OPEN
Let $q_1<q_2<\cdots$ be a sequence of primes such that $q_{i+1}\equiv 1\pmod{q_i}$. Is it true that
\[\lim_k q_k^{1/k}=\infty?\]
Does there exist such a sequence with
\[q_k \leq \exp(k(\log k)^{1+o(1)})?\]
Linnik's theorem implies that there exists such a sequence of primes with
\[q_k \leq e^{e^{O(k)}}.\]
See also [696].
