Problem No. 12. Sum of squares

Find all representations of the number 730 as a sum of two squares? And as a sum of three squares?

Teacher tip

Suitable for implementing the elimination and reminder methods while practicing the square numbers up to $30^2$.

Solution

All possible representations of the number 730 as a sum of two squares are:

$$730 = 1^2 + 27^2,$$

$$730 = 17^2 + 21^2.$$

For the representations as a sum of three squares, we refer to the Legendre's three-square theorem which states that:

A natural number $n$ can be represented as the sum of three squares of integers if and only if $n$ is not of the form $n=4^a (8b+7)$ for some nonnegative integers $a$ and $b$.

Since the prime factorization of the number 730 is $730=2 \cdot 5 \cdot 73$, and none of it's prime factors is of the form $8b+7$, we conclude that there is a representation of the number 730 as a sum of three squares. All possible representations of the number 730 as a sum of three squares are:

$$730 = 8^2 + 15^2 + 21^2,$$

$$730 = 12^2 + 15^2 + 19^2.$$

Note that, there are

• 34 representations of the number 730 as a sum of four squares,
• 107 representations of the number 730 as a sum of five squares,
• 883 representations of the number 730 as a sum of six squares.

These last results are computer-aided obtained.

Additional question 

Is there a representation of the number 730 as a difference of two squares?