Problems & Puzzles: Puzzles

Puzzle 716 Problem 9 of...

Philippe Fondanaiche sent the following nice puzzle.
Does there exist infinitely natural integers n for which there are prime twins p and q = p + 2 such that the following members are also prime :
 
Q1 : 2^n + p and 2^n + q ?
Q2 : 3^n + 5^n + p and 3^n + 5^n + q ?
Q3 : 2^n + 3^n + 5^n + p and 2^n + 3^n + 5^n + q ?
 
(Source: according to the problem n9 of the 5th International Tournament of Young Mathematicians)

Q. Solve Q1, Q2 & Q3


Contributions came from Emmanuel Vantieghem, and Jahangeer Kholdi & Farideh Firoozbakht.

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Emmanuel wrote:

Here is my first finding about puzzle 716 :

First, take  p = 3, q = 5.

Then, there are 'sporadic' solutions :

n = 1  and  n = 3  for  Q1  and no other  n <= 8000
n = 1  and  n = 5  for  Q2  and no other  n <= 8000
n = 2  and  n = 62  for Q3  and no other  n <= 8000.

I have no idea how it could be proved or disproved that there are more or no more solutions ...

If  p > 3  and  q = p+2  are both primes then we have modulo 3 : p == -1  and  q == 1.

Hence, when  p > 3 :

Q1 : one of the two numbers  2^n + p  or  2^n + q  is divisible by 3.

Q2 : because  5 === 2 mod 3, one of the two numbers  3^n +5^n+ p  or  3^n +5^n+ q  is divisible by 3.

Q3 : because  2^n+5^n === 2^(n+1) mod 3, one of the two numbers in question is divisible by 3.

Later he added:

I have increased the bound '8000' to  '10000'.

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Jahangeer Kholdi & Farideh Firoozbakht wrote:
The answers to all three questions is unknown.
Each pair in Q1, Q2, and Q3 is of the form (m, m+2), to have infinitely
many of such pairs of  primes, there must exists infinitely many twin primes
(p, p+2). But  we all know this fact is an unsolved problem. 

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