Erdős Problem 820

### What is the conjecture

https://www.erdosproblems.com/820

Let $H(n)$ be the smallest integer $l$ such that there exist $k0$ such that, for all $\epsilon>0$,
$$H(n) > \exp(n^{(c-\epsilon)/\log\log n})$$
for infinitely many $n$ and
$$H(n) < \exp(n^{(c+\epsilon)/\log\log n})$$
for all large enough $n$?

Does a similar upper bound hold for the smallest $k$ such that $(k^n-1,2^n-1)=1$?

Status: open

### Choose either option
- [ ] I plan on working on this conjecture
- [x] This issue is up for grabs: I would like to see this conjecture added by somebody else