Infinite divisibility chains under weak density assumptions
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Does a set A of positive upper doubly logarithmic density contain an infinite strictly increasing divisibility chain? A quantitative target asks that the chain’s upper count through x, divided by log log x, retain at least A’s density.
Why it matters and what counts as progress
Why it matters: A chain retaining the original density quantitatively strengthens existence of an infinite divisibility chain.
What counts as progress: Build a chain meeting the density bound or weaken the density assumption while preserving an infinite chain.
How the problem developed
- 1966 · Conjecture [1]
Erdős, Sárközi and Szemerédi Their paper on divisibility properties of integer sequences stated conjectures about long chains inside sets with logarithmic density.
- 1 May 2026 · Reported solution [2]
Alexeev, Barreto, Li, Lichtman, Price, Shah, Tang and Tao The preprint proves an infinite divisibility chain when the set has positive upper doubly logarithmic density, with a quantitative lower bound on the chain’s count through x.
AI contribution
Primitive sets and von Mangoldt chains: Erdős Problem #1196 and beyond
If A has positive upper doubly logarithmic density Δ, then A contains an infinite strictly increasing divisibility chain whose upper count through x, divided by log log x, is at least Δ.
Sources
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Publication dates and editorial changes.
AI-origin proof yields an infinite divisibility chain at weaker density