Some numbers can be expressed as the difference of two perfect squares:
$20 = 6^2 - 4^2$
$21 = 5^2 - 2^2$
$36 = 6^2 - 0^2$
$165 = 13^2-2^2$
How many of the numbers from $1$ to $30$ can you express as the difference of two perfect squares?
Here are some questions you might like to consider:
What is special about the difference between squares of consecutive numbers? Why?
What about the difference between the squares of two numbers which differ by $2$? By $3$? By $4$...?
When is the difference between two square numbers odd?
And when is it even?
What do you notice about the numbers you CANNOT express as the difference of two perfect squares?
Can you prove any of your findings?