Erdős Problem #696
Resolved in some way other than a straight proof or disproof — usual for open-ended questions, or for multi-part problems whose parts went different ways. A solution has also been checked by machine, in Lean.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
A strictly-increasing chain of natural numbers \(d_1 < d_2 < \dots\) such that consecutive elements satisfy \(d_{i+1} \equiv 1 \pmod{d_i}\), and every element divides \(n\) and satisfies the auxiliary predicate \(P\).
\(h(n)\) is the largest \(\ell\) such that there is a sequence of primes \(p_1<\cdots < p_\ell\) all dividing \(n\) with \(p_{i+1}\equiv 1\pmod{p_i}\).
\(H(n)\) is the largest \(u\) such that there is a sequence of integers \(d_1<\cdots < d_u\) all dividing \(n\) with \(d_{i+1}\equiv 1\pmod{d_i}\).
Attempts
submit one →If you take a run at this one — with your own AI subscription or without — the guide says how, and the submission form gets you an honest read of what you found.