The smallest squares containingk 77's :
5776 = 76²,   17757796 = 4214²,   39077777761 = 197681²,
7732770777796 = 2780786²,   5777777776773025 = 76011695².

77 = (1² + 2² + 3² + ... + 10²) / (1² + 2²).

77² = 5929, a zigzag square.

77² = 5929, 5 + 9 + 2 + 9 = 5²,
77² = 5929, 59³ + 2³ + 9³ = 454².

77² = 18² + 19² + 20² + 21² + ... + 28².

5² + 29² + 53² + 77² = 98².

(1² + 2² + 3² + ... + 77²) = 155155, the unique 6-digit integer which consists of odd digits.

77² = 5929, 59 + 2 * 9 = 77.

77² = 1! + 4! + 4! + 5! + 6! + 7!.

77⁸ = 1235736291547681, 1 + 2 + 3 + 5736 + 2 + 9 + 15 + 4 + 76 + 81 = 77²,
  = 1 + 2 + 3 + 5736 + 2 + 9 + 154 + 7 + 6 + 8 + 1 = 77² (and more 17 equations).

154^k + 490^k + 1708^k + 3577^k are squares for k = 1,2,3 (77², 3997², 225547²).
294^k + 1288^k + 1617^k + 2730^k are squares for k = 1,2,3 (77², 3437², 163513²).
385^k + 1232^k + 2156^k + 2156^k are squares for k = 1,2,3 (77², 3311², 148225²).
462^k + 1034^k + 2200^k + 2233^k are squares for k = 1,2,3 (77², 3333², 151613²).
924^k + 1078^k + 1617^k + 2310^k are squares for k = 1,2,3 (77², 3157², 136367²).

Komachi equations:
77² = - 1 - 2 - 34 * 5 + 678 * 9 = 987 * 6 + 5 + 4 - 3 + 2 - 1
  = 987 * 6 + 5 - 4 + 3 + 2 + 1 = 987 * 6 + 5 - 4 + 3 * 2 */ 1
  = 9 * 8 * 7 - 6 + 5432 - 1 = 987 * 6 - 5 - 4 + 3 * 2 + 10,
77² = - 12² - 3² * 4² * 5² + 67² + 8² * 9² = - 9² + 8² * 7² - 6² + 54² - 3² + 2² - 1²
  = - 98² + 7² - 6² * 5² + 4² * 32² */ 1² = 9² - 8² + 76² + 5² + 4² - 3² + 2² + 10²,
77² = 9³ - 8³ - 7³ + 6³ + 54³ / 3³ + 2³ - 1³.

(77² + 5) = (8² + 5)(9² + 5).

385² = 28² + 29² + 30² + ... + 77².

1² + 2² + 3² + 4² + ... + 77² = 155155, which consists of odd digits 1 and 5.

(1 + 2 + ... + 45)(46 + 47 + ... + 69)(70 + 71 + ... + 77) = 28980².

(1³ + 2³ + ... + 7³)(8³ + 9³ + ... + 13³)(14³ + 15³ + ... + 77³) = 7277088²,
(1² + 2² + ... + 5²)(6² + 7²)(8² + ... + 22²)(23² + ... + 77²) = 1608200²,
(1² + 2² + ... + 8²)(9² + 10² + ... + 17²)(18² + 19² + ... + 61²)(62² + 63² + ... + 77²) = 43547064²,
(1² + 2² + ... + 10²)(11² + 12² + ... + 38²)(39² + 40² + ... + 45²)(46² + 47² + ... + 77²) = 104824720²,
(1² + 2² + ... + 22²)(23² + 24² + ... + 27²)(28² + 29² + ... + 54²)(55² + 56² + ... + 77²) = 237946500².

77² = 5929 appears in the decimal expressions of π and e:
  π = 3.14159•••5929••• (from the 23041st digit),
  e = 2.71828•••5929••• (from the 4374th digit).

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Page of Squares : First Upload February 23, 2004 ; Last Revised January 31, 2011
by Yoshio Mimura, Kobe, Japan