For which primes p is $p^2+2^p$ also prime?

Question

for 3: $3^2+2^3=17$

so 3 is the only solution?

thanks!

Answer 1

If $p=6m\pm1$

$$p^2+2^p\equiv(\pm1)^2+(-1)^{6m\pm1}\equiv0\pmod3$$