I was looking for numbers who can be expressed as sum of exactly four squares and not less. And I think I have found them. They are all the integers of the form
I have no idea how to prove this statement and I wonder if ALL the numbers which need four squares are this kind of numbers.
Thanks to the comments and the answer the statement can be more precise
A number can be expressed as the sum of four squares and not less if and only if it has the form