Problem 695

Erdős Problem #695: declared status 'open'. Formalized: yes. Let $p_1<p_2<\cdots$ be a sequence of primes such that $p_{i+1}\equiv 1\pmod{p_i}$. Is it true that$$\lim_k p_k^{1/k}=\infty?$$Does there exist such a sequence with$$p_k \leq \exp(k(\log k)^{1+o(1)})?$$ Current best: If we take the obvious 'greedy' chain with $2=p_1$ and $p_{i+1}$ is the smallest prime $\equiv 1\pmod{p_i}$ then Linnik's theorem implies that this sequence grows like$$p_k \leq e^{e^{O(k)}}.$$It is conjectured that, for any prime $p$, there is a prime $p'\leq p(\log p)^{O(1)}$ which is congruent to $1\pmod{p}$, which would imply this sequence grows like$$p_k\leq \exp(k(\log k)^{1+o(1)}).$$An extensive study of the growth of finite prime chains was carried out by Ford, Konyagin, and Luca [FKL10]. Prize: no. OEIS: A061092. Tags: number theory.