The smallest squares containingk 81's :
81 = 9²,   3818116 = 1954²,   818188816 = 28604²,
58181781681 = 241209²,   2608818148181281 = 51076591².

The squares which begin with 81 and end in 81 are
8173881 = 2859²,   81162081 = 9009²,   81739681 = 9041²,   811737081 = 28491²,
812763081 = 28509²,...

The square of 9, and a square with a decreasing sequence of digits.

The sum of the divisors of 81 is 11², a square.

81 = (8 + 1)².

81² = 6561, a zigzag square.

8173881 = 2859².

8281 = 91².

81² = 9³ + 18³ = 2⁴ + 2⁴ + 4⁴ + 4⁴ + 5⁴ + 6⁴ + 8⁴ = 3⁷ + 3⁷ + 3⁷.

A 3-by-3 magic square consisting of different squares with constant 81²:

81^k + 136^k + 304^k + 704^k are squares for k = 1,2,3 (35², 783², 19495²).

Komachi equations:
81² = 1 * 2 * 3 / 4 * 56 * 78 + 9 = 1 - 234 + 5 + 6789
  = - 1 + 23 * 4 * 5 + 678 * 9 = 987 * 6 + 5 * 4 * 32 - 1
  = 9 + 8 * 765 + 432 * 1 = 9 + 8 * 765 + 432 / 1
  = - 9 + 876 * 5 / 4 * 3 * 2 * 1 = - 9 + 876 * 5 / 4 * 3 * 2 / 1
  = 9 + 8 - 7 + 6543 - 2 + 10 = 987 * 6 + 5 + 4 + 3 * 210
  = - 9 + 8 + 7 + 6543 + 2 + 10,
81² = 1² * 2² + 3² - 4² * 5² + 6² * 7² + 8² * 9² = - 1² - 2² + 3² + 4² * 5² - 6² * 7² + 89²
  = - 12² - 34² + 5² - 6² - 7² + 89² = 9² * 8² + 7² * 6² - 5² * 4² + 3² + 2² */ 1²
  = 9² - 8² + 7² - 6² * 5² + 43² * 2² - 1² = 9² + 87² - 6² + 54² - 3² * 21²
  = - 98² + 7² * 6² + 5² * 4² * 3² * 2² + 1² = 9² * 8² * 7² / 6² - 5² * 4² + 3² - 2² - 10²
  = 98² / 7² * 6² - 5² * 4² + 3² - 2² - 10² = 9² * 8² - 7² * 6² + 54² + 3² / 2² * 10²,
81² = 12⁴ - 3⁴ * 4⁴ + 56⁴ - 7⁴ * 8⁴ + 9⁴ = 12⁴ - 3⁴ * 4⁴ + 56⁴ / 7⁴ - 8⁴ + 9⁴
  = 12⁴ - 3⁴ * 4⁴ + 56⁴ / 7⁴ / 8⁴ * 9⁴ = 12⁴ - 3⁴ * 4⁴ - 56⁴ + 7⁴ * 8⁴ + 9⁴
  = 12⁴ - 3⁴ * 4⁴ - 56⁴ / 7⁴ + 8⁴ + 9⁴ = 12⁴ - 3⁴ * 4⁴ * 56⁴ / 7⁴ / 8⁴ + 9⁴
  = 12⁴ - 3⁴ * 4⁴ / 56⁴ * 7⁴ * 8⁴ + 9⁴ = 12⁴ / 3⁴ - 4⁴ + 56⁴ - 7⁴ * 8⁴ + 9⁴
  = 12⁴ / 3⁴ - 4⁴ + 56⁴ / 7⁴ - 8⁴ + 9⁴ = 12⁴ / 3⁴ - 4⁴ + 56⁴ / 7⁴ / 8⁴ * 9⁴
  = 12⁴ / 3⁴ - 4⁴ - 56⁴ + 7⁴ * 8⁴ + 9⁴ = 12⁴ / 3⁴ - 4⁴ - 56⁴ / 7⁴ + 8⁴ + 9⁴
  = 12⁴ / 3⁴ - 4⁴ * 56⁴ / 7⁴ / 8⁴ + 9⁴ = 12⁴ / 3⁴ - 4⁴ / 56⁴ * 7⁴ * 8⁴ + 9⁴
  = 12⁴ / 3⁴ / 4⁴ - 56⁴ / 7⁴ / 8⁴ + 9⁴ = 12⁴ / 3⁴ / 4⁴ * 56⁴ - 7⁴ * 8⁴ + 9⁴
  = 12⁴ / 3⁴ / 4⁴ * 56⁴ / 7⁴ - 8⁴ + 9⁴ = - 12⁴ + 3⁴ * 4⁴ - 56⁴ + 7⁴ * 8⁴ + 9⁴
  = - 12⁴ + 3⁴ * 4⁴ - 56⁴ / 7⁴ + 8⁴ + 9⁴ = - 12⁴ + 3⁴ * 4⁴ + 56⁴ - 7⁴ * 8⁴ + 9⁴
  = - 12⁴ + 3⁴ * 4⁴ + 56⁴ / 7⁴ - 8⁴ + 9⁴ = - 12⁴ + 3⁴ * 4⁴ + 56⁴ / 7⁴ / 8⁴ * 9⁴
  = - 12⁴ + 3⁴ * 4⁴ * 56⁴ / 7⁴ / 8⁴ + 9⁴ = - 12⁴ + 3⁴ * 4⁴ / 56⁴ * 7⁴ * 8⁴ + 9⁴
  = - 12⁴ / 3⁴ + 4⁴ - 56⁴ + 7⁴ * 8⁴ + 9⁴ = - 12⁴ / 3⁴ + 4⁴ - 56⁴ / 7⁴ + 8⁴ + 9⁴
  = - 12⁴ / 3⁴ + 4⁴ + 56⁴ - 7⁴ * 8⁴ + 9⁴ = - 12⁴ / 3⁴ + 4⁴ + 56⁴ / 7⁴ - 8⁴ + 9⁴
  = - 12⁴ / 3⁴ + 4⁴ + 56⁴ / 7⁴ / 8⁴ * 9⁴ = - 12⁴ / 3⁴ + 4⁴ * 56⁴ / 7⁴ / 8⁴ + 9⁴
  = - 12⁴ / 3⁴ + 4⁴ / 56⁴ * 7⁴ * 8⁴ + 9⁴ = - 12⁴ / 3⁴ / 4⁴ + 56⁴ / 7⁴ / 8⁴ + 9⁴
  = - 12⁴ / 3⁴ / 4⁴ * 56⁴ + 7⁴ * 8⁴ + 9⁴ = - 12⁴ / 3⁴ / 4⁴ * 56⁴ / 7⁴ + 8⁴ + 9⁴.

