multiple choice question on number theory

Question

Let $a\in \mathbb{Z}$ be such that $a=b^2+c^2$ where, $b,c \in \mathbb{Z}-\{0\}$. Then $a$ cannot be written as

1. $pd^2$ where $d \in \mathbb{Z}$ and $p$ is prime with $p \equiv 1 \pmod4$
2. $pd^2$ where $d \in \mathbb{Z}$ and $p$ is prime with $p \equiv 3\pmod4$
3. $pqd^2$ where $d \in \mathbb{Z}$ and $p,q$ are primes with $p \equiv 1\pmod4$ and $q \equiv 3\pmod4$
4. $pqd^2$ where $d \in \mathbb{Z}$ and $p,q$ are primes with $p,q \equiv 3\pmod4$

The similar question with some part missing is asked here, but I want to discuss additional options too.

Here, I got 1) is false Since $13=3^2+2^2$ For other options What should I do

Answer 1

So I can conclude like this, by Sum of two squares theorem options, 2,3,4 are the correct options. For first option we have counter example, 5=$1^2+2^2$.

Answer 2

(2) Is correct since we know that every sum of two squares must be divisible by primes of the form $4n+1$ or of the form $4n+3$ if they are raised to an even exponent.
Check here for more details.