The smallest squares containingk 66's :
11664 = 108²,   1666681 = 1291²,   19666696644 = 140238²,
6666166299664 = 2581892²,   24806656666666681 = 157501291².

The second integer which is the sum of 5 distinct squares : 1² + 2² + 3² + 4² + 6².

66² = 4356 with different digits.

66² = 4356, 435 + 6 = 21².

66² = 4356, 4 * 3 * 5 + 6 = 66.

66² = 1³ + 2³ + 3³ + 4³ + 5³ + 6³ + 7³ + 8³ + 9³ + 10³ + 11³,
66² = (1 + 2)(3 + 4 + 5)(6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16).

66² = (2² + 2)(3² + 2)(8² + 2).

The sum of divisors of 66 is 12².

154^k + 814^k + 1298^k + 2090^k are squares for k = 1,2,3 (66², 2596², 108900²).
42^k + 66^k + 108^k + 145^k are squares for k = 1,2,3 (19², 197², 2161²).
66^k + 210^k + 1230^k + 1410^k are squares for k = 1,2,3 (54², 1884², 68364²).

Komachi equations:
66² = - 1 - 2 * 3 + 4 + 56 * 78 - 9 = - 1 * 2 + 3 - 4 + 56 * 78 - 9
  = 9 + 876 * 5 - 4 * 3 - 21 = 9 + 8 + 7 + 6 + 5 + 4321
  = 9 + 8 * 7 - 6 * 5 + 4321 = 9 * 8 - 7 - 6 * 5 + 4321
  = 9 * 8 - 7 * 6 + 5 + 4321 = - 9 + 876 * 5 - 4 * 3 - 2 - 1
  = 98 - 7 * 6 + 5 * 43 * 2 * 10 = 98 + 7 * 654 - 32 * 10
  = - 9 + 876 * 5 - 4 - 3 + 2 - 10 = - 9 * 8 * 7 + 6 * 54 * 3 / 2 * 10
  = - 9 + 8 + 7 + 6 * 5 + 432 * 10 = - 9 + 8 + 7 * 6 - 5 + 432 * 10
  = - 9 + 8 * 7 - 6 - 5 + 432 * 10,
66² = 12² * 3² + 4² + 5² - 6² + 7² * 8² - 9² = - 1² + 2² - 3² * 4² + 5² + 67² + 8² - 9²
  = 9² + 8² - 7² + 65² + 4² * 3² / 2² - 1² = 9² * 8² - 7² - 6² * 5² + 4² + 3² - 2² + 10²
  = - 98² - 7² + 6² * 5² * 4² + 3² - 2² * 10²,
66² = 1³ / 2³ * 34³ + 5³ + 6³ + 7³ - 8³ - 9³.

(66² - 6) = (4² - 6)(21² - 6) = (8² - 6)(9² - 6).

1² + 2² + 3² + 4² + ... + 66² = 98021, which consists of different digits.

(1)(2 + 3)(4 + 5 + ... + 66) = 105²,
(1)(2 + 3 + ... + 14)(15 + 16 + ... + 66) = 468²,
(1)(2 + 3 + ... + 28)(29 + 30 + ... + 66) = 855²,
(1)(2 + 3 + ... + 50)(51 + 52 + ... + 66) = 1092²,
(1)(2 + 3 + ... + 52)(53 + 54 + ... + 66) = 1071²,
(1 + 2 + ... + 9)(10 + 11 + ... + 28)(29 + 30 + ... + 66) = 5415²,
(1 + 2 + ... + 12)(13 + 14)(15 + 16 + ... + 66) = 2106²,
(1 + 2 + ... + 13)(14)(15 + 16 + ... + 66) = 1638²,
(1 + 2 + ... + 17)(18 + 19 + ... + 52)(53 + 54 + ... + 66) = 12495²,
(1 + 2 + ... + 18)(19 + 20 + ... + 38)(39 + 40 + ... + 66) = 11970²,
(1 + 2 + ... + 20)(21 + 22 + ... + 48)(49 + 50 + ... + 66) = 14490²,
(1 + 2 + ... + 31)(32 + 33 + ... + 61)(62 + 63 + ... + 66) = 14880²,
(1 + 2 + ... + 49)(50 + 51 + 52)(53 + 54 + ... + 66) = 12495².

(1² + 2² + 3² + ... + 66²) = 98021, which consists of different digits.

(1² + 2² + ... + 14²)(15² + 16² + ... + 62²)(63² + 64² + ... + 66²) = 1165220²,
(1² + 2² + ... + 17²)(18² + 19² + ... + 35²)(36² + 37² + ... + 49²)(50² + 51² + ... + 66²) = 185550750².

66⁸ = 360040606269696, 3 + 60 + 04060 + 62 + 69 + 6 + 96
  = 36 + 004060 + 62 + 6 + 96 + 96 = 3600 + 4 + 0606 + 2 + 69 + 69 + 6
  = 3600 + 40 + 6 + 0626 + 9 + 69 + 6 = 3600 + 40 + 60 + 626 + 9 + 6 + 9 + 6
  = 3600 + 40 + 606 + 26 + 9 + 69 + 6 = 3600 + 406 + 06 + 269 + 69 + 6 = 66²,
66⁹ = 23762680013799936, 237 + 6 + 268 + 001 + 3799 + 9 + 36 = 66² and more 3 equations.

66² = 4356 appears in the decimal expressions of π and e:
  π = 3.14159•••4356••• (from the 5476th digit),
  e = 2.71828•••4356••• (from the 10575th digit).

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Page of Squares : First Upload February 9, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan