My hope is that this question is "trivial," but it is outside my knowledge base, so I'd appreciate some advice.

Given positive integers $m$ and $n$, find the least prime $p$ such that $p-1 = mnk$ for some $k \geq 1$.

For what I am trying to do, I need an explicit algorithm to find $p$, as opposed to an approximation. Is there a best one known? What is the upper bound on how much larger $p$ might be than $mn$? I am happy to assume that $m$ and $n$ are "sufficiently large" for the algorithm to have nice properties, if that helps.

Thank you. Hopefully the answer is obvious to everyone but me. :-)

some$k\ge1$" or "forevery$k\ge1$" ?:-( $\endgroup$