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Claim. Let $g(n)=\#\{m\ge1:\phi(m)=n\}$. Theorem 1 of the paper: for a fixed nonzero integer $a$, $\theta=0.2961$ and $x\ge x_0$, more than $x/(\log x)^{C_1}$ primes $p\le x$ have every prime factor of $p-a$ at most $x^{\theta}$, with $C_1$ an absolute constant. Corollary 1, the Erdős--Pomerance transfer with $\beta=\theta$: the integers $m$ with more than $m^{1-\beta}$ solutions $n$ of $\phi(n)=m$ form an infinite sequence $m_1<m_2<\cdots$ with $\log m_{i+1}/\log m_i\to1$. So

$$g(n)>n^{0.7039}$$

for infinitely many $n$, which answers the question of Problem 821 for every $\epsilon\ge0.2961$, since then $n^{1-\epsilon}\le n^{0.7039}$. The transfer from smooth shifted primes to large fibers is the product-and-pigeonhole argument of Erdős and Pomerance: products of many primes $p$ whose predecessors $p-1$ draw their prime factors from a small pool collide under $\phi$. The statements are on the card Baker and Harman 1998; the proof, which counts solutions of $p-a=lmn$ with Harman's sieve and the Bombieri--Friedlander--Iwaniec equidistribution results, is not reconstructed in this repository. The exponent $0.2961$ improved Friedlander's $1/(2\sqrt e)+\epsilon=0.3032\ldots$ and was improved in turn by Lichtman's $0.2844$ (Lichtman 2022).

Covers. The range $\epsilon\ge0.2961$: infinitely many $n$ with $g(n)>n^{0.7039}\ge n^{1-\epsilon}$. The question for smaller $\epsilon$, and so the problem as posed, is not addressed.

Depends on. Nothing in this wiki.

Acceptance. Refereed: Acta Arith. 83 (1998), no. 4, 331--361, a refereed journal. The site's commentary names Baker and Harman only as the exponent that Lichtman improved and labels the problem OPEN (page last edited 1 October 2025), so no reviewed evidence is listed. The journal volume gives the year 1998 and no month or day; the page's date carries the first day of that year.