The square of 7.

49 is the sum of m squares for m = 3, 4, ..., 35.

49 = 7², 4 = 2², 9 = 3².

The smallest squares containing k 49's :
49 = 7²,   499849 = 707²,   498494929 = 22327²,
849149349049 = 921493²,   404949496949449 = 20123357².

The squares which begin with 49 and end in 49 are
499849 = 707²,   49098049 = 7007²,   49603849 = 7043²,   49801249 = 7057²,
490312449 = 22143²,...

1 + 2 + 3 + ... + 49 = 35².

49² = 5³ + 6³ + 9³ + 11³.

The first square which is the sum of 3 distinct squares: 7² = 2² + 3² + 6²

The 3rd integer which is the sum of 8 squares in just 5 ways.
The 1st integer which is the sum of 9 squares in just 3 ways.
The 1st integer which is the sum of 10 squares in just 6 ways (see 48).
The 1st integer which is the sum of 11 squares in just 6 ways.

49² = 2401 is a zigzag square whose digits are distinct.

49² = 2401, 2³ + 40³ + 1³ = 253²,   49² = 2401, 24 + 0 + 1 = 5².

49² = 12² + 24² + 41² = 14² + 42² + 21².

49² - 1 = 5! + 5! + 6! + 6! + 6!

499849 = 707².

The alternating sum of the first 49 squares is a square: 1² - 2² + 3² - ... + 49² = 35².

49^k + 98^k + 170^k + 212^k are squares for k = 1,2,3 (23², 293², 3937²).

Komachi equations:
49² = 123 * 4 * 5 + 6 + 7 - 8 * 9 = 123 * 4 * 5 + 6 - 7 * 8 - 9
 = 123 * 4 * 5 - 6 * 7 - 8 - 9,
49² = - 98 + 7 * 6 * 5 * 4 * 3 - 21,
49² = - 9 * 8 - 7 * 6 - 5 + 4 * 3 * 210 = - 9 * 87 - 6 - 5 * 4 + 3210
49² = - 9 * 87 - 6 * 5 + 4 + 3210,
49² = 1² + 23² * 4² - 5² - 6² - 78² + 9² = 12² * 3² / 4² + 56² * 7² / 8² - 9²
  = 12² - 3² / 4² + 5² + 6² * 7² / 8² * 9²,
49² = 98³ / 7³ - 6³ / 54³ * 3³ * 21³,
49² = 98⁴ / 7⁴ * 6⁴ * 5⁴ / 4⁴ / 3⁴ * 2⁴ / 10⁴ = 9⁴ * 8⁴ + 7⁴ - 6⁴ * 5⁴ * 4⁴ * 3⁴ * 2⁴ / 10⁴
 = 9⁴ / 8⁴ * 7⁴ / 6⁴ / 5⁴ * 4⁴ / 3⁴ * 2⁴ * 10⁴, and more 10 equations.

(49² + 7) = (6² + 7)(7² + 7),   (49² - 7) = (7² - 7)(8² - 7).

(1)(2 + 3 + ... + 25)(26 + 27 + ... + 49) = 540²,
(1 + 2 + ... + 8)(9 + 10 + ... + 16)(17 + 18 + ... + 49) = 1980²,
(1 + 2 + ... + 8)(9 + 10 + ... + 25)(26 + 27 + ... + 49) = 3060².

(1² + 2² + ... + 24²)(25² + 26² + ... + 48²)(49²) = 624260²,
(1² + 2² + 3²)(4² + 5² + ... + 24²)(25²)(26² + 27² + ... + 49²) = 1221500²,
(1² + 2² + ... + 8²)(9² + 10² + ... + 17²)(18² + 19² + ... + 26²)(27² + 28² + ... + 49²) = 6981696²,
(1³ + 2³ + ... + 8³)(9³ + 10³ + ... + 20³)(21³ + 22³ + ... + 49³) = 8988840²,
(1³ + 2³ + ... + 48³)(49³) = 403368².

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Page of Squares : First Upload January 13, 2004 ; Last Revised August 4, 2013
by Yoshio Mimura, Kobe, Japan