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Statement

Setting (pp. 1, 4). $H(p)$ is the length of the longest prime chain $p_1\prec\cdots\prec p_k=p$, where $a\prec b$ means $b\equiv1\pmod a$; equivalently the height of the Pratt tree of $p$. Trivially $H(p)\le\frac{\log p}{\log2}+1$ (p. 4).

Theorem 4 (p. 5, quoted). "We have $H(p)\leqslant(\log p)^{0.9503}$ for almost all $p$."

The paper says (p. 5) that before this work it was unknown whether some infinite sequence of primes has $H(p)=o(\log p)$. The proof gives an exceptional set of size $O(x\exp\{-(\log x)^\delta\})$ among primes up to $x$ for some $\delta>0$ (p. 19).

Proof pointer

Section 5, pp. 12--19. A sieve upper bound for prime $k$-tuples uniform in $k$ (Lemma 5.1, p. 12) and averages of the singular series (Lemmas 5.2--5.4) give Theorem 7 (p. 16): few primes $p\le x$ have a chain $p_{rl}\prec\cdots\prec p_0=p$ with $p_{rl}>x^{(r+1)^{-\eta}}$. The proof of Theorem 4 (p. 19) takes $\eta=0.15718$, $l=\lfloor(\log x)^\varepsilon\rfloor$ and $r=\lfloor(\log x)^\beta\rfloor$: outside the exceptions of Theorem 7 every prime at level $rl$ of the tree is below $x^{(r+1)^{-\eta}}$, so the trivial bound applied there gives $H(p)\ll(\log x)^{0.95022}$.

Read depth

Claims checked: the statement was read on the print (p. 5), and the statement of Theorem 7 and the deduction of Theorem 4 (p. 19) were followed. The sieve lemmas of Section 5 were not checked. 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

• Problem 695: context only. A sequence $p_1<p_2<\cdots$ with $p_{i+1}\equiv1\pmod{p_i}$ makes $p_1\prec\cdots\prec p_k$ a prime chain, so $H(p_k)\ge k$; were $H(p)\le(\log p)^{0.9503}$ true for every prime, $\log p_k\ge k^{1/0.9503}$ would follow and the first question would be answered yes. Theorem 4 holds only outside an exceptional set of primes, and the terms of one chain may all lie in it, so it decides neither question.