The smallest squares containingk 75's :
7569 = 87²,   18757561 = 4331²,   7575787521 = 87039²,
75753754756 = 275234²,   6757757547547561 = 82205581².

The third integer which is the sum of 5 squares : 1² + 2² + 3² + 5² + 6².

75² = 5625, a zigzag square.

75² = 5625, 5 + 6 + 25 = 6²,
75² = 5625, 56 + 25 = 9².

75² = 5625, 5 * 6 / 2 * 5 = 75.

990^k + 1020^k + 1785^k + 1830^k are squares for k = 1,2,3 (75², 2925², 117675²).

Komachi Fraction : 75² = 4910625/873.

Komachi equations:
75² = 1234 * 5 - 67 * 8 - 9 = 123 * 45 - 6 + 7 + 89
  = 9 * 8 * 76 - 54 - 3 + 210 = - 9 + 87 * 65 - 4 + 3 - 2 * 10,
75² = - 1² - 2² - 3² - 4² * 5² + 6² + 78² - 9² = 9² + 87² - 6² * 5² / 4² * 3² * 2² */ 1²
  = 9² - 8² - 7² * 6² - 5² + 43² * 2² + 1² = - 9² - 8² + 76² - 5² - 4² + 3² * 2² - 1²
  = - 9² + 87² - 6² + 5² - 43² - 2² + 1² = 9² * 8² + 7² - 6² + 54² / 3² + 2² + 10².

75 and 76 are consecutive integers having square factors (the 6th case).

(75² + 3) = (8² + 3)(9² + 3) = (2² + 3)(3² + 3)(8² + 3),
(75² + 7) = (5² + 7)(13² + 7) = (2² + 7)(3² + 7)(5² + 7).

(1)(2 + 3 + ... + 12)(13 + ... + 75) = 462².

(1²)(2² + 3² + ... + 5²)(6² + ... + 39²)(40² + ... + 75²) = 368730²,
(1² + 2² + 3² + 4²)(5² + 6² + 7²)(8² + 9² + ... + 29²)(30² + 31² + ... + 75²) = 1935450².

75⁷ = 13348388671875, 1 + 3348 + 388 + 6 + 7 + 1875 = 75²,
75⁷ = 13348388671875, 13 + 3 + 4838 + 8 + 671 + 87 + 5 = 75²,
75⁹ = 75084686279296875,
    7+508+4686+27+9+296+87+5 = 7+5084+6+8+62+79+296+8+75
  = 7+5084+68+6+2+79+296+8+75 = 7+5084+68+6+279+2+96+8+75
  = 7+5084+68+62+7+9+296+87+5 = 750+8+4686+2+79+2+9+6+8+75 = 75².

75² = 5625 appears in the decimal expressions of π and e:
  π = 3.14159•••5625••• (from the 3085th digit),
  e = 2.71828•••5625••• (from the 1962nd digit).

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Page of Squares : First Upload February 23, 2004 ; Last Revised December 14, 2013
by Yoshio Mimura, Kobe, Japan