Manuscript OpenAI@2026-10-07 · OpenAI.tex · remark · 6a931f8f
Remark: 011-01. Weighted dilation graphs, smooth shifted primes and totient fibers
Manuscript map · lines 2645–2654
OpenAI@2026-10-07#rem:oai-p-011-01
Authors: OpenAI. Family 011.
Title: Weighted dilation graphs, smooth shifted primes and totient fibers.
Abstract: We prove Erdős's conjecture on the largest fibers of Euler's totient function: for every ε > 0, infinitely many positive integers n have more than $n{}{1-{}varepsilon }$ preimages. We also show that, for every fixed δ > 0, there are at least $x{}{1-o(1)}$ primes p in $2x{}lt p{}le5x$ whose predecessors have no prime factor exceeding xδ.
This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.
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