 + 5 + 8 + 2 + 7 + 5 + 1 + 1 + 386 + 6 + 4 + 1 = 21² and more 35 equations.

(4² + 1)(5² + 1) = 21² + 1,
(1² + 7)(7² + 7) = 21² + 7,
(1² + 9)(6² + 9) = (3² + 9)(4² + 9) = 21² + 9,
(5² - 9)(6² - 9) = 21² - 9.

29² = 20² + 21² + 22² + ... + 21².

21² + 22² + 23² + ... + 116² = 724².

(1 + 2 + 3)(4 + 5)(6) = 18²,
(1)(2)(3)(4 + 5)(6) = 18²,
(1 + 2)(3)(4 + 5 + 6 + 7 + 8 + 9 + 10) = 21²,
(1 + 2)(3)(4 + 5 + ... + 21) = 45²,
(1 + 2 + 3)(4 + 5)(6 + 7 + ... + 21) = 108²,
(1 + 2 + ... + 8)(9 + 10 + ... + 18)(19 + 20 + 21) = 540².

(1³ + 2³ + 3³ + 4³ + 5³ + 6³) = 21²,
(1³)(2³ + 3³ + ... + 13³)(14³ + 15³ + ... + 21³) = 19320²,
(1³ + 2³ + ... + 6³)(7³ + 8³ + ... + 14³)(15³ + 16³ + ... + 21³) = 444528².

Komachi Fractions : (2/21)² = 8460/932715, (21/95)² = 7938/162450.

Komachi: 21² = 1 - 234 + 5 + 678 - 9 and more 26 equations.
Komachi: 21² = 98 + 76 - 54 + 321 and more 30 equations.
Komachi: 21² = 1² + 2² * 3² + 4² * 5² + 6² - 7² - 8² + 9².
Komachi: 21² = 9² - 8² - 7² + 6² + 5² * 4² + 3² * 2² + 1².

1² + 2² + 3² + ... + 21² = 3311, which consists of 2 kinds of odd digits (the first 4-digit sum).

21² = 441 appears in the decimal expressions of π and e:
π = 3.14159•••441••• (from the 726th digit), (441 is the 6th 3-digit square in the expr. of π,)
e = 2.71828•••441••• (from the 1288th digit).

Squares : First Upload October 6, 2003 ; Last Revised May 8, 2006