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

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Discussion Overview

The discussion revolves around identifying pairs of prime numbers (p, q) such that both expressions 2p + 3q and 2q + 3p yield prime numbers. The scope includes mathematical reasoning and exploration of prime properties.

Discussion Character

  • Exploratory, Mathematical reasoning

Main Points Raised

  • One participant suggests that the pairs (1,1), (1,2), (2,3), and (2,6) are potential solutions.
  • Another participant points out that 6 is not a prime number, questioning the validity of the proposed pairs.
  • A subsequent reply corrects the earlier claim about the pairs, stating that the valid solutions found are (2,2), (2,3), (3,5), and (3,13).
  • There is a note that 1 is also not a prime, further refining the acceptable pairs.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the initial list of pairs, as some are identified as invalid. However, there is an agreement on a revised list of pairs that includes (2,2), (2,3), (3,5), and (3,13).

Contextual Notes

The discussion highlights the need for careful verification of prime status for the numbers involved, as well as the importance of adhering to forum policies regarding showing attempts.

rrronny
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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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You have to show your attempts to receive help. This is a forum policy.
 
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]
 
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 [tex](p,q)[/tex] that I found are [tex](2,2),(2,3),(3,5), (3,13).[/tex]
 

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