# Square number

From Wikipedia
In mathematics, a **square number**, sometimes also called a **perfect square**, is an integer that can be written as the square of some other integer. (In other words, a number whose square root is an integer.) So for example, 9 is a square number since it can be written as 3 × 3. If rational numbers are included, then the ratio of two square integers is also a square number (e.g. 2/3 × 2/3 = 4/9).

The number *m* is a square number if and only if one can arrange *m* points in a square.

The first 50 squares A000290 are:

- 1
^{2}= 1 - 2
^{2}= 4 - 3
^{2}= 9 - 4
^{2}= 16 - 5
^{2}= 25 - 6
^{2}= 36 - 7
^{2}= 49 - 8
^{2}= 64 - 9
^{2}= 81 - 10
^{2}= 100

- 11
^{2}= 121 - 12
^{2}= 144 - 13
^{2}= 169 - 14
^{2}= 196 - 15
^{2}= 225 - 16
^{2}= 256 - 17
^{2}= 289 - 18
^{2}= 324 - 19
^{2}= 361 - 20
^{2}= 400

- 21
^{2}= 441 - 22
^{2}= 484 - 23
^{2}= 529 - 24
^{2}= 576 - 25
^{2}= 625 - 26
^{2}= 676 - 27
^{2}= 729 - 28
^{2}= 784 - 29
^{2}= 841 - 30
^{2}= 900

- 31
^{2}= 961 - 32
^{2}= 1024 - 33
^{2}= 1089 - 34
^{2}= 1156 - 35
^{2}= 1225 - 36
^{2}= 1296 - 37
^{2}= 1369 - 38
^{2}= 1444 - 39
^{2}= 1521 - 40
^{2}= 1600

- 41
^{2}= 1681 - 42
^{2}= 1764 - 43
^{2}= 1849 - 44
^{2}= 1936 - 45
^{2}= 2025 - 46
^{2}= 2116 - 47
^{2}= 2209 - 48
^{2}= 2304 - 49
^{2}= 2401 - 50
^{2}= 2500

n^{2} equals to the sum of the first *n* odd numbers ([math]n^2 = 2(n-1)^2-(n-2)^2+2[/math]). A square number is also the sum of two consecutive triangular numbers. The sum of two consecutive square numbers is a centered square number. Every odd square is also a centered octagonal number.

An easy way to find square numbers is to find two numbers which have a mean of it, 21^{2}:20 and 22, and then multiply the two numbers together and add the square of the distance from the mean: 22x20=440+1^{2}=441. This works because of the identity

- (x-y)(x+y)=x
^{2}–y^{2}

known as the difference of two squares. Thus (21–1)(21+1)=21^{2}–1^{2}=440, if you work backwards.