The smallest squares containingk 63's :
16384 = 128²,   16386304 = 4048²,   16356363664 = 127892²,
36386363116321 = 6032111²,   2636363656363024 = 51345532².

63² is the first square which is the sum of 4 distinct fourth powers : 2⁴ + 4⁴ + 6⁴ + 7⁴.

63² = 3969, a zigzag square.

63² = 3969, 3 - 9 + 69 = 63.

63² = 3969, 3 + 9 + 69 = 9².

63² = 1³ + 8³ + 12³ + 12³ = 8³ + 9³ + 10³ + 12³ = 2⁴ + 4⁴ + 6⁴ + 7⁴ = 3⁴ + 6⁴ + 6⁴ + 6⁴.

63² = 21² + 42² + 42², 24² + 24² + 12² = 36².

63² = (1 + 2)(3 + 4 + 5 + 6 + 7 + 8 + ... + 51)
  = (1 + 2 + 3 + 4 + 5 + 6)(7)(8 + 9 + 10) = (1 + 2 + 3 + 4 + 5 + 6)(7 + 8 + 9 + ... + 20).

A 3-by-3 magic square consisting of different squares with constant 63²:

Komachi Fraction : (31/63)² = 8649/35721,

Komachi equations:
63² = 12 * 3 / 4 * 56 * 7 / 8 * 9 = - 12 * 34 + 56 * 78 + 9
  = - 9 * 8 + 76 * 54 - 3 * 21 = 9 * 8 + 76 * 54 + 3 - 210
  = 98 + 7 + 654 + 3210,
63² = 1² + 2² - 34² - 56² / 7² + 8² * 9² = 1² - 23² + 4² - 5² + 67² - 8² + 9²
  = - 1² * 2² - 3² * 4² + 5² * 6² + 7² * 8² + 9² = - 1² + 2² * 3² + 45² + 6² * 7² + 8² + 9²
  = - 12² + 3² - 45² - 6² + 78² + 9² = - 1² + 23² - 4² + 5² - 67² + 89²
  = 9² + 8² * 7² + 6² * 5² - 4² * 3² - 2² */ 1² = 9² - 8² + 7² - 6² + 54² + 32² - 1²
  = - 9² / 8² * 7² * 6² + 5² / 4² * 3² * 21² = 9² + 87² - 65² + 4² * 3² + 2² * 10²
  = 9² * 87² * 6² / 54² - 3² * 2² * 10²,
63² = 9³ + 87³ / 6³ + 5³ + 4³ + 3³ / 2³ - 1³.

63 and 64 are consecutive integers having square factors (the 5th case).

(63² + 7) = (7² + 7)(8² + 7),   (63² - 9) = (8² - 9)(9² - 9).

(1 + 2 + ... + 5)(6 + 7 + ... + 23)(24 + 25 + ... + 63) = 2610²,
(1 + 2 + ... + 11)(12 + 13 + ... + 24)(25 + 26 + ... + 63) = 5148²,
(1 + 2 + ... + 17)(18 + 19 + ... + 38)(39 + 40 + ... + 63) = 10710²,
(1 + 2 + ... + 31)(32 + 20 + ... + 61)(62 + 63) = 9300².

63⁶ = 62523502209, 6 + 252 + 3502 + 209 = 63²,
63⁷ = 3938980639167, 3 + 9 + 3 + 8 + 9 + 8 + 06 + 3916 + 7 = 63²,
    3 + 9 + 3898 + 06 + 39 + 1 + 6 + 7 = 39 + 3898 + 06 + 3 + 9 + 1 + 6 + 7 = 63²,
63⁸ = 248155780267521, 2 + 481 + 5 + 5 + 780 + 2675 + 21 = 63²,
63⁸ = 248155780267521, 2481 + 5 + 5 + 780 + 2 + 675 + 21 = 63²,
63⁹ = 15633814156853823, 1+5+6+3+3+8+1+41+5+68+5+3823 = 63² and more 45 equations.

63² = 3969 appears in the decimal expressions of π and e:
  π = 3.14159•••3969••• (from the 35623rd digit),
  e = 2.71828•••3969••• (from the 973rd digit),
  (3969 is the tenth 4-digit square in the expression of e.)

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Page of Squares : First Upload February 2, 2004 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan