Normal orders of the functions in Erdős Problem 696
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For an integer n, let h(n) be the longest chain of prime divisors p₁<⋯<p_ℓ with pᵢ₊₁≡1 mod pᵢ, and let H(n) allow arbitrary increasing divisors with the same congruence. What are their normal orders, and does H(n)/h(n) tend to infinity for almost all n?
Why it matters and what counts as progress
Why it matters: Normal orders distinguish prime-divisor chains from arbitrary-divisor chains and decide whether their typical ratio diverges.
What counts as progress: Prove matching almost-everywhere bounds, determine the log-star leading constants or establish a finite limiting ratio instead of divergence.
How the problem developed
- 1979 · Open problem [1]
Paul Erdős Introduced Problem 696 on divisor-chain functions and described the expected normal order of h(n) as log* n.
- 14 Oct 2025 · Partial result [2]
Wouter van Doorn A comment records a proof that h(n) tends to infinity for almost all n, a qualitative result short of its expected normal order.
- April 2026 · Reported claim [3]
David Turturean The project reports h(n)=(1/2+o(1))log* n and H(n)=(1+o(1))log* n almost everywhere, implying H(n)/h(n) tends to 2.
AI contribution
Normal orders of the Erdős chain functions h(n) and H(n)
Turturean’s write-up claims h(n)=(1/2+o(1))log* n and H(n)=(1+o(1))log* n for almost all n, so H(n)/h(n) tends to 2. A GPT-5.5 Pro subset-product successor lemma supports the composite-divisor lower bound.
Sources
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