The smallest squares containingk 62's :
625 = 25²1625625 = 1275²,   626250625 = 25025²,
16262625625 = 127525²,   336262622626225 = 18337465².

The second integer which is the sum of a square and a prime in 4 ways :
  1² + 61, 3² + 53, 5² + 37, 7² + 13.

1 / 62 = 0.016129..., 16129 = 127².

62² is the 4th square which is the sum of 8 fourth powers : [2,2,2,5,5,5,5,6].

62^k + 73^k + 266^k + 440^k are squares for k = 1,2,3 (29², 523², 10229²).

Komachi Fraction : (9/62)² = 3645/172980.

Komachi equations:
62² = - 9 * 8 - 7 + 654 * 3 * 2 - 1 = 9 - 8 - 7 + 6 * 5 * 4 * 32 + 10
62² = 98 / 7 + 6 * 5 * 4 * 32 - 10,
62² = 1² + 23² - 4² + 56² + 7² + 8² + 9² = - 12² * 3² + 4² + 5² - 6² - 7² + 8² * 9²
  = - 1² + 2² + 3² + 4² * 5² - 67² + 89² = 9² + 8² + 7² + 65² - 4² * 3² * 2² + 1²
  = 98² / 7² + 65² - 4² * 3² * 2² - 1² = 9² * 8² + 7² + 6² - 5² * 4² - 32² - 1²
  = 98² - 76² + 5² / 4² * 32² / 10² = 9² - 8² - 7² + 6² + 54² + 32² - 10²
  = - 9² + 8² - 7² * 6² + 5² / 4² * 3² * 2² * 10² = - 9² + 8² * 7² + 6² * 5² - 4² + 3² - 2² - 10².

62² = 1³ + 5³ + 7³ + 15³.

(62² - 4) = (4² - 4)(18² - 4),
(62² + 6) = (4² + 6)(13² + 6) = (7² + 6)(8² + 6) = (1² + 6)(2² + 6)(7² + 6).

1² + 2² + 3² + 4² + ... + 62² = 81375, which consists of different digits.

(1 + 2)(3 + 4 + ... + 17)(18 + 19 + ... + 62) = 900²,
(1 + 2 + 3)(4 + 5 + ... + 12)(13 + 14 + ... + 62) = 900²,
(1 + 2 + 3)(4 + 5 + ... + 17)(18 + 19 + ... + 62) = 1260²,
(1 + 2 + ... + 6)(7 + 8 + ... + 42)(43 + 44 + ... + 62) = 4410²,
(1 + 15 + ... + 16)(17)(18 + 19 + ... + 62) = 2040²,
(1 + 2 + ... + 17)(18 + 19 + ... + 22)(23 + 24 + ... + 62) = 5100²,
(1 + 2 + ... + 24)(25 + 26 + ... + 33)(34 + 35 + ... + 62) = 10440²,
(1 + 2 + ... + 36)(37)(38 + 39 + ... + 62) = 5550².

(1² + 2² + 3² + ... + 62²) = 81375, which consists of different digits.

(1²)(2² + 3² + ... + 6²)(7² + 8² + ... + 61²)(62²) = 163680²,
(1² + 2² + ... + 7²)(8² + ... + 24²)(25² + ... + 39²)(40² + ... + 62²) = 25180400²,
(1² + 2² + ... + 11²)(12² + 13² + ... + 16²)(17² + 18² + ... + 39²)(40² + 41² + ... + 62²) = 24090660²,
(1² + 2² + ... + 25²)(26² + 27² + ... + 50²)(51² + 52² + ... + 61²)(62²) = 165794200².

62² = 3844 appears in the decimal expressions of π and e:
  π = 3.14159•••3844••• (from the 123rd digit),
  (3136 is the first 4-digit square in the expression of π,)
  e = 2.71828•••3844••• (from the 4563rd digit).

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Page of Squares : First Upload February 2, 2004 ; Last Revised January 31, 2011
by Yoshio Mimura, Kobe, Japan