Is it known whether any sufficiently large positive integer can be written as the sum of four different squares? I know that every positive integer can be written as the sum of four not necessarily different squares.
Furthermore, OEIS has a sequence of numbers that can be written as the sum of any number of different squares: A003995. The largest integer not in the sequence is 128.
Is there anything known about this? If so, can someone give a reference to the proof or give a sketch how to proof it?
Update (Thanks, Mark Bennet!) :
From Jacobi's four-square theorem it follows that there are only 24 ways to write a power of two as the sum of four squares. However, if all those squares are different, there are at least 16*24>24 ways, since there are 16 ways to choose sign and 24 ways to choose order. Therefore a power of two cannot be written as the sum of four different squares. Is this correct?
If so, I want to restate my question. Is it, for any integer $n$, known whether any sufficiently large positive integer can be written as the sum of $n$ different squares?