I would like to know if it has been proved that :
- There are no $a$, $b$ and $c$, all prime numbers, such that $a^2 + b^2 = c^2$
- There are no $a$, $b$, $c$ and $d$, all prime numbers, such that $a^2 + b^2 + c^2 = d^2$
- There are no $a$, $b$, $c$, $d$ and $e$, all prime numbers, such that $a^2 + b^2 + c^2 + d^2 = e^2$
Thanks for your answers.
It turns out that the (full) solution is longer than a comment. So here it goes.
Suppose that for $a,b,c,d$ primes, $$a^2+b^2+c^2=d^2$$ Obviously $d=3$ is not a solution. That implies $$a^2+b^2+c^2\equiv 1\pmod{3}.$$ So exact two of $a,b,c$ are equal to $3$. We are left to solve $$d^2-a^2=(d-a)(d+a)=18.$$ But $d-a$ and $d+a$ are of the same parity and $18$ cannot be factorized into such a product.
For the third equations, note that $n^2\equiv 1\pmod{4}$ if $n$ is odd. Thus if $$a^2+b^2+c^2+d^2=e^2,$$ then exactly $3$ of $a,b,c,d$ are $2$. We are left to solve $$(e-d)(e+d)=12=6\cdot 2.$$ It follows that $e+d=6$, and $e-d=2$. That means $e=4$, not a prime.