Breakthrough Tracker record
Erdős Problems #1196 and #1217: primitive sets and divisibility chains
The authors resolve 1966 Erdős–Sárközy–Szemerédi conjectures concerning dense primitive subsets of sufficiently large integers and long divisibility chains. They develop a Markov-chain and von Mangoldt framework for the proofs.
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- Stable ID
math-erdos-1196-1217-primitive-sets-2026- Revision
math-erdos-1196-1217-primitive-sets-2026.v1- Field
- Mathematics · Number theory and combinatorics
- Evidence
- Tier 1 · Peer reviewed: No
- Record state
- Current · Expert-authored preprint
- Last checked
AI role
The paper credits GPT-5.4 Pro with suggesting the Markov-chain idea; the human authors developed and checked the proofs.
Record details
- Problem or result
- Erdős Problems #1196 and #1217
- Authors
- Boris Alexeev, Kevin Barreto, Yanyang Li, Jared Duker Lichtman, Liam Price, Jibran Iqbal Shah, Quanyu Tang and Terence Tao
- Institutions
- Multi-institution collaboration
- Result date
- Preprint submitted May 1, 2026
Why it matters
The work settles two named Erdős problems and introduces machinery that may apply beyond them.
Limits
The paper also gives a short proof related to the already resolved Erdős primitive-set conjecture #164. That result must not be presented as a third newly solved problem.
Sources
- Primary: Alexeev et al., Primitive sets and von Mangoldt chains
- Primary: Terence Tao's technical discussion
Correction and revision history
- Added on 2026-07-22 with #1196 and #1217 separated from the paper's alternate treatment of already resolved #164.
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