Erdős Problem #696

The problem

Let $h(n)$ be the largest $\ell$ such that there is a sequence of primes $p_1<\cdots < p_\ell$ all dividing $n$ with $p_{i+1}\equiv 1\pmod{p_i}$. Let $H(n)$ be the largest $u$ such that there is a sequence of integers $d_1<\cdots < d_u$ all dividing $n$ with $d_{i+1}\equiv 1\pmod{d_i}$.

Estimate $h(n)$ and $H(n)$. Is it true that $H(n)/h(n)\to \infty$ for almost all $n$?

Statement quoted from erdosproblems.com/696, maintained by Thomas Bloom. Original source: [Er79e,p.81]. Current status there: solved (Lean).

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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