The smallest squares containingk 91's :
2916 = 54²,   69189124 = 8318²,   19109191696 = 138236²,
19191829199104 = 4380848²,   8913919178910916 = 94413554².

91 is the first integer which is the sum of 6 distinct squares : 1² + 2² + 3² + 4² + 5² + 6².

91 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 = 1² + 2² + 3² + 4² + 5² + 6².

91² = 8281, a zigzag square.

91² = 8281, 82 and 81 are consecutive.

91² = 8281, 8 + 281 = 17².

91² is the first square which is the sum of 4 cubes in 4 ways:
    1³ + 4³ + 6³ + 20³ = 3³ + 4³ + 11³ + 19³ = 8³ + 13³ + 13³ + 15³ = 9³ + 12³ + 12³ + 16³.
91² is the first square which is the sum of 6 cubes in 5 ways:
    [1,2,3,6,13,18], [1,4,6,7,14,17], [1,5,10,11,12,16], [3,4,7,12,14,15], [6,8,9,10,12,16].

The alternating sum of the first 25 consecutive cubes is a square : 1³ - 2³ + 3³ - ... + 25³ = 91².

52^k + 91^k + 117^k + 169^k + 247^k are squares for k = 1,2,3,4 (26², 338², 4732², 69290²).

Kaprekar number : 91² = 8281, 8 + 2 + 81 = 82 + 8 + 1 = 91.

A 3-by-3 magic square consisting of different squares with constant 91²:

260^k + 286^k + 1742^k + 5993^k are squares for k = 1,2,3 (91², 6253², 469651²).
17^k + 91^k + 299^k + 377^k are squares for k = 1,2,3 (28², 490², 9004²).

Komachi equations:
91² = 1² * 23² * 4² - 5² + 6² - 7² - 8² - 9² = - 1² + 2² - 3² + 4² + 56² - 7² + 8² * 9²
  = 98² - 7² * 6² + 5² - 4² - 3² + 21² = 98² - 7² * 6² - 5² + 4² + 3² + 21².

91² = 1³ + 4³ + 6³ + 20³ = 3³ + 4³ + 11³ + 19³ = 8³ + 13³ + 13³ + 15³ = 9³ + 12³ + 12³ + 16³.

(91² + 1) = (9² + 1)(10² + 1).

91² + 92² + 93² + ... + 332² = 3465².

91² = (2 + 3 + 4 + 5 + 6 + 7 + 8)² + (9 + 10 + 11 + 12 + 13 + 14 + 15)².

(1 + 2 + ... + 7)(8 + 9 + ... + 55)(56 + 57 + ... + 91) = 10584².

(1² + 2² + ... + 22²)(23² + 24² + ... + 45²)(46² + 47² + ... + 69²)(70² + 71² + ... + 91²) = 1099791000².

1³ + 2³ + 3³ + 4³ + 5³ + 6³ + 7³ + 8³ + 9³ + 10³ + 11³ + 12³ + 13³ = 91².

91⁷ = 51676101935731, 516 + 7610 + 1 + 93 + 57 + 3 + 1 = 91²,
91⁹ = 427929800129788411,
    4+2+7+9+29+8001+2+97+88+41+1 = 4+2+7+92+9+8001+29+7+88+41+1,
  = 4+2+79+2+9+8001+2+9+78+84+11 = 4+2+79+29+8001+29+7+88+41+1,
  = 4+2+79+298+001+2+9+7884+1+1 = 4+2+7929+80+0129+7+88+41+1,
  = 4+27+9+2+9+8001+2+97+88+41+1 = 4+27+92+9+8001+2+9+7+88+41+1,
  = 4+279+2+9+80+01+2+9+7884+11 = 4+279+2+9+80+012+9+7884+1+1,
  = 4+279+2+98+001+2+9+7884+1+1 = 42+7+9+29+8001+2+97+8+84+1+1,
  = 42+7+9+29+8001+2+97+88+4+1+1 = 42+7+9+29+8001+29+78+84+1+1,
  = 42+7+9+298+001+29+7884+11 = 42+7+92+9+8001+2+97+8+8+4+11,
  = 42+7+92+9+8001+29+7+8+84+1+1 = 42+7+92+9+8001+29+7+88+4+1+1,
  = 42+7+92+9+8001+29+78+8+4+11 = 42+79+2+9+8001+2+9+7+88+41+1,
  = 42+79+29+8001+2+97+8+8+4+11 = 42+79+29+8001+29+7+8+84+1+1,
  = 42+79+29+8001+29+7+88+4+1+1 = 42+79+29+8001+29+78+8+4+11,
  = 42+7929+8+00129+78+84+11 = 42+7929+80+01+2+97+88+41+1,
  = 42+7929+80+0129+7+8+84+1+1 = 42+7929+80+0129+7+88+4+1+1,
  = 42+7929+80+0129+78+8+4+11 = 4279+2980+0129+7+884+1+1 = 91².

91² = 8281 appears in the decimal expressions of π and e:
  π = 3.14159•••8281••• (from the 5392nd digit),
  e = 2.718281828459045••• (from the 3rd digit),
  (8281 is the first 4-digit square in the expression of e.)

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Page of Squares : First Upload March 15, 2004 ; Last Revised February 3, 2011
by Yoshio Mimura, Kobe, Japan