The smallest squares containing k 57's :
576 = 24²,   25765776 = 5076²,   757570576 = 27524²,
5757057975769 = 2399387²,   5757957575776849 = 75881207².

57 = (1² + 2² + 3² + ... + 9²) / (1² + 2²).

The 7th integer which is the sum of 4 distinct squares: 1² + 2² + 4² + 6².
The 2nd integer which is the sum of 6 squares in just 6 ways.
  [1,1,1,1,2,7], [1,1,1,2,5,5], [1,1,1,3,3,6], [1,2,2,4,4,4], [1,2,3,3,3,5], [2,2,2,2,4,5].
The third integer which is the sum of 7 squares in just 5 ways.
The first integer which is the sum of 9 squares in just 9 ways.

57² is the 6th squares which is the sum of 5 fourth powers : [2,2,5,6,6].

57² = 3249 is a zigzag square, whose digits are distinct.

57² = (1 + 2 + ... + 18)(19).

57² = 3249, 3 + 24 + 9 = 6²,
57² = 3249, 32 + 49 = 9².

57² = 3249, 324 = 18², 9 = 3².

The sum of 29 consecutive odd cubes is a square: 1² + 3² + 5² + ... + 57² = 1189².

Cubic polynomial (X + 44²)(X + 57²)(X + 144²) = X³ + 161²X² + 10668²X + 361152².

57⁷ = 1954897493193, 1 + 9 + 5 + 4 + 8 + 9 + 7 + 4 + 9 + 3193 = 57²,
57⁸ = 111429157112001, 1 + 114 + 2915 + 7 + 11 + 200 + 1 = 57²,
    1 + 1142 + 91 + 5 + 7 + 1 + 1 + 2001 = 111 + 4 + 2915 + 7 + 11 + 200 + 1 = 57².

A 3-by-3 magic square consisting of different squares with constant 57² (the least):

10^k + 44^k + 57^k + 58^k are squares for k = 1,2,3 (13², 93², 683²).
57^k + 102^k + 222^k + 348^k are squares for k = 1,2,3 (27², 429², 7371²).
30^k + 57^k + 168^k + 474^k are squares for k = 1,2,3 (27², 507², 10557²).

Komachi Fraction : (57/4)² = 935712/4608.

Komachi equations:
57² = - 12 - 3 + 456 * 7 + 8 * 9 = 9 - 8 - 7 + 6 * 543 - 2 - 1
  = - 9 - 8 + 7 + 6 * 543 + 2 - 1 = - 9 + 8 - 7 + 6 * 543 - 2 + 1
  = 9 + 8 + 7 + 6 + 5 + 4 + 3210 = 9 + 8 + 7 * 6 - 5 * 4 + 3210
  = 9 + 8 * 7 - 6 - 5 * 4 + 3210 = 9 + 8 * 7 - 6 * 5 + 4 + 3210
  = 9 * 8 - 7 - 6 - 5 * 4 + 3210 = 9 * 8 - 7 - 6 * 5 + 4 + 3210
  = 9 * 8 - 7 * 6 + 5 + 4 + 3210 = 9 + 8 + 76 - 54 + 3210
  = - 9 - 87 * 6 + 54 / 3 * 210 = - 9 - 8 + 76 - 5 * 4 + 3210,
57² = 1² + 2² * 3² - 4² - 5² + 6² + 7² * 8² + 9² = 12² * 3² / 4² + 56² + 7² + 8² - 9²
  = 1² * 23² + 45² - 67² + 8² * 9² = - 1² * 2² - 3² - 4² + 5² + 6² + 7² * 8² + 9²
  = - 1² * 2² + 3² + 4² - 5² + 6² + 7² * 8² + 9² = - 1² * 2² + 3² - 45² + 6² + 7² + 8² * 9²
  = - 12² * 3² / 4² + 56² + 7² + 8² + 9² = - 1² * 2² + 34² - 56² + 7² + 8² * 9²
  = 9² + 8² * 7² + 6² + 5² - 4² - 3² - 2² */ 1² = 9² + 8² * 7² + 6² - 5² + 4² + 3² - 2² */ 1²
  = 9² + 8² * 7² + 6² - 5² - 4² + 3² * 2² + 1² = 98² / 7² - 6² * 5² - 4² + 3² * 21²
  = - 9² + 8² + 7² + 65² + 4² - 32² */ 1² = 9² + 8² * 7² + 6² / 5² * 4² + 3² - 2² / 10²
  = 9² * 8² * 7² * 6² / 54² + 3² + 2² + 10².

(1² + 2² + 3² + ... + 57²) = 63365, which consists of 3 kinds of digits.

(57² + 1) = (3² + 1)(18² + 1) = (7² + 1)(8² + 1)
    = (2² + 1)(3² + 1)(8² + 1) = (1² + 1)(2² + 1)(18² + 1),
(57² + 6) = (1² + 6)(3² + 6)(5² + 6),
(57² + 7)= (1² + 7)(20² + 7) = (2² + 7)(17² + 7).

(1 + 2 + ... + 9)(10 + 11 + ... + 17)(18 + 19 + ... + 57) = 2700².

57² = 3249 appears in the decimal expressions of π and e:
  π = 3.14159•••3249••• (from the 4122nd digit),
  e = 2.71828•••3249••• (from the 3037th digit).

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Page of Squares : First Upload January 26, 2004 ; Last Revised January 31, 2011
by Yoshio Mimura, Kobe, Japan