Powers of two are not sums of squares of different integers greater than zero It seems that non-trivial sums of squares of different non-zero integers aren't powers of two. Tested for sums of $2,3,4$ terms.
Is there something known about this topic? Are there counter-examples?

Couldn't find a counter-example for up to $4$ terms as well (maybe someone can show why), but for more terms there are for example
$$
1^2+5^2+7^2+9^2+10^2=2^8\\
2^2+3^2+5^2+7^2+13^2=2^8\\
3^2+5^2+6^2+9^2+19^2=2^9\\
\vdots \\
1^2+5^2+6^2+7^2+8^2+9^2=2^8\\
\vdots \\
$$
