1:
+ x
4:
x + x x + + x x
9:
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 formula for the nth square number is n^{2}. 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, 5^{2} = 25 = 1 + 3 + 5 + 7 + 9.
Lagrange's foursquare 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). This is generalized by Waring's problem.
A positive integer that has no perfect square divisors except 1 is called squarefree.
See also:
Search Encyclopedia

Featured Article
