The smallest squares containing k 40's :
400 = 20²,   40401 = 201²,   404090404 = 20102²,
40409040400 = 201020²,   402404402400400 = 20060020².

40² is the third square which is the sum of 5 fourth powers : 2⁴ + 2⁴ + 2⁴ + 4⁴ + 6⁴.

254^k + 362^k + 394^k + 590^k are squares for k = 1,2,3 (40², 836², 18176²).

Komachi equations:
40² = 12 + 3 + 4 * 56 * 7 + 8 + 9 = - 1 + 2 / 3 * 4 * 567 + 89
  = - 1 + 2 + 3 * 45 * 6 + 789,
40² = 9 * 8 + 7 * 654 / 3 + 2 * 1 = 9 + 8 + 76 * 5 * 4 + 3 * 21
  = 98 + 76 * 5 * 4 + 3 - 21, and more 3 equations,
40² = 98 + 7 * 6 * 54 / 3 * 2 - 10,
40² = 9² * 8² + 7² - 6² - 5² * 4² * 3² + 2² - 1² = 9² * 8² - 7² * 6² + 5² - 43² + 2² * 1²
  = 9² * 8² - 7² * 6² + 5² - 43² + 2² / 1²,
40² = 12³ + 3³ + 4³ + 5³ + 6³ - 7³ + 8³ - 9³ = - 1³ - 2³ * 3³ + 4³ + 56³ / 7³ + 8³ + 9³.

(40² - 2) = (6² - 2)(7² - 2).

40² = (1² + 4)(2² + 4)(6² + 4) = (2² + 4)(14² + 4).

40² = 4³ + 8³ + 8³ + 8³.

(1)(2 + 3 + ... + 22)(23 + 24 + ... + 40) = 378²,
(1 + 2 + ... + 4)(5 + 6 + ... + 40) = 90²,
(1 + 2 + ... + 5)(6 + 7 + ... + 15)(16 + 17 + ... + 40) = 1050²,
(1 + 2 + ... + 7)(8 + 9 + ... + 13)(14 + 15 + ... + 40) = 1134²,
(1 + 2 + ... + 7)(8 + 9 + ... + 22)(23 + 24 + ... + 40) = 1890²,
(1 + 2 + ... + 7)(8 + 9 + ... + 34)(35 + 36 + ... + 40) = 1890²,
(1 + 2 + ... + 8)(9 + 10 + ... + 40) = 168²,
(1 + 2 + ... + 14)(15)(16 + 17 + ... + 40) = 1050²,
(1 + 2 + ... + 24)(25 + 26 + ... + 39)(40) = 2400².

(1² + 2² + ... + 5²)(6²)(7²)(8² + 9² + ... + 40²) = 46200².

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Page of Squares : First Upload December 15, 2003 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan