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

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Let [tex]\mathbb{P}[/tex] the set of primes. Let's [tex]p,q \in \mathbb{P}[/tex] and [tex]p \le q.[/tex] Find the pairs [tex](p,q)[/tex] such that [tex]2^p+3^q[/tex] and [tex]2^q+3^p[/tex] 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: [tex](1,1), (1,2), (2,3), (2,6).[/tex]
 
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 [tex](p,q)[/tex] that I found are [tex](2,2),(2,3),(3,5), (3,13).[/tex]
 

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