I am wondering about the following question.
Do there exist infinitely many prime numbers $p$ such that there exist integers $m,n$ with $5(m+1) \geq 7n$ and $$5m < p < 7n?$$
If not, what other numbers would we have to replace $5$ and $7$ by for there to be? If so, what is the minimal set of numbers we would have to replace $5$ and $7$ by? This leads onto my second question below.
Find numbers $a,b$ with $|a-b|$ minimal such that there exist infinitely many primes $p$ with the property that there exist integers $m,n$ with $a(m+1) \geq bn$ and $$am < p < bn.$$