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Statement

Setting (p. 20). Factor $n=\prod_{j=1}^{\Omega(n)}p_j(n)$ with $p_1(n)\ge p_2(n)\ge\cdots$, put $p_j(n)=1$ for $j>\Omega(n)$, and let

$$S(n)=\left(\frac{\log p_1(n)}{\log n},\frac{\log p_2(n)}{\log n},\ldots\right).$$

Probabilities over a set of integers are natural densities: by footnote 3 (p. 20), for $\mathcal B\subseteq\mathcal A\subseteq\mathbb N$, $\mathbf P(n\in\mathcal B\mid n\in\mathcal A)=\alpha$ means that $|\{n\in\mathcal B:n\le x\}|/|\{n\in\mathcal A:n\le x\}|\to\alpha$. The Poisson--Dirichlet distribution $PD(1)$ is the law of the decreasing rearrangement of $U_1,(1-U_1)U_2,(1-U_1)(1-U_2)U_3,\ldots$ for independent $U_i$ uniform on $[0,1]$ (Donnelly--Grimmett, (6.1)). Over all integers, $S(n)$ has $PD(1)$ distribution in the sense the paper states: for each $j$, the first $j$ components of $S(n)$ are distributed as the first $j$ components of $PD(1)$ (Billingsley, 1972; p. 20).

Conjecture 5 (p. 20, quoted). "As $p$ runs over the set of primes, $S(p-1)$ has $PD(1)$ distribution."

The paper calls the conjecture widely believed and "a simple consequence of EH" (the Elliott--Halberstam conjecture), without proof (p. 20). It then assumes, beyond Conjecture 5, that the vectors $S(q-1)$ for the primes $q\mid p-1$ are independent and so on down the Pratt tree, which turns the tree into a random fragmentation process, equivalently a branching random walk, the model behind Conjectures 2 and 3 (pp. 20--21).

Proof pointer

None: a conjecture. The paper cites Lamzouri for a result assuming EH: the largest prime at each fixed level of the Pratt tree has the distribution the model predicts (p. 21).

Read depth

Claims checked: the definitions, footnote 3 and Conjecture 5 were read clause by clause on the print (p. 20). Nothing here is independently reviewed.

Dependencies

None in the corpus.

Source. Kevin Ford, Sergei V. Konyagin and Florian Luca, Prime chains and Pratt trees, Geom. Funct. Anal. 20 (2010), no. 5, 1231--1258, doi:10.1007/s00039-010-0089-0, arXiv:0904.0473; page numbers are those of the arXiv version 4 named on the source card.

Bears on

None among the problems the corpus links to this paper. The 2026 manuscript on the Poisson--Dirichlet law of prime predecessors states that its Theorem 1.1 resolves this conjecture.