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Statement

Setting (pp. 1--2, 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 $p$. Hypothesis (1.1) (p. 2) with parameters $Q$ and $R$ is

$$\sum_{m\leqslant Q}\max_{y\leqslant x}\left|\pi(y;m,1)-\frac{\mathrm{li}(y)}{\phi(m)}\right|\ll R .$$

Bombieri--Vinogradov gives (1.1) with $Q=x^{1/2}(\log x)^{-B}$ and $R=x(\log x)^{-A}$; the Elliott--Halberstam conjecture (EH) is (1.1) with $Q=x^\theta$ and $R=x(\log x)^{-A}$ for any $\theta<1$ and $A>0$ (p. 2). $\Lambda=\limsup_{p\to\infty}H(p)/\log_2p$ (1.6), p. 4.

Theorem 3 (p. 5, quoted). "(a) If (1.1) holds with $Q=x^\theta$ and $R=o(x/\log x)$, then for any $c<\frac{1}{\mathrm{e}^{-1}-\log\theta}$, $H(p)>c\log_2p$ for almost all primes $p$; (b) If (1.1) holds with $Q=x^\theta$ and $R=x(\log x)^{-A}$ for every $A>1$, then for every $c<\frac{1}{-\log\theta}$, there is a $K$ so that $H(p)>c\log_2p$ for $\gg x/(\log x)^K$ primes $p\leqslant x$. Consequently, $\Lambda\geqslant\frac{1}{-\log\theta}$."

Corollary 1 (p. 5, quoted). "EH implies that for every $c<\mathrm{e}$, $H(p)>c\log_2p$ for almost all $p$."

The paper presents Theorem 3 as a version of Kátai's theorem (an unconditional $H(p)\ge c\log_2p$ for almost all primes, for some constant $c>0$, with $o(x/\log x)$ exceptions up to $x$) with the constant made explicit in terms of the level of distribution (p. 5). Remark 1 (p. 5) notes that (1.1) holds unconditionally with $Q=x^{1-\varepsilon}$ and $R=O_\varepsilon(x/\log x)$, which is not $o(x/\log x)$. The paper calls the constant $\mathrm e^{-1}$ in (a) likely best possible (p. 5).

Proof pointer

Section 4, pp. 9--12. Part (b) is an induction over dyadic intervals: a prime $p$ with a factor $q\mid p-1$ of size about $p^\theta$ already in the set inherits the height bound, and (1.1) counts such $p$ (pp. 9--10). Part (a) iterates $k$ levels of the chain at once, counts chains with $p_{j+1}\le p_j^\theta$ by (1.1) and removes pairs of chains with the same top by a sieve bound, getting a positive proportion of primes with $H(p)>h\log_2p$; Theorem 6 (p. 6, proved p. 22) then upgrades a positive proportion to almost all primes (pp. 10--12).

Read depth

Claims checked: Theorem 3, Corollary 1 and Remark 1 were read clause by clause on the print (p. 5); the proof of (b) was followed and that of (a) read in outline. 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. The bounds hold for almost all primes or for many primes, under hypotheses on primes in progressions; they do not constrain a single infinite chain and answer neither of the problem's questions.