Let $\varphi(n)$ denote the totient function and let $k$ be a positive even integer.
Define $f(k)$ to be the smallest positive integer $n$ satisfying $$\varphi(n+k)-\varphi(n)=k$$
If for every even positive integer $k$ , there is a prime $p$ such that $p+k$ is prime as well , which is strongly conjectured , but not proven , then we always have a prime number as a solution.
For $k\le 10^7$ , the smallest solution is always a prime number. If the above conjectre holds , can we assume that this is always the case ?