The smallest squares containing k 47's :
4761 = 69²,   474721 = 689²,   4747347801 = 68901²,
47474744769 = 217887²,   1474747474758481 = 38402441².

47 is the sum of m squares for m = 3, 4, ..., 33.

The second integer which is the sum of 8 squares in just 5 ways.
The first integer which is the sum of 9 squares in just 5 ways (see 45).

The third number which is the sum of a square and a prime in 3 ways : 2² + 43, 4² + 31, 6² + 11.

47² is the fourth square which is the sum of 5 fourth powers : 2⁴ + 2⁴ + 4⁴ + 5⁴ + 6⁴.

47² - 1 = 4! + 4! + 6! + 6! + 6!

47⁸ = 23811286661761, 238 + 1 + 1286 + 6 + 617 + 61 = 47²,
    238 + 112 + 86 + 6 + 6 + 1761 = 238 + 1128 + 666 + 176 + 1 = 47².

Komachi Fractions : 47² = 2107386/954, (8/47)² = 1728/59642,
  (9/47)² = 1458/39762 = 2187/59643.

Komachi equations:
47² = - 9 * 8 + 76 * 5 * 4 * 3 / 2 + 1,
47² = 9 - 8 + 7 * 6 * 54 - 3 * 2 * 10 = 9 + 8 + 7 * 6 + 5 * 432 - 10
  = 9 + 8 * 7 - 6 + 5 * 432 - 10, and mor 4 equations,
47² = 9² - 8² * 7² + 65² + 4² + 32² - 1²,
47² = - 1⁴ + 2⁴ + 3⁴ + 4⁴ + 5⁴ + 6⁴ + 7⁴ + 8⁴ - 9⁴,
47² = - 9⁴ + 8⁴ + 7⁴ + 6⁴ + 5⁴ + 4⁴ + 3⁴ + 2⁴ - 1⁴.

(47² + 1) = (2² + 1)(21² + 1) = (1² + 1)(4² + 1)(8² + 1) = (2² + 1)(4² + 1)(5² + 1),
(47² + 5) = (6² + 5)(7² + 5) = (1² + 5)(2² + 5)(6² + 5),
(47² - 9) = (7² - 9)(8² - 9).

(1 + 2 + 3 + 4 + 5)(6 + 7 + ... + 15)(16 + 17 + ... + 47) = 1260²,
(1 + 2 + ... + 8)(9 + 10 + 11 + 12)(13 + 14 + ... + 47) = 1260²,
(1 + 2 + ... + 14)(15)(16 + 17 + ... + 47) = 1260².

1² + 2² + 3² + 4² + ... + 47² = 35720, which consists of different digits.

(1² + 2² + 3² + 4²)(5² + 6² + ... + 18²)(19² + 20² + ... + 25²)(26² + 27² + ... + 47²) = 2536380²,
(1² + 2² + ... + 9²)(10² + 11² + ... + 15²)(16² + 17² + ... + 37²)(38² + 39² + ... + 47²) = 8981775².

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Page of Squares : First Upload January 5, 2004 ; Last Revised May 21, 2010
by Yoshio Mimura, Kobe, Japan