The smallest squares containing k 50's :
2500 = 50²,   5053504 = 2248²,   508502500 = 22550²,
65050502500 = 255050²,   150505050802500 = 12268050².

50 is the sum of m squares for m = 2, 3, ..., 36.

50 is the sum of m distinct squares for m = 2, 3, 4:
  50 = 1² + 7² = 3² + 4² + 5² = 1² + 2² + 3² + 6².

The first integer which is the sum of 12 squares in just 6 ways (see 49).

The first integer which is the sum of two squares in two ways: 50 = 1² + 7² = 5² + 5².

50² = (1)(2)(3 + 4 + 5 + 6 + 7)(8 + 9 + ... + 12),
50² = (1²)(2²)(3² + 4²)(5²).

50² is the 5th square which is the sum of 5 fourth powers : 1⁴ + 1⁴ + 2⁴ + 3⁴ + 7⁴.

50² = (2² + 1)(3² + 1)(7² + 1) = (4² + 4)(11² + 4).

50² = 5⁴ + 5⁴ + 5⁴ + 5⁴.

50² = (2² + 1)(3² + 1)(7² + 1).

10^k + 50^k + 190^k + 320^k + 330^k are squares for k = 1,2,3,4. (30², 500², 8700², 153800²)

In the set {50,2450,4439,8786} the sum of any two members is a square.

26^k + 338^k + 910^k + 1226^k are squares for k = 1,2,3 (50², 1564², 51332²).
106^k + 266^k + 814^k + 1314^k are squares for k = 1,2,3 (50², 1572², 53180²).
146^k + 154^k + 654^k + 1546^k are squares for k = 1,2,3 (50², 1692², 63100²).
170^k + 370^k + 830^k + 1130^k are squares for k = 1,2,3 (50², 1460², 45500²).

Komachi equations:
50² = - 1 - 2 + 3 * 4 * 5 * 6 * 7 - 8 - 9 = - 1 + 2345 + 67 + 89
  = 9 - 8 + 7 * 6 * 5 * 4 * 3 - 21 = - 9 - 8 + 7 * 6 * 5 * 4 * 3 - 2 - 1
  = - 9 * 8 - 7 + 6 * 5 * 43 * 2 - 1 = 9 + 8 - 7 - 6 * 5 + 4 * 3 * 210
  = 9 + 8 - 7 * 6 + 5 + 4 * 3 * 210,
50² = 1² - 23² - 4² + 5² - 6² + 7² * 8² - 9² = - 9² + 8² * 7² - 6² + 5² - 4² * 3² - 2² * 10²,
50² = 98³ / 7³ - 6³ + 5³ + 4³ - 3³ * 2³ - 1³.

(50² - 6) = (7² - 6)(8² - 6),   (50² + 8) = (6² + 8)(7² + 8).

50² = (1² + 2² + 3² + 4²)(1² + 2² + 3² + ... + 19²).

195² = 25² + 26² + 27² + ... + 50².

50² + 51² + 52² + ... + 15674² = 1133000²,
50² + 51² + 52² + ... + 171² = 1281².

(1 + 2 + ... + 6)(7 + 8 + ... + 12)(13 + 14 + ... + 50) = 1197²,
(1 + 2 + ... + 7)(8 + 9 + ... + 21)(22 + 23 + ... + 50) = 2436²,
(1 + 2 + ... + 20)(21 + 22 + ... + 29)(30 + 31 + ... + 50) = 6300²,
(1 + 2 + ... + 27)(28 + 29 + ... + 41)(42 + 43 + ... + 50) = 8694².

(1² + 2² + ... + 24²)(25² + 26² + ... + 50²) = 13650²,
(1² + 2² + ... + 8²)(9²)(10² + 11² + ... + 17²)(18² + 19² + ... + 50²) = 1009800².

50² = 2500 appears in the decimal expressions of π and e:
  π = 3.14159•••2500••• (from the 13374th digit),
  e = 2.71828•••2500••• (from the 13932nd digit).

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Page of Squares : First Upload January 19, 2004 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan