Contributions came from Emmanuel Vantieghem, and Jahangeer Kholdi & Farideh Firoozbakht.
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
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.
The answers to all three questions is
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.