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Weighted dilation graphs, smooth shifted primes and totient fibers

OpenAI · September 24, 2026

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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.

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Notes

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work in progress

Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.

Follow-up works

Comments and discussion

  • Thomas Bloom, Erdős problems in the OpenAI proofs (erdosproblems.com blog, 2026-10-08). The erdosproblems.com maintainer lists this paper's claim as a resolution of Problem #821, notes the previous best exponent was $0.715\ldots$, and says the proof goes through results on the prime factors of $p-1$.

Formalization

  • OpenAI, lean/OAI/NumberTheory/TotientFibers/FiniteFiber.lean (openai/math, 2026-10-06). The release's own Lean file, proving finiteness of totient fibers and a pigeonhole transfer that assumes a large set of primes with $t$-smooth predecessors; it does not prove the smooth-shifted-prime count (Theorem 1.2).
  • Yao Xu (@cadamcat), totient-fibers-lean4 (GitHub, v1.0.0, verification recorded 2026-10-14). An independent Lean 4 formalization of Theorems 1.1 (Erdős #821) and 1.2 following the paper's structure, with a statement comparison against Formal Conjectures; per its verification record, Erdos821.erdos_821 depends only on propext, Classical.choice and Quot.sound. [AI-generated per source]
  • Yao Xu (@cadamcat), equal-divisor-sums-lean4 (GitHub, accessed 2026-10-21). A Lean 4 proof of Erdős #824 that combines Li–Zhuang's conditional reduction with a proved supply of smooth shifted primes $p+1$, reusing the sign-free parts of totient-fibers-lean4. [AI-generated per source]
  • Shisheng Li and Zhong Zhuang, Erdős Problem 824: coprime pairs with equal sum of divisors (Lean 4 development, 2026). Formalizes the implication from an explicit smooth-successor hypothesis to the pair-count lower bound, excluding the weighted-dilation argument and the squarefree addendum. [AI-generated per source]
  • google-deepmind/formal-conjectures, ErdosProblems/824.lean (formal-conjectures). The formal statement of Erdős #824, marked solved; it cites this paper and Li–Zhuang and links equal-divisor-sums-lean4 as a formal proof.
  • Plasma AI, erdos-408 (Lean 4) (GitHub, 2026-10-19). Lean development for Erdős #408 whose graph inputs come from a vendored copy of totient-fibers-lean4, i.e. formalized versions of this paper's results. [family-level]

News and blogs

  • Patrick White with Claude, Erdős #821: OpenAI totient fibers, refereed (Erdős Problem a Day, 2026-10-11). A machine review by ten AI "seats" that reports no mathematical gap but lists drafting issues (a sketched bijection in Lemma 3.7, an unwritten inequality in §5, a $4\delta<\varepsilon$ versus $3\delta<\varepsilon$ mismatch in §8), and states that no human has verified the paper. [AI-generated per source]

Elsewhere

Independent sites; their content is not reviewed here.

Cite

@misc{OAI:Weighted-Dilation-Graphs-Smooth-Shifted-Primes-and-Totient-Fibers-September-24-2026,
  author = {{OpenAI}},
  title = {{Weighted dilation graphs, smooth shifted primes and totient fibers}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Weighted-Dilation-Graphs-Smooth-Shifted-Primes-and-Totient-Fibers-September-24-2026/paper.pdf}{OAI:Weighted-Dilation-Graphs-Smooth-Shifted-Primes-and-Totient-Fibers-September-24-2026}},
  year = {2026}
}

Family 011: Prime-factor statistics of $p-1$

Proves that the normalized ordered logarithms of the prime factors of $p-1$, counted with multiplicity, converge jointly to the Poisson–Dirichlet law $\mathrm{PD}(1)$ as p ranges uniformly over primes up to x and $x\to\infty$. This resolves the Ford–Konyagin–Luca conjecture. It also proves that infinitely many integers n have more than $n^{1-\varepsilon}$ totient preimages, for every ε > 0.

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