We know that every prime $1\bmod 4$ can be written in an unique way as $a^2+b^2$ form where $a,b\in\Bbb N$.
Is there a comprehensive list of other statements of form "every prime $d\mod r$ can be represented written in an unique way as $ga^2+hb^2+ic^2$ where $g,h,i\in\Bbb Z$, $a,b,c,d,r\in\Bbb N$"?
At least what are some of famous ones?
I think it will be good to have such a detailed list and some references.
See http://arxiv.org/pdf/1207.0172.pdf.
As ending of page $6$ says, all representations in Theorem $1.1$ are unique.
$1.2$: $(I), (IV), (IX)$ are unique, all the others aren't. Same for $1.3$.
To prove uniqueness, simply observe $$k^2a^2-b^2=(ka+b)(ka-b)=p\iff \begin{cases}ka+b=p\\ ka-b=1\end{cases}$$
for all $k\in\{1,2,3\}, a,b\ge 0$. Same for $a^2-k^2b^2$.
To disprove uniqueness, find counterexamples with a program.
$1.4$: all of them are unique for $p\le 10\, 000$.