Square number

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In mathematics, a square number, sometimes also called a perfect square, is a positive integer that can be written as the square of some other integer. So for example, 9 is a square number since it can be written as 3×3. By convention, the first square number is 1. The number m is a square number if and only if one can arrange m points in a square:

+ x

x +             x x

+ + x x

x x +           x x x

x x +           x x x

+ + + x x x

16:

x x x +         x x x x

x x x +         x x x x

x x x +         x x x x

+ + + + x x x x

25:

x x x x +       x x x x x

x x x x +       x x x x x

x x x x +       x x x x x

x x x x +       x x x x x

+ + + + + x x x x x

The first 50 squares are:

   1    4    9   16   25    36   49   64   81  100

 121  144  169  196  225   256  289  324  361  400

 441  484  529  576  625   676  729  784  841  900

 961 1024 1089 1156 1225  1296 1369 1444 1521 1600

1681 1764 1849 1936 2025 2116 2209 2304 2401 2500

The formula for the nth square number is n². This is also equal to the sum of the first n odd numbers, as can be seen in the above pictures, where a square results from the previous one by adding an odd number of points (marked as '+'). So for example, 5² = 25 = 1 + 3 + 5 + 7 + 9.

A square number is also the sum of two consecutive triangular numbers.

Lagrange's four-square theorem states that any positive integer can be written as the sum of at most 4 perfect squares. 3 squares are not sufficient for numbers of the form 4^k(8l + 7). A positive integer can be represented as a sum of two squares precisely if its prime factorization contains no odd powers of primes of the form 4k+3. This is generalized by Waring's problem.

A positive integer that has no perfect square divisors except 1 is called square-free.

See also:

Triangular number
Polygonal number
triangular square number
Euler's four-square identity
Automorphic number
http://sources.wikipedia.org/wiki/Square_Numbers_1-1000