**Question:** Is the conjecture as follows true or false?



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> > For any integer $n>1$, there always exists at least one prime number $p$ with
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> > $$n < p< n+\left(\ln\Big(\frac{n}{\ln n}\Big)+1\right)^2$$
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The conjecture was checked true with $n$ up to $10^8$ and some [The 80 known maximal prime gaps](https://en.wikipedia.org/wiki/Prime_gap#Numerical_results)



For any integer $n>1$, there always exists at least one prime number $p$ with $n < p< n+\left(\ln\Big(\frac{n}{\ln n}\Big)+1\right)^2$
False.


Let $n=1693182318746371$. The next prime after $n$ is $1693182318747503$.


$(\ln(\frac{n}{\ln n})+1)^2 \le1057$, but the prime gap is $1132$.


