Paper: Primitive sets and von Mangoldt chains: Erdős…
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Paper: Primitive sets and von Mangoldt chains: Erdős Problem #1196 and beyond
Primitive sets and von Mangoldt chains: Erdős Problem #1196 and beyond
arXiv:2605.00301 \math.NT\]: [ https://arxiv.org/abs/2605.00301 https://arxiv.org/abs/2605.00301
Boris Alexeev, Kevin Barreto, Yanyang Li, Jared Duker Lichtman, Liam Price, Jibran Iqbal Shah, Quanyu Tang, Terence Tao
Abstract: 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.
#science
Primitive sets and von Mangoldt chains: Erdős Problem #1196 and beyond
arXiv:2605.00301 \math.NT\]: [ https://arxiv.org/abs/2605.00301 https://arxiv.org/abs/2605.00301
Boris Alexeev, Kevin Barreto, Yanyang Li, Jared Duker Lichtman, Liam Price, Jibran Iqbal Shah, Quanyu Tang, Terence Tao
Abstract: 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.
#science
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