0 reactions
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
0 reactions
Gauss from Math, Inc. has formalized the proof of Erdős Problem #1196 . The initial proof was 7.2K lines of Lean, done in ~5 hours. Subsequent golfing has compressed it down to 4K lines.
Github repo for the code:: https://github.com/math-inc/Erdos1196/tree/main
From Math, Inc. on 𝕏: https://x.com/mathematics\_inc/status/2044717899944960037 https://x.com/mathematics_inc/status/2044717899944960037
#science
Github repo for the code:: https://github.com/math-inc/Erdos1196/tree/main
From Math, Inc. on 𝕏: https://x.com/mathematics\_inc/status/2044717899944960037 https://x.com/mathematics_inc/status/2044717899944960037
#science