multiple choice question on number theory 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

*

*$pd^2$ where $d \in \mathbb{Z}$ and $p$ is prime with $p \equiv 1 \pmod4$

*$pd^2$ where $d \in \mathbb{Z}$ and $p$ is prime with $p \equiv 3\pmod4$

*$pqd^2$ where $d \in \mathbb{Z}$ and $p,q$ are primes with $p \equiv 1\pmod4$ and $q \equiv 3\pmod4$

*$pqd^2$ 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

(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.
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