# The set of numbers that are not the sum of three non-zero squares

In thinking about this question sum of 3 squares that has now been marked as a duplicate

I wrote

"If the numbers $(8k+7)$, $1$, $2$, $5$, $10$, $13$, $25$, $37$, $58$, $85$ and $130$ are all the primitives that that cannot be written as the sum of $3$ [non-zero squares], then the full list of numbers that cannot be written as the sum of three [non-zero squares] squares is $(8k+7)\times4^m$, $1\times4^m$, $2\times4^m$, $5\times4^m$, $10\times4^m$, $13\times4^m$, $25\times4^m$, $37\times4^m$, $58\times4^m$, $85\times4^m$ and $130\times4^m$ "

Does this second list represent the entire set of numbers that cannot be written as the sum of three non-zero squares?

If this is not the entire set can it at least be shown that the number of primitives not of the form $(8k+7)$ is finite.

[Note: I have not proved that $4^m$ is the only such multiplier of primitives so the above statement in quotes is a conjecture.]

• We can write $1\times 4^m$ as $0^2+0^2+(2^m)^2$. – mathlove Dec 11 '17 at 4:29
• @mathlove It looks like this list is intended to be integers that are not the sum of $3$ non-zero squares. – Erick Wong Dec 11 '17 at 8:31
• Seems that the intent is to calssify the positive integers that cannot be written as the sum of EXACTLY three NON-ZERO-squares. – Peter Dec 11 '17 at 10:51
• In this case, the given list seems to be the full list. I did not find further primitive values with PARI/GP yet. – Peter Dec 11 '17 at 11:02
• Edited the question to make it clear this is about the sum of three non-zero squares. – James Arathoon Dec 11 '17 at 14:14