TITLE: Prove or disprove that $p_n > e^{p_n - p_{n-1}}$ for large enough $n$.
QUESTION [6 upvotes]: Let $p_n$ denote the $n$-th prime. Prove or disprove that for large enough $n$ we have $$p_n > e^{p_n - p_{n-1}}.$$

The inequality trivially holds for all the twin primes larger than $7$. With $n$ larger than some $N$, does it always hold? I think not, but don't really know how to prove it.

REPLY [2 votes]: A disproof: Rankin showed that there are infinitely many $n$ for which
$$
p_n-p_{n-1} > \log n (\log\log n)^{1/2}
$$
(in fact his result was rather stronger than this). In particular, there are infinitely many $n$ such that
$$
p_n-p_{n-1} > 2\log n > \log p_n
$$
(the latter inequality is just asserting that $p_n < n^2$ for sufficiently large $n$). Exponentiation both sides yields
$$
e^{p_n-p_{n-1}} > p_n
$$
for infinitely many $n$.
When one exponentiates an expression (like $p_n-p_{n-1}$), a tiny difference between it being of size $0.99\log n$ and $1.01\log n$ gets magnified so much that it makes all the difference in the world compared to $p_n$.