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Math-2795 · mathematics · Number theory

Poisson-Dirichlet factorization of prime predecessors

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For each prime $p>=3$ list the prime factors of p-1, counting multiplicity, in descending order and divide their logarithms by log(p-1), padding with zeros. Do the first k coordinates, for every fixed k, converge in distribution under ordinary uniform prime counting $p<=X$ to the decreasing rearrangement of the stick-breaking sequence U1,(1-U1)U2,(1-U1)(1-U2)U3,... for independent uniform Uj on (0,1)?

Short English statement

For each prime $p>=3$ list the prime factors of p-1, counting multiplicity, in descending order and divide their logarithms by log(p-1), padding with zeros. Do the first k coordinates, for every fixed k, converge in distribution under ordinary uniform prime counting $p<=X$ to the decreasing rearrangement of the stick-breaking sequence U1,(1-U1)U2,(1-U1)(1-U2)U3,... for independent uniform Uj on (0,1)?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2795-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2795

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 011; 2026-09-24).

Poisson-Dirichlet factorization of prime predecessors

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 011; 2026-09-24), the asserted result is that the logarithmically normalized ordered prime factors of $p-1$, for uniformly sampled primes, converge jointly to the parameter-one Poisson--Dirichlet partition.

Mathematical statement of the claimed result

Manuscript 1.
For each prime $p\ge3$, order prime factors $q_j(p)$ of $p-1$ decreasingly with multiplicity, pad by ones, and set $V_j(p)=\log q_j(p)/\log(p-1)$. Take independent uniform $U_j\in(0,1)$, form stick fragments $B_j=(\prod_{i<j}U_i)(1-U_j)$, and let $(L_j)$ be their decreasing rearrangement, the Poisson--Dirichlet law with parameter one. For every fixed $k\ge1$ and bounded continuous $F:[0,1]^k\to\mathbb R$, the claimed ordinary prime-average limit is \[ \lim_{x\to\infty}\frac1{\pi(x)-1}\sum_{3\le p\le x}F(V_1(p),\ldots,V_k(p)) =\mathbb EF(L_1,\ldots,L_k). \] The sum ranges over primes with equal weights, and the limit is through all real $x$.

Outline of the argument

For manuscript 1 (family 011; 2026-09-24), marked divisor moments are expressed using a determinant expansion and paired endpoint weights. Independent residue lines and a symmetric memory space control the signed graph error, allowing the finite-dimensional factor statistics to match the ordinary-integer model.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 011; 2026-09-24).

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

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