The smallest squares containingk 74's :
7744 = 88²,   7447441 = 2729²,   7497481744 = 86588²,
7174474747441 = 2678521²,   7479774747747481 = 86485691².

74 is the sum of m squares for m = 2, 3, ..., 59.

74² = 5476, a zigzag square with different digits.

74² is the third square which is the sum of 7 fourth powers in just 2 ways :
      [1,2,2,2,5,7,7], [3,3,3,4,4,5,8].
74² is the first square which is the sum of 8 fourth powers in just 3 ways :
      [1,2,2,2,3,5,5,8], [1,2,4,4,5,5,6,7], [1,3,4,5,5,6,6,6].

74⁷ = 12151280273024, 1² + 21² + 51² + 28² + 027² + 30² + 2² + 4² = 74².
74⁸ = 899194740203776, 8 + 9 + 919 + 4 + 740 + 20 + 3776 = 74².

10^k + 74^k + 98^k + 142^k are squares for k = 1,2,3 (18², 188², 2052²).

Komachi Fraction : 74² = 3104892/567.

Komachi equations:
74² = 123 * 45 + 6 + 7 - 8 * 9 = 123 * 45 + 6 - 7 * 8 - 9
  = 123 * 45 - 6 * 7 - 8 - 9 = - 123 * 4 + 5 + 67 * 89
  = 9 * 8 * 76 + 5 + 4 - 3 - 2 */ 1 = 9 * 8 * 76 + 5 + 4 - 3 * 2 + 1
  = 9 * 8 * 76 - 5 + 4 + 3 + 2 */ 1 = 9 * 8 * 76 - 5 + 4 + 3 * 2 - 1
  = 9 * 8 * 76 - 5 + 4 * 3 - 2 - 1 = 9 + 8 + 7 * 65 * 4 * 3 - 2 + 1
  = 9 - 8 + 7 * 6 + 5432 + 1 = 9 * 8 * 76 - 5 - 4 * 3 + 21
  = 9 * 8 * 76 - 5 * 4 + 3 + 21 = 9 * 8 * 76 + 5 + 4 + 3 + 2 - 10
  = 9 * 8 * 76 + 5 * 4 - 3 * 2 - 10 = 9 * 8 * 76 - 5 + 4 - 3 - 2 + 10
  = 9 * 8 * 76 + 5 / 4 * 32 / 10,
74² = 9² * 8² - 7² + 6² + 5² * 4² + 3² - 2² - 10² = 98² + 7² - 65² + 4² * 3² + 2² - 10²
  = 98² + 7² - 65² - 4² - 3² * 2² + 10².

(1² + 2² + 3² + ... + 74²) = 137825, the first 6-digit integer which consists of different digits.

(74² + 2) = (8² + 2)(9² + 2) = (2² + 2)(3² + 2)(9² + 2),
(74² - 2) = (3² - 2)(28² - 2) = (2² - 2)(5² - 2)(11² - 2) = (3² - 2)(5² - 2)(6² - 2),
(74² + 5) = (4² + 5)(16² + 5).

74² + 75² + 76² + ... + 36554² = 4035066².

1² + 2² + 3² + 4² + ... + 74² = 137825, which consists of different digits.

(1 + 2)(3 + 4 + ... + 24)(25 + 26 + ... + 74) = 1485²,
(1 + 2 + 3 + 4 + 5)(6 + 7 + ... + 69)(70 + 71 + ... + 74) = 3600²,
(1 + 2 + ... + 10)(11 + 12 + ... + 24)(25 + 26 + ... + 74) = 5775².

(1² + 2² + ... + 24²)(25² + 26²... + 73²)(74²) = 1849260².

74² = 5476 appears in the decimal expressions of π and e:
  π = 3.14159•••5476••• (from the 19726th digit),
  e = 2.71828•••5476••• (from the 7557th digit).

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Page of Squares : First Upload February 16, 2004 ; Last Revised January 31, 2011
by Yoshio Mimura, Kobe, Japan