For different integers there exists an integer so that exactly one of the sums is prime. Let $i,j$ be distinct natural numbers. In a comuter science class the lecturer said

Since the prime numbers are not regularly distributed, there is some natural number $k$ such that $i+k$ is prime and $j+k$ is not.

I found this reasoning a bit unsatisfying, so I tried to show the claim, however it turned out to be not that trivial. If lets say $i$ has a prime factor $p$ that doesn't divide $j$ then by Dirichlet there is a prime $q\equiv j\pmod p$, $q>j$. Then choose $k=q-j$, so $j+k$ is prime but $i+k$ has the prime factor $p$, hence is not prime. But this reasoning doesn't work if $i,j$ have the same prime factors. I also feel like this can be proven more elementary but I don't see how.
Edit: Thanks everyone for the help, I learned something from every answer. Unfortunately I can only accept one of them, so I hope I didn't offend anyone for not accepting their answer.

An elementary proof
Suppose $i<j$ and let $p$ be any prime greater than $i$. Let $a=j-i$.
Consider the sequence $p,p+a,p+2a,...,p+pa$. The initial number is prime and the final number is composite and so there is a successive pair of these numbers $u<v$ where precisely one is prime.
Then $k=u-i$ solves the problem.
