On the primality of 2^p+3^q

[rrronny]

Let \mathbb{P} the set of primes. Lets 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]

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[rrronny]

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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... :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).