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:

1² = 1

2² = 4

3² = 9

4² = 16

5² = 25

6² = 36

7² = 49

8² = 64

9² = 81

10² = 100

11² = 121

12² = 144

13² = 169

14² = 196

15² = 225

16² = 256

17² = 289

18² = 324

19² = 361

20² = 400

21² = 441

22² = 484

23² = 529

24² = 576

25² = 625

26² = 676

27² = 729

28² = 784

29² = 841

30² = 900

31² = 961

32² = 1024

33² = 1089

34² = 1156

35² = 1225

36² = 1296

37² = 1369

38² = 1444

39² = 1521

40² = 1600

41² = 1681

42² = 1764

43² = 1849

44² = 1936

45² = 2025

46² = 2116

47² = 2209

48² = 2304

49² = 2401

50² = 2500

n² equals to the sum of the first n odd numbers ([math]\displaystyle{ 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, 21²:20 and 22, and then multiply the two numbers together and add the square of the distance from the mean: 22x20=440+1²=441. This works because of the identity

(x-y)(x+y)=x²–y²

known as the difference of two squares. Thus (21–1)(21+1)=21²–1²=440, if you work backwards.

External links

• Java applet that decomposes a natural number into a sum of up to four squares
• Square number