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Updated Oct 7

Problem 695

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)})?

Status. Open.

Source. erdosproblems.com/695, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #695, https://www.erdosproblems.com/695.

References.

[FKL10] Ford, Kevin and Konyagin, Sergei V. and Luca, Florian, Prime chains and Pratt trees. Geom. Funct. Anal. (2010), 1231-1258.

Formalization. Statement in formal-conjectures.

Progress

Not yet compiled.

Known Results

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Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (6)

Sequences and Densities of Integers integer_sequences/ford_2010_prime_chains_pratt_trees Section 1.3 (p. 4): the chain of least primes 2 = q_1, q_{j+1} the least prime = 1 mod q_j Theorem 1 (p. 2): N(x;p) <= x exp{log x (log_3 x + O(1))/log_2 x}Theorem 3 and Corollary 1 (p. 5): conditional lower bounds H(p) > c log_2 p from a level of distribution theta Theorem 4 (p. 5): H(p) <= (log p)^0.9503 for almost all p

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