The smallest squares containing k 48's :
484 = 22²,   4800481 = 2191²,   1448487481 = 38059²,
1648483948489 = 1283933²,   4874899148148484 = 69820478².

48 is the sum of m squares for m = 3, 4, ..., 34.

The third integer which is the sum of 8 squares in just 3 ways (see 46).
The first integer which is the sum of 9 squares in just 6 ways:
  [1,1,1,1,1,1,1,4,5], [1,1,1,1,1,3,3,3,4], [1,1,1,1,2,2,2,4,4], [1,1,1,2,2,2,2,2,5], [1,2,2,2,2,2,3,3,3],
  [2,2,2,2,2,2,2,2,4]

48² is the fifth square which is the sum of 3 cubes : 4³ + 8³ + 12³.

48² = 2304, a zigzag square whose digits are distinct.

48² = 2304, 2 + 3 + 0 + 4 = 3²,
48² = 2304, 2 + 30 + 4 = 6².

48² = (1³ + 2³ + 3³)(4³).

48² = 4! + 5! + 6! + 6! + 6!

48² = (3² - 1)(17² - 1)

48, 49 and 50 are three consecutive integers having square factors (the first case).

2² + 5² + 8² + 11² + ... + 26² = 48².

48⁶ = 12230590464, 1 + 2230 + 59 + 04 + 6 + 4 = 48²,
48⁹ = 1352605460594688, 1 + 3 + 52 + 605 + 4 + 605 + 946 + 88 = 48²,
48⁹ = 1352605460594688, 1 + 352 + 605 + 4 + 60 + 594 + 688 = 48².

48^k + 177^k + 186^k + 1614^k are squares for k = 1,2,3 (45², 1635², 64935²).
82^k + 182^k + 722^k + 1318^k are squares for k = 1,2,3 (48², 1516², 51696²).

Komachi Fraction : (48/103)² = 20736/95481.

Komachi equations:
48² = 1 * 2345 - 6 * 7 - 8 + 9 = 1 + 2 - 3 + 4 * 56 / 7 * 8 * 9
48² = 123 * 4 * 5 - 67 - 89, and more 7 equations,
48² = 9 + 8 * 765 / 4 * 3 / 2 * 1 = 9 + 8 * 765 / 4 * 3 / 2 / 1
 = - 98 * 7 * 6 + 5 * 4 * 321,
48² = 9 * 8 * 7 + 6 * 5 * 4 * 3 / 2 * 10 = - 9 - 87 + 6 * 5 / 4 * 32 * 10
 = - 9 * 8 * 7 + 65 * 432 / 10 = - 98 / 7 * 65 + 4 + 3210,
48² = 1² * 2² * 34² - 56² * 7² / 8² + 9² = 1² * 2² - 3² + 4² * 5² + 6² * 7² + 8² + 9²
  = 1² + 2² + 3² * 4² - 5² * 6² + 7² * 8² - 9²,
48² = 9² + 8² + 7² * 6² + 5² * 4² - 3² + 2² * 1² = 9² + 8² + 7² * 6² + 5² * 4² - 3² + 2² / 1²,
48² = 98³ / 7³ - 6³ - 5³ - 4³ - 3³ - 2³ * 1³ = 98³ / 7³ - 6³ - 5³ - 4³ - 3³ - 2³ / 1³
 = - 98³ / 7³ + 6³ * 5³ - 4³ / 3³ * 21³.

(48² + 6) = (1² + 6)(18² + 6) = (2² + 6)(15² + 6) = (6² + 6)(7² + 6),
(48² + 6) = (1² + 6)(3² + 6)(4² + 6),
(48² - 8) = (7² - 8)(8² - 8).

143² = 38² + 39² + 40² + ... + 48²,
182² = 25² + 26² + 27² + ... + 48².

(1 + 2 + ... + 26)(27 + 28 + ... + 39)(40 + 41 + ... + 48) = 7722².

1² + 2² + 3² + 4² + ... + 48² = 38024, which consists of different digits.

(1² + 2² + ... + 24²)(25² + 26² + ... + 48²) = 12740².

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Page of Squares : First Upload January 13, 2004 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan