Can every even integer be expressed as the difference of two primes? If so, is there any elementary proof?
This is listed as an open question at the Prime Pages: http://primes.utm.edu/notes/conjectures/
This follows from Schinzel's conjecture H. Consider the polynomials $x$ and $x+2k$. Their product equals $2k+1$ at 1 and $4(k+1)$ at 2, which clearly do not have any common divisors. So if Schinzel's conjecture holds, there are infinitely many numbers $n$ such that the polynomials are both prime at $n$, and so subtracting gives the result.