I have constructed the number $7131372917538397234773191167617941438959$, which is prime as shown here, using all primes under $100$ which contains two digits by randomly ordering them as $31,37,29,\ldots, 59$ except $13$. I have got that number $7131372917538397234773191167617941438959$ satisfies the following properties:
1.- The sum of its digits is also prime: it is equal to $193$.
3.- The number is of the form $6n+1$
4.- This number can't be written as a sum of $3$ squares.
Now my question here is:
Could be this : $7131372917538397234773191167617941438959$ written as $x^{2}+y^{2}$ with $x, y$ integers?
No, it cannot. Note that for all $x \in \mathbb{Z}$, $$x^2 \equiv 0, 1 \pmod {4}$$ Thus a sum of two squares can only be $0,1,2$ modulo $4$.
However, note that $$7131372917538397234773191167617941438959 \equiv 59 \equiv 3 \pmod {4}$$ Thus, it is not a sum of two squares. We are done!
For primes that are the sum of two squares, see Fermat's Theorem on the Sum of two Squares. For a prime $p$ to be of the form $$p=x^2+y^2$$ Then $p \equiv 1 \pmod {4}$ or $p=2$.