(81² - 6)= (3² - 6)(5² - 6)(11² - 6),
(81² + 9) = (8² + 9)(9² + 9),   (81² - 9) = (9² - 9)(10² - 9).

(1 + 2)(3 + 4 + ... + 18)(19 + 20 + ... + 81)=1260²,
(1 + 2 + ... + 6)(7 + 8 + ... + 18)(19 + 20 + ... + 81) = 3150²,
(1 + 2 + ... + 9)(10 + 11 + ... + 74)(75 + 76 + ... + 81) = 8190²,
(1 + 2 + ... + 12)(13 + 14 + ... + 35)(36 + 37 + ... + 81) = 10764²,
(1 + 2 + ... + 31)(32 + 33 + ... + 73)(74 + 75 + ... + 81) = 26040²,
(1 + 2 + ... + 32)(33 + 34 + ... + 72)(73 + 74 + ... + 81) = 27720²,
(1 + 2 + ... + 39)(40 + 41 + ... + 48)(49 + 50 + ... + 81) = 25740²,
(1 + 2 + ... + 44)(45 + 46 + ... + 66)(67 + 68 + ... + 81) = 36630².

(1³ + 2³ + ... + 80³)(81³) = 2361960².

81⁶ = 282429536481, 2 + 8 + 2 + 4 + 2 + 9 + 53 + 6481 = 81²,
81⁸ = 1853020188851841, 185 + 302 + 01 + 888 + 5184 + 1 = 81²,
81⁹ = 150094635296999121,
    1+5+009+4+6352+9+69+99+12+1 = 1+5+0094+6352+9+69+9+9+12+1
  = 1+50+09+4+6352+9+6+9+9+91+21 = 1+50+09+4+6352+9+6+9+99+1+21
  = 1+50+09+4+6352+9+6+99+9+1+21 = 1+50+09+4+6352+96+9+9+9+1+21
  = 1+50+094+6352+9+6+9+9+9+1+21 = 1+500+9+4635+296+999+121
  = 1+5009+4+6+3+529+6+9+991+2+1 = 1+5009+4+6+3+529+6+999+1+2+1
  = 1+5009+4+6+352+969+99+121 = 1+5009+463+5+2+969+99+12+1
  = 1+5009+463+52+9+6+9+991+21 = 1+5009+463+52+9+6+999+1+21
  = 15+009+4+6352+9+69+9+91+2+1 = 15+009+4+6352+9+69+99+1+2+1
  = 15+0094+6352+9+69+9+9+1+2+1 = 150+09+4+6352+9+6+9+9+9+1+2+1
  = 150+094+6+3+5296+99+912+1 = 150+094+6+3+5296+999+12+1
  = 1500+9+4635+296+9+99+12+1 = 1500+9+4635+296+99+9+12+1 = 81².

81² = 6561 appears in the decimal expressions of π and e:
  π = 3.14159•••6561••• (from the 3798th digit),
  e = 2.71828•••6561••• (from the 15228th digit).

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Page of Squares : First Upload March 1, 2004 ; Last Revised February 3, 2011
by Yoshio Mimura, Kobe, Japan