On extracting primes from coprimes

Question

Proof or disprove the following statement -

There exists infinitely many $a$ and $b$ which are pair of co-prime integers , either $ab+1$ or $ab-1$ is prime.

Motivation- Looking at some twin prime pairs and calculating a lot - I guessed something like above.

Answer 1

Take any odd prime number $p$. Let $a$ be the greatest power of $2$ which divides $p-1$, and $b=\cfrac {p-1}a$ then $ab+1=p$ and $(a,b)=1$.

You can do the same for $ab-1$ by taking factors of $p+1$.

Since $p$ was arbitrary, and there are infinitely many primes, there are infinitely many pairs $(a,b)$ satisfying your criterion.

Answer 2

The statement is trivial from the infinitude of primes. If $p$ is prime, let $a=p+1$ and $b=1$, then $ab-1$ is prime.