The smallest hypotenuse of the integral Pythagorean triangles.

The smallest integer which is the sum of two distinct squares : 5 = 1² + 2².

The smallest squares containing k 5's are :
25 = 5²,   5625 = 75²,   55225 = 235²,   555025 = 745²,
505575225 = 22485²,   525555625 = 22925²,   55255554225 = 235065²
255555525625 = 505525²,   5552545555456 = 2356384².

The squares which begin with 5 and end in 5 are
5625 = 75²,   50625 = 225²,   55225 = 235²,   511225 = 715²,   525625 = 725²,...

Every integer n is the sum of 5 nonzero squares unless n = 1, 2, 3, 4, 6, 7, 9, 10, 12, 15, 18, 33.

5 is the third number which is not the sum of a square and a prime (They are 1, 2, 5, 13, ...).

(1 + 2) x (3 + 4 + 5) = 6²,  (1 + 2) x (3) x (4 + 5) = 9²,  (1) x (2 + 3) x (4) x (5) = 10².

(1²) x (2²) x (3² + 4²) x (5²) = 50².

1² + 2² + 3² + 4² + 5² = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10) = 55.

1³ + 2³ + 3³ + 4³ + 5³ = (1 + 2 + 3 + 4 + 5)².

5² = 3² + 4², the first square which is the sum of 2 squares.
5² is the first square which is the sum of 7 squares in just 2 ways.
5² is the third square which is the sum of 4 squares: 5² = 1² + 2² + 2² +4².
5² is the second square which is the sum of 4 cubs: 5² = 1³ + 2³ + 2³ + 2³.

5² is the only square that is 2 less than a cube, 3³.

5² = 25, the second square with an increasing sequence of digits.

(5² - 1) = (2² - 1)(3² - 1),   (5² + 3) = (1² + 3)(2² + 3),
(5² - 7) = (3² - 7)(4² - 7).

5⁸ = 390625, 3 + 9 + 06 + 2 + 5 = 5².

4! + 1 = 5²,   5! + 1 = 11²,   4! + 5! = 12².

1⁵ + 2⁵ + 3⁵ + ... + n⁵ = n²(n + 1)²(2n² + 2n - 1),
where n_k+2 = 10n_k+1 - n_k + 4, k = 0,1,2,..., n₀ = 1, n₁ = 13
Examples: 1⁵ + 2⁵ + 3⁵ + ... + 13⁵ = 1001²,   1⁵ + 2⁵ + 3⁵ + ... + 133⁵ = 971299².

The sum of five consecutive odd cubes is 35².   1³ + 3³ + 5³ + 7³ + 9³ = 35²

The alternating sum of five consecutive cubes is 9².   5³ - 4³ + 3³ - 2³ + 1³ = 9²

Komachi Fractions: 5² = 172350/6894 = 217350/8694 = 237150/9486,
Komachi Fractions: (3/5)² = 29403/81675,   (4/5)² = 8496/13275
(5/6)² = 27864/19350,   (5/36)² = 6075/314928,   (5/26)² = 675/1924803,
(5/336)² = 675/3048192,   (5/546)² = 675/8049132,   (5/632)² = 450/7189632,
(5/664)² = 450/7936128,   (5/714)² = 450/9176328.

Komachi equations:
5² = 1 + 2 + 3 - 4 + 5 - 6 + 7 + 8 + 9 = 1 + 2 - 3 + 4 + 5 + 6 - 7 + 8 + 9
  = 1 + 2 * 3 + 4 * 5 + 6 - 7 + 8 - 9, and more 255 equations,
5² = 9 + 8 + 7 + 6 + 5 - 4 - 3 - 2 - 1 = 9 + 8 + 7 + 6 + 5 - 4 - 3 * 2 * 1
  = 9 + 8 + 7 + 6 + 5 - 4 * 3 + 2 * 1, and more 277 equations,
5² = 9 + 8 + 7 + 6 - 5 + 4 + 3 * 2 - 10 = 9 + 8 + 7 + 6 - 5 + 4 * 3 - 2 - 10
  = 9 + 8 + 7 + 6 - 5 - 4 - 3 * 2 + 10, and more 189 equations,
5² = 1² + 2² + 3² + 4² + 5² + 6² - 7² + 8² - 9² = 1² - 2² - 3² + 4² + 5² - 6² + 7² + 8² - 9²
  = 1² * 2² * 3² - 4² - 5² - 6² + 7² - 8² + 9², and more 8 equations,
5² = 9² - 8² / 7² * 6² - 5² + 4² + 3² / 21² = 9² + 8² - 7² - 6² - 5² - 4² + 3² - 2² + 1²
  = 9² - 8² + 7² + 6² - 5² - 4² - 3² * 2² * 1², and more 10 equations,
5² = - 9² + 8² - 7² - 6² - 5² + 4² + 3² * 2² + 10² = - 9² + 8² - 7² + 6² - 5² + 4² - 3² * 2² + 10².

5² = 25 appears in the decimal expressions of π and e:
  π = 3.14159•••25••• (from the 89th digit), 25 is the 5th 2-digit square in the expression of π,
  e = 2.71828•••25••• (from the 92th digit), 25 is the 4th 2-digit square in the expression of e.

---

Page of Squares : First Upload August 17, 2003 ; Last Revised May 18, 2010
by Yoshio Mimura, Kobe, Japan