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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 $p_1\prec p_2\prec\cdots\prec p_k$, and $N(x;p)$ is the number of prime chains with $p_1=p$ and $p_k/p_1\le x$ ($k$ variable). Here $\log_k x$ is the $k$-fold iterated logarithm (p. 2).

Theorem 1 (p. 2, quoted). "For $p\geqslant2$ and $x\geqslant20$, we have the effective estimate $N(x;p)\leqslant x\exp\left\{\frac{\log x(\log_3x+O(1))}{\log_2x}\right\}$. In particular, for every $\varepsilon>0$ there is an effective constant $C(\varepsilon)$ so that $N(x;p)\leqslant C(\varepsilon)x^{1+\varepsilon}$."

The bound is uniform in $p$. The paper notes (p. 3) that it is nearly best possible, since $N(x;p)\ge\pi(px;p,1)$, and poses Conjecture 1 (p. 3): $N(x;p)\ll x$. Before this theorem, iterating the Brun--Titchmarsh inequality gave only $N(x;p)\ll x^{O(\log_3x)}$ (p. 2).

Proof pointer

Section 2, pp. 6--8. The proof relaxes primality to coprimality with the product $r$ of the primes up to $y$, and counts chains $n_1\prec\cdots\prec n_k$ through their links $m_j$ with $n_{j+1}=m_jn_j+1$. Weighting each tuple of links by $(m_1\cdots m_{k-1})^{-s}$, the count is bounded by $x^s$ times column sums of powers of a matrix indexed by the reduced residues modulo $r$. Its row sums are computed exactly, the largest being $(2^s-1)^{-1}\prod_{p>y}(1-p^{-s})^{-1}$ (2.2), and the choice $y=\log x/\log_2x$, $s=1+\log_2y/\log y$ (p. 8) gives the theorem.

Read depth

Claims checked: the statement was read clause by clause on the print (p. 2) and the proof in Section 2 (pp. 6--8) 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

• Problem 695: context only. Theorem 1 counts all prime chains from a given starting prime; it says nothing about the growth of a single infinite chain and answers neither of the problem's questions.