We are defining square factorization as representation a positive natural number as sum of squares of different positive, integer numbers. For example $5 = 1^2 +2^2$ and $5$ has no more representation. But one number can possess more representations, eg. $30$.
$$30 = 1^2 + 2^2 + 5^2 = 1^2 +2^2 + 3^2 + 4^2$$
Sometimes $n$ has no representation, eg. $8$. If square factorization for $n$ is impossible, we call $n$ indecomposable, so $8$ is indecomposable. Because it is a significant part of other proof I began to wonder, how can we know there is finite number of indecomposable numbers. Of course, it is well know fact, but I have still problem with proving that.
How to show that any number greater than $128$ can be written as sum of distinct squares — for all $n > 128$ square factorization exist?