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Statement

Setting (p. 1). Write $a\prec b$ when $b\equiv1\pmod a$. A prime chain is a sequence of primes $p_1\prec\cdots\prec p_k$; $f(p)$ is the number of prime chains with $p_k=p$, and $N(x)$ the number of prime chains with $p_k\le x$ ($k$ variable). Equivalently (pp. 3--4) $f(p)$ is the number of nodes of the Pratt tree of $p$, and $f(2)=1$, $f(p)=1+\sum_{q\mid p-1}f(q)$ (1.4).

Theorem 2 (p. 3, quoted). "(i) We have $f(p)\geqslant0.378\log p$ for almost all primes $p$. Hence, $N(x)\gg x$. (ii) For all $x\geqslant3$ and any positive integer $h$, $|\{p\leqslant x:f(p)=h\}|\leqslant\left(\frac{6\log x}{h}\right)^h$."

The paper records the matching trivial upper bound $f(p)\le\frac{2\log p}{\log2}-1$ for all $p$ (1.5), which gives $N(x)\ll x$ (p. 3), and notes that (ii) makes the primes with $f(p)=o(\log p)$ number $x^{o(1)}$ up to $x$ (p. 3).

Proof pointer

Section 3, pp. 8--9. With $l(n)=\prod_{p^a\parallel n}p^{a-1}$, the product of $l(q-1)$ over the prime labels of the Pratt tree of an odd prime $p$ is at most $p\,2^{-f(p)/2}$ (3.1), since half its nodes are labelled $2$. Fixing the shape of the subtree of odd labels, the labels are recovered from the numbers $l_j$, and summing $(x2^{-h/2}/l_1\cdots l_h)^\beta$ over shapes, with Cayley's count of labelled trees, gives the bound (3.3) on $|\{p\le x:f(p)=h\}|$ for every $\beta>0$. The choice $\beta=0.37$ gives (i) and $\beta=h/\log x$ gives (ii) (p. 9).

Read depth

Claims checked: the statement was read clause by clause on the print (p. 3) and the proof in Section 3 (pp. 8--9) was followed. 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. The theorem counts chains ending at a prime, not the growth of one infinite prime chain.