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

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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.
 
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Borek said:
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).
 
wait... 6 is not a prime...
 
quantumdoodle said:
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.
 
quantumdoodle said:
wait... 6 is not a prime...
Sorry... :blushing: 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).
 

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