# Is $29$ the only prime of the form $p^p+2$?

I searched for primes of the form $$p^p+2$$, where $$p$$ is prime for a range of $$p \le 10^5$$ on PARI/GP and found that 29 is the only prime of this form in this range.

Questions:

$$(1)$$ Is $$29$$ the only prime of the form $$p^p+2$$, where $$p$$ is prime?

$$(2)$$ If not, then are there a finite number of primes of the form $$p^p+2$$? Can you prove/disprove this?

Edit: Since $$p^p$$ grows really fast and primes get rarer and are spread farther out for large numbers,

I conjecture that $$29$$ is the only prime of the form $$p^p+2$$ where $$p$$ is a prime.

With pfgw, I checked $$p^p+2$$ for the primes from $$\ 3\$$ to $$\ 24\ 001\$$
The only prime occured for $$\ p=3\$$ Hence if another prime of this form exist, it must have more than $$\ 100\ 000\$$ digits