Primitive sets and von Mangoldt chains: Erdős Problem #1196 and beyond
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arXiv
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| Auteurs principaux: | , , , , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866911638160932864 |
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| author | Alexeev, Boris Barreto, Kevin Li, Yanyang Lichtman, Jared Duker Price, Liam Shah, Jibran Iqbal Tang, Quanyu Tao, Terence |
| author_facet | Alexeev, Boris Barreto, Kevin Li, Yanyang Lichtman, Jared Duker Price, Liam Shah, Jibran Iqbal Tang, Quanyu Tao, Terence |
| contents | A set of integers is primitive if no number in the set divides another. We introduce a new method for bounding Erdős sums of primitive sets, suggested from output of GPT-5.4 Pro, based on Markov chains with von Mangoldt weights. The method leads to a host of applications, yet seems to have been overlooked by the prior literature since Erdős's seminal 1935 paper.
As applications, we prove two 1966 conjectures of Erdős-Sárközy-Szemerédi, on primitive sets of large numbers (#1196) and on divisibility chains (#1217). The method also provides a short proof of the Erdős Primitive Set Conjecture (#164), as well as the related claim that 2 is an ''Erdős-strong'' prime. Moreover, the method resolves a revised form of the Banks-Martin conjecture, which has long been viewed as a unifying `master theorem' for the area. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_00301 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Primitive sets and von Mangoldt chains: Erdős Problem #1196 and beyond Alexeev, Boris Barreto, Kevin Li, Yanyang Lichtman, Jared Duker Price, Liam Shah, Jibran Iqbal Tang, Quanyu Tao, Terence Number Theory Combinatorics Probability Primary: 11B83, Secondary: 11A05, 11N05, 11N37, 06A07 A set of integers is primitive if no number in the set divides another. We introduce a new method for bounding Erdős sums of primitive sets, suggested from output of GPT-5.4 Pro, based on Markov chains with von Mangoldt weights. The method leads to a host of applications, yet seems to have been overlooked by the prior literature since Erdős's seminal 1935 paper. As applications, we prove two 1966 conjectures of Erdős-Sárközy-Szemerédi, on primitive sets of large numbers (#1196) and on divisibility chains (#1217). The method also provides a short proof of the Erdős Primitive Set Conjecture (#164), as well as the related claim that 2 is an ''Erdős-strong'' prime. Moreover, the method resolves a revised form of the Banks-Martin conjecture, which has long been viewed as a unifying `master theorem' for the area. |
| title | Primitive sets and von Mangoldt chains: Erdős Problem #1196 and beyond |
| topic | Number Theory Combinatorics Probability Primary: 11B83, Secondary: 11A05, 11N05, 11N37, 06A07 |
| url | https://arxiv.org/abs/2605.00301 |