# 2^p+3^q, p and q in P

Let $\mathbb{P}$ the set of prime numbers in $\mathbb{N}.$ Lets $p, q$ in $\mathbb{P}$ and $p
To find the pairs $(p, q)$ such that $2^p +3^q$ and $2^q +3^p$ are primes simultaneously.
Interesting question. It's easy for $p \equiv q \pmod4$ since $2^p+3^q \equiv 2^p + (-2)^q \equiv 0 \pmod5$ iff $p \equiv q \pmod4$. When $p \not\equiv q \pmod4$ it seems to be a lot harder.