The smallest squares containing k 45's :
13456 = 116²,   1245456 = 1116²,   45454564 = 6742²,
1451745453456 = 1204884²,   4544534545664569 = 67413163².

45 is the sum of m squares for m = 2, 3, ..., 31.

The first integer which is the sum of 7 squares in just 5 ways:
  [1,1,1,1,1,2,6], [1,1,1,1,3,4,4], [1,1,1,2,2,3,5], [1,2,2,3,3,3,3], [2,2,2,2,2,3,4],
The second integer which is the sum of 10 squares in just 5 ways.

45 = (1² + 2²)(3²).

45² = 1³ + 8³ + 8³ + 10³ = 2³ + 7³ + 7³ + 11³.

45² = 10³ + 4⁵ + 1⁷.

45² = (2² - 1)(26² - 1).

The third Kaprekar number : 45² = 2025, 20 + 25 = 45. (Cf. 1, 9, 55, 99, ...)

21^k + 45^k + 201^k + 633^k are squares for k = 1,2,3 (30², 666², 16182²).
6^k + 408^k + 690^k + 921^k are squares for k = 1,2,3 (45², 1221², 34317²).
42^k + 129^k + 660^k + 1194^k are squares for k = 1,2,3 (45², 1371², 44631²).
48^k + 177^k + 186^k + 1614^k are squares for k = 1,2,3 (45², 1635², 64935²).
138^k + 417^k + 582^k + 888^k are squares for k = 1,2,3 (45², 1149², 31185²).
204^k + 294^k + 561^k + 966^k are squares for k = 1,2,3 (45², 1173², 33345²).

Komachi Fractions : (2/45)² = 1764/893025, (31/45)² = 17298/36450.

Komachi equations:
45² = 1 * 2 + 345 * 6 - 7 * 8 + 9 = 12 - 3 + 4 * 567 * 8 / 9,
45² = 9 + 8 * 7 + 654 * 3 - 2 * 1 = 9 + 8 * 7 + 654 * 3 - 2 / 1
  = 9 * 8 - 7 + 654 * 3 - 2 * 1 = 9 * 8 - 7 + 654 * 3 - 2 / 1,
45² = 98 - 7 + 6 * 54 * 3 * 2 - 10 = 9 + 8 * 7 / 6 * 5 * 432 / 10
  = - 9 + 8 + 7 / 6 * 54 * 32 + 10 = - 9 * 87 + 65 * 432 / 10,
45² = 1² + 2² - 3² + 45² + 6² - 7² - 8² + 9² = 1² - 2² + 3² * 4² - 5² + 6² * 7² + 8² + 9²
  = 1² - 2² + 3² + 4² * 5² + 6² * 7² - 8² - 9²,
45² = 9² + 8² + 7² * 6² - 5² + 4² * 3² - 2² + 1² = 9² + 8² - 7² + 6² + 54² - 32² + 1²,
45² = 1³ + 2³ + 3³ + 4³ + 5³ + 6³ + 7³ + 8³ + 9³,
45² = 9³ + 8³ + 7³ + 6³ + 5³ + 4³ + 3³ + 2³ + 1³.

(45² + 3) = (6² + 3)(7² + 3).

1² + 2² + 3² + 4² + ... + 45² = 31395, which consists of odd digits (the 2nd 5-digit sum).

45² = (8 + 9 + 10)² + (11 + 12 + 13)².

45² = (1 + 2)(3)(4 + 5 + ... + 21) = (1 + 2)(3)(4 + 5 + 6)(7 + 8)
    = (1 + 2)(3 + 4 + 5 + 6 + 7)(8 + 9 + 10).

45² = 1³ + 2³ + 3³ + 4³ + 5³ + 6³ + 7³ + 8³ + 9³.

(1 + 2)(3)(4 + 5 + 6)(7 + 8) = 45²,
(1 + 2)(3 + 4 + 5 + 6 + 7)(8 + 9 + 10) = 45²,
(1 + 2)(3)(4 + 5 + ... + 21) = 45².

(1 + 2)(3 + 4 + 5 + 6)(7 + 8 + ... + 45) = 234²,
(1 + 2)(3 + 4 + ... + 17)(18 + 19 + ... + 45) = 630²,
(1 + 2 + 3)(4 + 5 + ... + 17)(18 + 19 + ... + 45) = 882²,
(1 + 2 + ... + 9)(10 + 11 + ... + 20)(21 + 22 + ... + 45) = 2475²,
(1 + 2 + ... + 9)(10 + 11 + ... + 34)(35 + 36 + ... + 45) = 3300²,
(1 + 2 + ... + 14)(15 + 16 + ... + 35)(36 + 37 + ... + 45) = 4725²,
(1 + 2 + ... + 16)(17)(18 + 19 + ... + 45) = 1428².

(1³ + 2³ + ... + 9³)(10³ + 11³ + ... + 17³)(18³ + 19³ + ... + 45³) = 6735960²,
(1³ + 2³ + ... + 19³)(20³ + 21³ + ... + 29³)(30³ + 31³ + ... + 45³) = 69825000².

(1² + 2²... + 5²)(6²)(7² + 8² + ... + 12²)(13² + 14² + ... + 45²) = 184470²,
(1² + 2² + ... + 21²)(22²)(23² + 24² + ... + 34²)(35² + 36² + ... + 45²) = 16753660².

45² = 2025, 2 + 0 + 2 + 5 = 3².

45⁸ = 16815125390625, 1681 + 51 + 253 + 9 + 06 + 25 = 45²,
45⁹ = 756680642578125, 7 + 5 + 668 + 0642 + 578 + 125 = 45²,
    7 + 566 + 806 + 42 + 578 + 1 + 25 = 75 + 66 + 806 + 4 + 257 + 812 + 5
  = 756 + 6 + 8 + 06 + 425 + 7 + 812 + 5 = 756 + 6 + 806 + 425 + 7 + 8 + 12 + 5 = 45².

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Page of Squares : First Upload December 29, 2003 ; Last Revised November 30, 2013
by Yoshio Mimura, Kobe, Japan