The smallest squares containingk 86's :
8649 = 93²,   868624 = 932²,   7886860864 = 88808²,
4865866868641 = 2205871²,   866686868686116 = 29439546².

86² = 2⁴ + 2⁴ + 2⁴ + 5⁴ + 5⁴ + 6⁴ + 7⁴ + 7⁴.

86² = 7396, a zigzag square with different digits.

86² = 7396, a square with odd digits except the last digit.

86² = 7396, 7 + 3 + 9 + 6 = 5²,
86² = 7396, 73 + 96 = 13².

11² + 29² + 47² + 65² = 86².

Komachi Fractions : 86² = 6057324/819 = 6190452/837.

Komachi equations:
86² = 98 * 76 - 54 + 3 - 2 + 1 = 98 * 76 - 5 * 4 - 32 * 1
  = 98 * 76 - 5 * 4 - 32 / 1 = 98 * 76 - 5 * 4 * 3 - 2 + 10,
86² = - 1² * 23² + 4² - 5² - 6² + 7² + 89² = - 9² + 87² + 6² - 5² - 4² + 3² + 2² - 10².

(86² - 4) = (9² - 4)(10² - 4).

(1 + 2 + 3 + 4)(5 + 6 + ... + 34)(35 + 36 + ... + 86) = 4290²,
(1 + 2 + ... + 7)(8 + 9 + ... + 39)(40 + 41 + ... + 86) = 7896²,
(1 + 2 + ... + 8)(9 + 10 + ... + 48)(49 + 50 + ... + 86) = 10260²,
(1 + 2 + ... + 11)(12 + 13 + ... + 23)(24 + 25 + ... + 86) = 6930²,
(1 + 2 + ... + 13)(14 + 15 + ... + 34)(35 + 36 + ... + 86) = 12012²,
(1 + 2 + ... + 14)(15 + 16 + ... + 41)(42 + 43 + ... + 86) = 15120²,
(1 + 2 + ... + 17)(18 + 19 + ... + 34)(35 + 36 + ... + 86) = 14586²,
(1 + 2 + ... + 18)(19 + 20 + ... + 38)(39 + 40 + ... + 86) = 17100²,
(1 + 2 + ... + 20)(21 + 22 + 23)(24 + 25 + ... + 86) = 6930²,
(1 + 2 + ... + 27)(28 + 29 + ... + 84)(85 + 86) = 14364²,
(1 + 2 + ... + 35)(36 + 37 + ... + 83)(84 + 85 + 86) = 21420²,
(1 + 2 + ... + 36)(37 + 38 + ... + 61)(62 + 63 + ... + 86) = 38850²,
(1 + 2 + ... + 40)(41)(42 + 43 + ... + 86) = 9840²,
(1 + 2 + ... + 49)(50 + 51 + ... + 61)(62 + 63 + ... + 86) = 38850²,
(1 + 2 + ... + 49)(50 + 51 + ... + 82)(83 + 84 + 85 + 86) = 30030²,
(1 + 2 + ... + 51)(52 + 53 + ... + 69)(70 + 71 + ... + 86) = 43758²,
(1 + 2 + ... + 69)(70 + 71 + ... + 74)(75 + 76 + ... + 86) = 28980²,
(1 + 2 + ... + 81)(82)(83 + 84 + 85 + 86) = 9594².

(1³ + ... + 56³)(57³ + ... + 75³)(76³ + ... + 86³) = 9132315192².

86⁸ = 2992179271065856, 2 + 99 + 21 + 7 + 92 + 7106 + 5 + 8 + 56 = 86²,
    2 + 99 + 21 + 7 + 92 + 7106 + 58 + 5 + 6 = 29 + 9 + 2 + 179 + 2 + 7106 + 5 + 8 + 56
  = 29 + 9 + 2 + 179 + 2 + 7106 + 58 + 5 + 6 = 29 + 9 + 217 + 9 + 2 + 7106 + 5 + 8 + 5 + 6
  = 29 + 92 + 1 + 7 + 92 + 7106 + 5 + 8 + 56 = 29 + 92 + 1 + 7 + 92 + 7106 + 58 + 5 + 6 = 86².

86² = 7396 appears in the decimal expressions of π and e:
  π = 3.14159•••7396••• (from the 1961st digit),
  e = 2.71828•••7396••• (from the 972nd digit).
  (7396 is the ninth 4-digit square in the expression of e.)

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Page of Squares : First Upload March 8, 2004 ; Last Revised August 4, 2013
by Yoshio Mimura, Kobe, Japan