Which Pairs (p,q) Satisfy 2^p+3^q and 2^q+3^p Being Prime?

[rrronny]

Let \mathbb{P} the set of primes. Let's p,q \in \mathbb{P} and p \le q. Find the pairs (p,q) such that 2^p+3^q and 2^q+3^p are simultaneously primes.

 

[Borek]

You have to show your attempts to receive help. This is a forum policy.

 

[rrronny]

You have to show your attempts to receive help. This is a forum policy.

Hi Borek,
I do not have a solution for this problem...
I found only four solutions: (1,1), (1,2), (2,3), (2,6).

 

[quantumdoodle]

wait... 6 is not a prime...

 

[ramsey2879]

wait... 6 is not a prime...

Neither is 1 so only one of the 4 pairs posted is acceptable. There is still another very obvious pair that was overlooked.

 

[rrronny]

wait... 6 is not a prime...

Sorry... I meant the sixth prime number.
In summary, then, the only solutions (p,q) that I found are (2,2),(2,3),(3,5), (3,13).