TITLE: Highest ratio between consecutive prime numbers
QUESTION [7 upvotes]: Let $r = p_2/p_1$;
where $p_1$, $p_2$ are consecutive prime numbers.
What is the highest possible value of $r?$
Are there any consecutive prime numbers such that $r > 5/3$?

REPLY [7 votes]: Quoting wiki

In 1952, Jitsuro Nagura proved that for  $n \ge 25$, there is always a prime between $n$ and $(1+\frac15)n$.

This set up an upper bound for the ratio $r = \frac{p_2}{p_1} \le 6/5$ for $p_1 \ge 25$. There is only a few pairs of primes remain to check. Yes, the highest ratio is $\frac{5}{3}$.