Find prime numbers $a, b, c$ such that $a^b+c$ is a prime Please help me find all $a, b, c \in \mathbb{P}$ such that $a^b+c$ is a prime 
Example I just can find: $2^3 + 5 = 13 \in \mathbb{P}$
 A: There are almost certainly infinitely many: classifying them is probably very difficult.
For example, take $a=b=2$. It has long been conjectured that there are infinitely many primes of the form $4+c$, where $c$ is prime. Tis problem is related to the famous twin primes problem. There is a great deal of numerical evidence suggesting that there are infinitely many primes $c$ such that $4+c$ is prime. But (so far) there is no proof. Examples are easy: $4+3$, $4+7$, $4+13$, and so on.
It is in fact conjectured that for any even number $e$, there are infinitely many primes of the form $e+c$, where $c$ is prime. That would mean that in particular there are infinitely many primes of the form $2^b+c$, where $c$ is any prime. Again, there are many examples: $2^3+3$, $2^3+5$, $2^3+11$, $2^5+5$, and so on. 
For odd $a$, the problem is different, for then the only possibility is $c=2$. But it is not known whether there are infinitely many primes of the form $a^b+2$, where $a$ is an odd prime and $b$ is prime. Examples are not hard to find, such as $3^2+2$, $3^3+2$, $5^3+2$, $3^{29}+2$.
