A perfect square is an integer than can be written as the square of another integer. For example:
• 25 is a perfect square, because it is the square of 5 (i.e., 5² = 25)
• 64 is a perfect square, because it is the square of 8 (i.e., 8² = 64)
Lagrange proved that any positive integer can be expressed as the sum of at most 4 positive perfect squares. For example:
1 = 1
2 = 1 + 1
3 = 1 + 1 + 1
4 = 4
5 = 4 + 1
6 = 4 + 1 + 1
7 = 4 + 1 + 1 + 1
8 = 4 + 4
9 = 9
10 = 9 + 1
11 = 9 + 1 + 1
12 = 9 + 1 + 1 + 1
13 = 9 + 4
14 = 9 + 4 + 1