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:
1:
+ x4:
x + x x + + x x9:
x x + x x x x x + x x x + + + x x x16:
x x x + x x x x x x x + x x x x x x x + x x x x + + + + x x x x25:
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 xThe 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 4k(8l + 7). This is generalized by Waring's problem.
A positive integer that has no perfect square divisors except 1 is called square-free.
See also: