The smallest squares containingk 82's :
8281 = 91²,   8208225 = 2865²,   82828201 = 9101²,
82182782582784 = 9065472²,   4824828298248256 = 69460984².

82 is the sum of m squares for m = 2, 3, ..., 68.

The 5th integer which is the sum of 5 squares.

The 7th Kaprekar number : 82² = 6724, 6 + 72 + 4 = 82.

1 / 82 = 0.0121..., 121 = 11².

8281 = 91².

82² = 6724, 6 + 72 + 4 = 82.

82² = 1⁴ + 3⁴ + 3⁴ + 9⁴ = 1⁴ + 5⁴ + 6⁴ + 7⁴ + 7⁴.

82² = 6724, a zigzag square with different digits.

82² = 2(2! + 5! + 6!) + 7!.

34^k + 82^k + 98^k + 110^k are squares for k = 1,2,3 (18², 172², 1692²).
82^k + 182^k + 722^k + 1318^k are squares for k = 1,2,3 (48², 1516², 51696²).
82^k + 298^k + 628^k + 673^k are squares for k = 1,2,3 (41², 971², 24073²).

Komachi equations:
82² = 1 - 2 * 3 + 4 * 5 * 6 * 7 * 8 + 9 = - 1 * 2 - 3 + 4 * 5 * 6 * 7 * 8 + 9
  = 9 + 8 * 7 * 6 * 5 * 4 - 3 - 2 * 1 = 9 + 8 * 7 * 6 * 5 * 4 - 3 - 2 / 1
  = 9 + 8 * 7 * 6 * 5 * 4 - 3 * 2 + 1 = 9 + 8 * 7 * 6 * 5 * 4 + 3 + 2 - 10
  = - 9 * 8 + 7 * 6 * 54 * 3 + 2 - 10 = - 9 - 8 + 7 * 6 / 5 / 4 * 3210,
82² = 1² - 2² - 34² - 5² + 6² - 7² + 89² = - 9² + 87² - 6² * 5² + 4² * 3² / 2² + 10²
  = - 9² + 87² + 6² - 5² * 4² * 3² / 2² + 10² = - 9² + 8² + 76² + 5² + 4² + 32² - 10²,
82² = - 9³ - 8³ + 7³ - 6³ - 5³ - 4³ + 3³ + 2³ * 10³.

(82² - 4) = (5² - 4)(18² - 4),   (82² - 8) = (9² - 8)(10² - 8),
(82² + 8) = (5² + 8)(14² + 8) = (1² + 8)(3² + 8)(6² + 8) = (2² + 8)(3² + 8)(5² + 8).

(1 + 2 + 3)(4 + 5 + ... + 46)(47 + 48 + ... + 82) = 3870²,
(1 + 2 + ... + 6)(7 + 8 + ... + 12)(13 + 14 + ... + 82) = 1995²,
(1 + 2 + ... + 11)(12 + 13 + ... + 38)(39 + 40 + ... + 82) = 10890²,
(1 + 2 + ... + 12)(13 + 14 + ... + 47)(48 + 49 + ... + 82) = 13650²,
(1 + 2 + ... + 14)(15 + 16 + ... + 19)(20 + 21 + ... + 82) = 5355²,
(1 + 2 + ... + 15)(16 + 17 + ... + 64)(65 + 66 + ... + 82) = 17640²,
(1 + 2 + ... + 17)(18 + 19 + ... + 67)(68 + 69 + ... + 82) = 19125²,
(1 + 2 + ... + 25)(26 + 27 + ... + 52)(53 + 54 + ... + 82) = 26325²,
(1 + 2 + ... + 29)(30 + 31 + ... + 57)(58 + 59 + ... + 82) = 30450²,
(1 + 2 + ... + 34)(35 + 36 + ... + 70)(71 + 72 + ... + 82) = 32130².

82² = 6724, 6⁴ + 7⁴ + 2⁴ + 4⁴ = 63²,
82² = 6724, 672 + 4 = 26².

(1³ + 2³ + ... + 8³)(9³ + 10³ + 11³ + 12³)(13³ + 14³ + ... + 82³) = 8474760².

82³ = 551368, 5⁴ + 5⁴ + 1⁴ + 3⁴ + 6⁴ + 8⁴ = 82²,
82⁶ = 304006671424, 3 + 04 + 006671 + 42 + 4 = 82²,
82⁶ = 304006671424, 3 + 040 + 06671 + 4 + 2 + 4 = 82²,
82⁷ = 24928547056768, 24 + 92 + 854 + 70 + 5676 + 8 = 82²,
82⁸ = 2044140858654976, 20 + 4 + 4 + 1 + 4 + 08 + 58 + 6549 + 76 = 82²,
    20+4+4+1+40+85+8+6549+7+6 = 20+4+41+4+085+8+6549+7+6
  = 20+44+1+4+085+8+6549+7+6 = 20+44+14+08+5+8+6549+76
  = 204+4+140+8+5865+497+6 = 2044+1+4085+86+5+497+6 = 82²,
82⁹ = 167619550409708032, 1 + 6 + 761 + 9 + 5 + 5040 + 97 + 0803 + 2 = 82²,
    1+6+761+95+5040+9+7+0803+2 = 1+67+61+9+5504+0970+80+32
  = 1+67+6195+5+0409+7+08+032 = 1+676+19+5+5040+970+8+03+2
  = 16+7+6195+5+0409+7+080+3+2 = 16+7+6195+50+409+7+08+032
  = 167+61+9+5504+0970+8+03+2 = 1676+1+95+50+4097+0803+2 = 82².

82² = 6724 appears in the decimal expressions of π and e:
  π = 3.14159•••6724••• (from the 5054th digit),
  e = 2.71828•••6724••• (from the 3473rd digit).

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