I´ve been dealing with this problem for almost a weak and I still can't find a solution:
Find a prime number p that is simultaneously expressible in the forms $x^2 +y^2 , a^2 + 2b^2 , t^2 + 3s^2 $ of course x,y,a,b,t and s are integers.
I've been reading some sources and I´ve found that a prime number have to fulfill some requierments in order to be expresed as one of the equations above, for example:
$ p=x^2+y^2 \iff p\equiv1(mod\ 4)$
$ p=x^2+2y^2 \iff p\equiv1,3(mod\ 8)$
$p=x^2+3y^2 \iff p\equiv1(mod\ 3)$
I tried to solve these linear congruences but I couldn't find anything, and I also tried to use the lengendre symbol to see if there was any relation I could conclude from there. At the end I was just trying by trial and error to see if maybe I could find it, but so far I only know that 41 and 17 can be express as $3^2 +(2)4^2 , 4^2 + 5^2$ and $1^2 + 4^2 , 3^2 + (2)2^2$ respectively.
I would really appreciate any hitn or advice, you cand porvide me in order to solve the problem. Thank you very much.
Your system of congruences is equivalent to $$ p \equiv 1 \mod 24$$
The first such prime $p$ is $p = 73$, and indeed, we have that $$\begin{align*} 73 & = 8^2 + 3^2 \\ & = 1^2 + 2 \cdot 6^2 \\ & = 5^2 + 3 \cdot 4^2 \end{align*}$$