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

1. ## 2^p+3^q, p and q in P

Let $\displaystyle \mathbb{P}$ the set of prime numbers in $\displaystyle \mathbb{N}.$ Lets $\displaystyle p, q$ in $\displaystyle \mathbb{P}$ and $\displaystyle p<q.$
To find the pairs $\displaystyle (p, q)$ such that $\displaystyle 2^p +3^q$ and $\displaystyle 2^q +3^p$ are primes simultaneously.

2. Interesting question. It's easy for $\displaystyle p \equiv q \pmod4$ since $\displaystyle 2^p+3^q \equiv 2^p + (-2)^q \equiv 0 \pmod5$ iff $\displaystyle p \equiv q \pmod4$. When $\displaystyle p \not\equiv q \pmod4$ it seems to be a lot harder.