Well, we all know the twin prime conjecture.
There are infinitely many primes p, such that p+2 is also prime.
Well, I actually got asked in a discrete mathematics course, to prove that there are infinitely many primes p such that p+2 is NOT prime.
Let p>3 be prime. If p+2 is not prime, we are happy. If p+2 is prime, then (p+2)+2 is not, since one of x,x+2,x+4 is divisible by 3.
Added: Dolda2000 noted that a more interesting question is whether there are infinitely many primes that are not members of a twin pair. For this we can use the fact that there are infinitely many primes of the form 15k±7. If p is such a prime, then one of p−2 or p+2 is divisible by 3, and the other is divisible by 5, so if p>7 then neither p−2 nor p+2 is prime.