---

Statement

Setting (pp. 2--3). $P^+(n)$ is the largest prime factor of $n$, and $\phi_k$ the $k$-th iterate of Euler's function $\phi$.

Theorem 5 (p. 5, quoted). "For every $\varepsilon>0$ and $\delta>0$, there is an integer $k$ so that for large $x$ and at least $(1-\delta)x$ integers $n\leqslant x$, $P^+(\phi_k(n))\leqslant x^\varepsilon$."

The paper says (p. 5) that the proof of Theorem 4 shows that for most primes all the primes at some bounded level of the Pratt tree are small, and that this settles Conjecture 2 of its reference [19]: P. Erdős, A. Granville, C. Pomerance and C. Spiro, On the normal behavior of the iterates of some arithmetic functions, in Analytic Number Theory (Proceedings of a conference in honor of Paul T. Bateman), Birkhäuser, Boston, 1990, 165--204.

Proof pointer

Section 5, pp. 19--20. If a prime $p>x^{\varepsilon/2}$ divides $\phi_k(n)$, then either the square of a prime $q>x^{\varepsilon/2}$ divides some $\phi_j(n)$ with $j\le k$, which the Brun--Titchmarsh inequality makes rare ($\ll_{\varepsilon,k}x^{1-\varepsilon/2}$ integers), or there is a prime chain $p=p_k\prec\cdots\prec p_0$ with $p_0\mid n$. Theorem 7 (p. 16) with $\eta=1/7$ and $k=rl$ bounds the second case by $\ll_\varepsilon x(2\eta\mathrm e^{1+\eta})^{-l/2}$, which is below $\delta x$ once $l$ is large.

Read depth

Claims checked: the statement was read on the print (p. 5) and the deduction from Theorem 7 (pp. 19--20) was 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

None. The theorem concerns iterates of phi at typical integers, not the growth of one infinite prime chain.