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Prime-factor statistics of \(p-1\)

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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.

The release has no Lean statement for this family, so there is no formal statement to compare with the result described above.

It bears on Erdős problem #821, as listed by Thomas Bloom, erdosproblems.com.

The paper states

Theorem 1.1

In Prime Predecessors with an Even Number of Prime Factors, 17 September 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

Three other possible matches are listed below the card, among them Theorem 1.1 in Weighted dilation graphs, smooth shifted primes and totient fibers and Theorem 1.1 in The Poisson-Dirichlet law for prime predecessors.

There are infinitely many primes p such that p-1 is squarefree and \Omega(p-1) is even. In particular, there are infinitely many such primes under either convention for counting prime factors.

Three more possible matches

  • Theorem 1.1 in Weighted dilation graphs, smooth shifted primes and totient fibers, 24 September 2026.
  • Theorem 1.1 in The Poisson-Dirichlet law for prime predecessors, 24 September 2026.
  • Theorem 1.2 in Weighted dilation graphs, smooth shifted primes and totient fibers, 24 September 2026.
Reported proof verification
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Papers and explanations
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Papers

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Weighted dilation graphs, smooth shifted primes and totient fibers

24 September 2026, current edition.

Read it herePDF

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.

The Poisson-Dirichlet law for prime predecessors

24 September 2026, current edition.

Read it herePDF

For a prime p chosen uniformly from \(3\le p\le x\), list the prime factors of \(p-1\) in decreasing order, with multiplicity. As \(x\to\infty\), their logarithms, divided by \(\log(p-1)\), converge in every finite joint distribution to the Poisson–Dirichlet distribution with parameter one. This proves the conjecture of Ford, Konyagin and Luca.

Lean statements

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Independent comparisons and proof-verification reports

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What people have said

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  • An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Career-defining publication”. The page says that nothing on it is verified.

    Author not stated. Source

  • Thomas Bloom, Erdős problems in the OpenAI proofs, 8 October 2026. Source

    Bloom lists Erdős 821 (totient values), says it is not formalised, and notes the PDF gets there through results on prime factors of p minus one.

  • eigendash, eigendash/openai-math-visualised: numerical probes of seven result families, 8 October 2026. Source

    Sieving all 1,857,858 primes up to 3 times 10^7, the first two normalised prime-factor statistics of p - 1 agree with the predicted PD(1) law within a few per cent and drift toward it, but components 3 onward exceed their predicted means by factors growing with the index (KS distances 0.27 to 0.43); the author reports the gap rather than calling it noise and suspects slow convergence.

Questions awaiting review

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