Problem Statement
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\[p_k \leq \exp(k(\log k)^{1+o(1)})?\]
Categories:
Number Theory
Progress
Such a sequence is sometimes called a prime chain.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].
See also [696].
Source: erdosproblems.com/695 | Last verified: January 16, 2026