Are there infinitely many primes p such that p + 2 is prime?

I do not know radio.  In theory at least I think there are infinitely many primes BUT

You have taken this a step further - I am not so sure your 'yes' would stand up to close scrutiny.  

Well numbers go on infinately and since primes also theoretically go on infinately, so yes, but the number of primes found by using p+2 will exponetially decrease.

This is known as "The Twin Prime Conjecture"......it is believed to be true, but hasn't been proved conclusively.......!!!

Here's more : https://en.wikipedia.org/wiki/Twin_prime