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 $r > 5/3$?
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Possibly relevant: http://arxiv.org/abs/1212.2785 – Charles Apr 27 '15 at 17:58
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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 $\frac{5}{3}$.
achille hui
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